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ON THE GALOIS STRUCTURE OF ARTIN'S LFUNCTION AT ITS CRITICAL POINTS By KENNETH WARD Bachelor of Arts of mathematics The University of Chicago Chicago, Illinois, United States of America 2004 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial ful llment of the requirements for the Degree of Master of Science May 2009 COPYRIGHT c By KENNETH WARD May 2009 ON THE GALOIS STRUCTURE OF ARTIN'S LFUNCTION AT ITS CRITICAL POINTS Thesis Approved: Dr. Anantharam Raghuram Thesis Advisor Dr. Dale Alspach Dr. Alan Noell Dr. A. Gordon Emslie Dean of the Graduate College iii ACKNOWLEDGMENTS I thank Anantharam Raghuram for introducing me to the problem in number theory that is the heart of this thesis. iv TABLE OF CONTENTS Chapter Page 1 Introduction 1 2 Representation theory 3 2.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Brauer's theorem on induction of characters . . . . . . . . . . . . . . 12 3 Artin's Lfunction 22 3.1 Artin's Lfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 The basic properties of this function . . . . . . . . . . . . . . . . . . 33 4 Class eld theory 37 4.1 Id eles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 The Artin symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5 The functional equation 50 5.1 Hecke's Lfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.2 The Artin conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.3 The functional equation . . . . . . . . . . . . . . . . . . . . . . . . . 69 6 The result of Coates and Lichtenbaum 77 6.1 Polyhedric cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.2 Shintani's unit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.3 Rewriting a particular kind of Lfunction . . . . . . . . . . . . . . . . 87 v 6.4 Siegel and Klingen's result on special values of Lfunctions . . . . . . 91 6.5 The value of Artin's Lfunction at negative integers . . . . . . . . . . 94 7 The main result: Behavior of special values under twisting 100 BIBLIOGRAPHY 109 vi CHAPTER 1 Introduction This work provides a relation between an Artin Lfunction for a character corre sponding to a representation of a Galois group of a Galois extension FjQ of number elds, and an Artin Lfunction for a character = of the same group, where is an even Dirichlet character. In particular, it is shown that, with ( ) denoting the Gauss sum of , the value of L(FjQ; ; s) is, up to an element of Q( ; ), equal to ( )dim( )L(FjQ; ; s), so long as s is an integer, at least 2, where the Euler factors at in nity for the Lfunction of are regular at 1 s. This naturally extends a work of Coates and Lichtenbaum [CL], which states that L(FjK; ; s) 2 Q( ) for a character of a Galois extension FjK of algebraic number elds if s is a negative integer. The work of Coates and Lichtenbaum employs a result of Siegel and Klingen [Neu, VII], which states that L(FjQ; ; s) 2 Q( ) if s is a negative integer and is a Dirichlet character. This work, as well as those mentioned above, employ the functional equation for the given Lfunction. It is of some importance that regularity is involved with the main result of this work, since poles emerge in the functional equation at other integer points that limit its use in determining the value of the Lfunction. Preparation is given to the basic topics necessary for understanding the main result. Proofs are provided, with exceptions made for class eld theory and the theory of LubinTate extensions. These omissions are for brevity, and references are provided where omissions have been made. The progression of this work is as follows. Chapter 2 proves Brauer's theorem 1 on induction of characters. Chapter 3 introduces Artin's Lfunction and its basic properties. Chapter 4 surveys the main results of class eld theory. Chapter 5 establishes the functional equation for Artin's Lfunction. Chapter 6 proves the above mentioned result of Coates and Lichtenbaum. Chapter 7 yields the main result. 2 CHAPTER 2 Representation theory 2.1 Basic notions This section introduces the basic notions of representation theory. Here, G will denote a nite group. A representation is a homomorphism : G ! GL(V ) where V denotes a vector space over C of nite dimension. Here, the notation g will be adopted instead of (g). The dimension of V will be called the dimension of the representation . A subspace W V will be called stable if gW W for all g 2 G. De ne a representation to be irreducible if may not be written as 1 M 2 : G ! G(V1) M G(V2) where i : G ! G(Vi) for each i 2 f1; 2g, where V = V1 L V2. Theorem 2.1 Given a representation of a nite group G, there is a decomposition V = Mr i=1 Vi; and likewise = Mr i=1 i; where i : G ! GL(Vi) and is irreducible, for each i 2 f1; 2; :::; rg. 3 Proof. First, consider a subspace W ( V that is stable and proper. Adding elements to a basis for W to obtain a basis for V , one may obtain a subspace W0 satisfying W L W0 = V . Consider the projection p of V onto W via this basis for V , and de ne the average p0 = 1 jGj X g2G gp 1 g : One notes that because W is stable, one must have p0 mapping V into W. Also, because W is stable, one must have gp 1 g x = x; for every x 2 W. Therefore p0x = x for every x 2 W, and so p0 is a projection of V onto W. Its kernel W0 satis es W M W0 = V: But also, one has hp0 1 h = 1 jGj X g2G h gp 1 g 1 h = 1 jGj X g2G hgp 1 hg = p0 for all h 2 G, and therefore x 2 W0 satis es p0x = 0, and thus p0pgx = 0. Thus pgx 2 W0; and W0 is a complement of W in V that is stable. By the de nition of irreducibility, the result then follows by induction on the dimension of V. For a representation , de ne the character as (g) = Tr( g): A few basic properties of this character function are as follows: 4 Theorem 2.2 (i) (1) = dim( ); (ii) is a class function from G to C; (iii) (g1) = (g). Proof. Properties (i) and (ii) are obvious. Property (iii) follows the observation that (g) = dXim( ) i=1 i where each i is a root of unity, and, in fact, an eigenvalue for g. Since g has nite order, one has that g is conjugate to a matrix with nonzero entries only along the diagonal, with each diagonal entry an eigenvalue for g. But then g1 is conjugate to the inverse of this matrix of eigenvalues. Thus ( g1) = dXim( ) i=1 1 i = dXim( ) i=1 i; with the last equality holding because each i is a root of unity, as G has nite order. As dXim( ) i=1 i = (g); the result follows. The following will now demonstrate various aspects of character theory that will be of use to us. In order to do this, one important de nition must be established. Two representations 1 : G ! GL(V1) and 2 : G ! GL(V2) are called isomorphic if there is a isomorphism of vector spaces f : V1 ! V2 such that f 1;g = 2;g f for all g 2 G. This de nition motivates the following. Theorem 2.3 (Schur's lemma) Suppose that 1 : G ! GL(V1) 5 and 2 : G ! GL(V2) are irreducible representations, and that f : V1 ! V2 is a linear map satisfying f 1;g = 2;g f for all g 2 G. (i) If 1 and 2 are not isomorphic, then f 0; (ii) if V1 = V2 and 1 = 2, then f(v) = v for some 2 C. Proof. For the rst part, note that the kernel of f must be stable, as one must have 0 = f 1;gx = 2;g fx = 2;g0 = 0: As V1 is irreducible, this kernel must be zero. Likewise, the image of f is stable. Therefore f is surjective and injective, hence an isomorphism, which contradicts the fact that 1 and 2 are isomorphic. (ii) As a linear map, f has an eigenvalue, say, f(v) = v for some nonzero element v 2 V = V1 = V2. With f = f 1V as a map from V to V , one has that f 1;g = 2;gf for all g 2 G, and therefore the nonzero kernel of this map must equal all of V , as V is irreducible. Therefore f . If 1 and 2 are two complexvalued functions on G, one de nes their inner product as h 1; 2i = 1 jGj X g2G 1(g) 2(g): The following result is of importance here. Theorem 2.4 (Schur's orthogonality of characters) 6 (i) If 1 and 2 correspond to representations 1 : G ! GL(V1) and 2 : G ! GL(V2), respectively, that are irreducible and not isomorphic, then h 1; 2i = 0; (ii) If 1 and 2 are irreducible and isomorphic, then h 1; 2i = 1. Proof. For (i), one considers a mapping, as in the proof of Theorem 2.1, de ned as f0 = 1 jGj X g2G ( 2;g)1f 1;g where f is a linear map from V1 to V2 as before. One may note that 2;gf0 = f0 1;g for all g 2 G. By Schur's lemma, f0 0. In this case, one considers f to be the linear map which, as a matrix, has an element 1 in a particular entry, and zeroes elsewhere. The inner product of 1 and 2, with 1 and 2 written in matrix form as 1;g = 1 ij(g) i;j and 2;g = 2 ij(g) i;j ; satis es h 1; 2i = X i;j h 1 ii; 2j ji: With f as the linear map mapping the jth basis element for V1 onto the ith basis element for V2 and all else to zero, one must have < 1 ii; 1j j >= 0 for all i and j. Therefore the inner product of 1 and 2 is equal to zero. For (ii), if 1 and 2 are irreducible and isomorphic, then one immediately identi es from the de nitions that 1 2. Thus to consider the inner product of these two characters, one may assume that V = V1 = V2, as has been done in the proof of Schur's lemma. With = 1 2, one has Tr(f0) = 1 jGj X g2G Tr( 1 g f g) = Tr(f) as matrices, and as, by the Schur lemma, f0 for some 2 C, it follows that = 1 dim( )Tr(f). With the above matrix notations, by considering f to be the linear map sending the jth basis element in V1 to the ith in V2, and all else to zero, one 7 must have h ii; jji = 1 dim( ) if i = j and zero otherwise. Therefore h 1; 2i = h 1; 1i = dXim( ) i=1 h ii; iii = dXim( ) i=1 1 dim( ) = 1: This yields immediately the uniqueness, up to isomorphism, of the decomposition of a representation into a direct sum of irreducible representations. In the sequel, it will also be useful to note the following fact. Lemma 2.1 (g) = dim( ) or (g) = dim( ) if and only if it holds respectively that g = I or that g = I, where I denotes the identity matrix. Proof. One may represent (g) as the trace of the diagonal matrix with entries along the diagonal consisting of the eigenvalues of g. The matrix g is conjugate to this diagonal matrix of eigenvalues. Also, because g has nite order, each of its eigenval ues is a root of unity, and thus has modulus one as a complex number. Therefore if (g) = dim( ), then g is conjugate, and hence equal, to the identity matrix, as each eigenvalue of g is equal to 1. If (g) = dim( ), then g is conjugate, and hence equal, to the diagonal matrix I, as each eigenvalue of g is equal to 1. The converse holds trivially. The following result will be necessary to the proof of Brauer's theorem on induction of characters. First, one must de ne the regular representation of G to be the C algebra C[G] which has elements of G as its basis, where the action of g on C[G] is de ned as 8 g X g02G cg0g0 = X g02G cg0gg0: Lemma 2.2 The irreducible characters of a nite group G form an orthonormal basis of the space of class functions of G. Proof. Consider a class function f, an irreducible representation ( ; G; V ), and the linear mapping (f) : V ! V de ned by (f) = X g2G f(g) g: One may notice that 1 g1 (f) g1 = X g2G f(g) 1 g1 g g1 = X g2G f(g) g1 1 gg1 = X g2G f(g1gg1 1 ) g = X g2G f(g) g = (f): Therefore, with denoting the character associated with , (f) is equivalent to a scalar = 1 dim(V ) X g2G f(g)Tr( g) = 1 dim(V ) X g2G f(g) (g) = jGj dim(V ) hf; i: Suppose then that f is a class function orthogonal to , for every character that corresponds to an irreducible representation of G. Considering an arbitrary representation ( ; G; V ) of G, one then has, by Schur's lemma and componentwise additivity of the inner product h ; i in each component, that (f) is identically zero. 9 Supposing now that this arbitrary representation is specially chosen to be the regular representation of G, one has (f) 1 = X g2G f(g)g in C[G]. As (f) is identically zero, one has (f) 1 = 0. Thus f(g) = 0 for all g 2 G, and f is identically zero. Therefore the characters from characters corresponding to irreducible repre sentations of G form an orthonormal basis for the set of class functions on G. If is such a character, then is also such a character, a fact which is immediate from the de nition of the inner product h ; i. Therefore the characters of the irreducible representations of G form an orthonormal basis for G. 2.2 Induction In this section, the notion of an induced representation is introduced. As before, suppose that G is a group. Where necessary a representation : G ! V for a group G will be denoted by ( ; V;G). Consider then H a subgroup of G, with a representation ( ;W;H). The induced representation ( ; IndG H(W);G) of ( ;W;H) is de ned as IndG H(W) = C[G] C[H] W; where elements of H act on W via the representation . The following wellknown result is of importance for the proof of Brauer's theorem on induced characters. It is due to Frobenius. In what follows, R will denote a set of left coset representatives for H in G, and G will denote a nite group. A character of H is viewed in the following theorem as a function on G with (g) = 0 for all g =2 H. Theorem 2.5 If ( ;W;H) has character , and ( ; V;G) is the induced representa 10 tion with character , then one has (g) = X r2R r1gr2H (r1gr) . Proof. Consider the direct sum V = M r2R rW; and some g 2 G. One may note that (g) consists of the sums of traces of those indices r 2 R for which grW = rW: Denoting by Rg the set of indices in R so xed by g, one has (g) = X r2Rg Tr rW( g;r) with here g;r denoting the restriction of g to the subspace rW. In fact, the set Rg consists exactly of those elements r 2 R for which r1gr 2 H, and thus, one may note that r is an isomorphism from W onto rW for every r 2 Rg, and therefore, as g r1gr = g;r r if r1gr 2 H, one has (r1gr) = TrW( r1gr) = Tr rW( g;r); and the result holds. This may be used to de ne for any class function f : H ! C the induced function IndG H(f) = X r2R r1gr2H (r1gr); where as with characters of H, f is viewed as a function extended to G by zero. With this machinery, one is now prepared to prove Brauer's theorem, whose nal steps are put forth in the last section of this chapter. 11 2.3 Brauer's theorem on induction of characters This section establishes Brauer's theorem on induction of characters. Lemma 2.3 Let p be a prime number, G a nite group, and g 2 G. Two elements g1 and g2 exist so that g = g1g2, the order of g1 is a power of p, the order of g2 is relatively prime to p, and g1 and g2 commute. An element of G commutes with g if and only if it commutes with both g1 and g2. Also, g1 and g2 are uniquely determined by the element g Proof. All of these facts follow easily from the following construction. Suppose that jgj = p m, where m and p are relatively prime. Then there exist integers a and b satisfying ap + bm = 1: In this case, one may set g1 = gbm and g2 = gap , and one will evidently have g = g1g2. Evidently, also g1 commutes with g2. The order of g1 is a power of p by its construction, and the order of g2 is relatively prime to p by its construction. That an element of G commutes with g if and only if it commutes with both g1 and g2 follows from the construction of g1 and g2 as powers of g. One must now establish uniqueness of g1 and g2. As p and m are relatively prime, one must have that g1 has order p and g2 has order m. Thus, one must have gm 1 = gm, and gp 2 = gp . Therefore, g1 = gbm 1 = gbm; and likewise g2 = gap 2 = gap : This proves the claim. De ne the function 1G(g) = 1; for all g 2 G: 12 Let the symbol p stand for a prime number dividing the order of G, let S denote the ring Z["], where " is a primitive mth root of unity, and m is a natural number where gm = 1 for all g 2 G. For a 2 G, denote the centralizer of a by CG(a) = fg 2 G j ga = agg: Then, suppose that R is a subring of C containing the integers. Denote by E the set of elementary subgroups of G, i.e., the subgroups of G of the form hai B where a 2 G is an element of order relatively prime to p, and B is a psubgroup of CG(a). Denote by Ch(G) the set of characters of representations of G. De ne the following three sets, ascending in the inclusion ordering: VR(G) = f j = X nite riIndG Ei( i);Ei 2 E; ri 2 R; for all ig; ChR(G) = f j = X nite ri i; i 2 Ch(G); ri 2 R; for all ig; UR(G) = f j jE 2 ChR(E) for each E 2 Eg: One may note that VR(G) is naturally an ideal of UR(G), for with 2 VR(G), 2 UR(G), one has, with = X nite riIndG Ei( i); that = X nite ri IndG Ei( i) = X nite riIndG Ei(( i)jEi); with the denoting multiplication. Of course, ( i)jEi = jEi i 2 Ch(Ei); for both jEi and i are characters of representations, and thus their product is the character of the tensor product of the representations for which jEi and i are characters. The following lemmas are necessary precursors to Brauer's theorem on induced characters. 13 Lemma 2.4 Let E be a supersolvable group. Then E is monomial, i.e., an irreducible representation of E is induced from a representation of degree one for a subgroup of E. Proof. This proof is by induction on the order of E. If jEj = 1, then the lemma is obvious. Also, if E is abelian, then an irreducible representation of E is automati cally of dimension one. Suppose then that jEj > 1, and that E is nonabelian. As E is supersolvable, E admits a tower 1 = E0 E1 E2 En = E where each Ei is a normal subgroup of E, and Ei+1=Ei is a cyclic group of prime order. One may then take the quotient of E with its center Z(E), and may thus obtain such a cyclic tower Z(E) = F0 F1 F2 Fn = E where, again, each Fi is a normal subgroup of E, and quotient groups of successive groups in this chain are cyclic of prime order. One may note that F1 must be an abelian, normal subgroup of E, and may suppose that F1 is not equal to Z(E). Suppose that the character corresponds to the irreducible representation of E. By induction, one may assume that is injective. Then, as F1 is not contained in the center of E, there must be some a 2 F1 so that (a) is not identical to a scalar. If the decomposition of the restriction of to F1 into irreducible representations consisted of a collection of pairwise isomorphic representations, then, as F1 is abelian, one would have a decomposition of as ( ; V; F1) = Mr i=1 ( i; Vi; F1) where Vi has dimension one, for each i 2 f1; 2; :::; rg. 14 Of course, as each representation in the direct sum is isomorphic to every other, one would have in particular for i; j 2 f1; 2; :::; rg some Fij : Vj ! Vi satisfying Fij j;f = i;f Fij ; for each f 2 F1. As each representation in the direct sum is also of one dimension, it must be that j;f = i;f for all i; j 2 f1; 2; :::; rg, for each f 2 F1 and thus f would be a scalar for every f 2 F1. This is a contradiction, and so the restriction of to F1 may not be decomposed in this way. Nonetheless, in restricting to F1, one may write ( ; V; F1) = Ms j=1 Mrj i=1 ( i;j ; Vi;j ; F1) where each collection Bj = f( i;j ; Vi;j ; F1)grj i=1 is a maximal collection of irreducible representations of F1 in its decomposition into irreducible representations. The linear transformation permutes the subspaces Vj = Mrj i=1 Vi;j : Considering some Vj , and the subgroup Hj E where hVj = Vj ; one notes that F1 Hj , and that is induced by the representation jHj when viewed as acting on Vj . Also, one must have Hj 6= E. This establishes the result. Theorem 2.6 A character of a representation of G is a Zlinear combination of characters of representations of G as = X nite IndG Hi( i) where, as before, Hi is a subgroup of G, and i is a character of a representation of dimension one for Hi. 15 Proof. One needs only to show that 1G 2 VZ(G). For, once this has been established, one may note that if 1G 2 VZ(G), then as Ch(G) UZ(G) and VZ(G) is an ideal of UZ(G), then so must Ch(G) VZ(G). Consider a character of a representation of G as a nite Zlinear combination of characters of representations G where each character in the linear combination is induced from a character of a representation of a particular element, say, E, of E. Writing = Xn j=1 njIndG Ej ( j) with Ej 2 E and nj 2 Z for each j 2 f1; 2; :::; ng, and with j = Xlj i=1 i;j the expression of j as a sum of characters from irreducible representations of Ej , for each j 2 f1; 2; :::; ng, it follows that = Xn j=1 njIndG Ej ( j) = Xn j=1 njIndG Ej ( Xlj i=1 i;j) = Xn j=1 Xlj i=1 njIndG Ej ( i;j); where the last equality holds because induction is additive on characters of representa tions, from Frobenius' theorem in Section 2.2. And each i;j is induced by a character of a representation of dimension one on a subgroup of Ej , for each i 2 f1; 2; :::; ljg, for each j 2 f1; 2; :::; ng, because each Ej is nilpotent, and thus supersolvable. Therefore will be the desired Zlinear combination. Noting this, let us now must only establish that 1G 2 VZ(G). Consider then an element E 2 E. Suppose, to begin, that E = hai; 16 and let n = jhaij. Suppose then that ! 2 S is a primitive nth root of unity. The class function de ned as (a) = jhaij and (ai) = 0 for all ai 6= a may, by the above, be written as = X nite ai!i where each !i denotes a character of an irreducible representation of E and each ai denotes a complex number. Here, one may, in particular, take !i to be the character uniquely de ned by !i(a) = !i. In this case, the set of these characters !i constitute all of the characters for irreducible representations of E. So written, one has explicitly that ai = h ; !ii = !i 2 S: Therefore one must have that 2 ChS(E), noting here that this de nition does apply because S = Z["] is a subring of C containing Z. Then, considering a general element of E, one may see that this argument extends to these groups of the form hai B where B need not be trivial, by de ning 0 to be equal to the function constructed immediately above on hai, and extended as a function on E that is constant in its second coordinate. Of course, one may do the same with the characters !i, and this yields a set of irreducible characters of E, denoted by !0i for each i, with, by the above, 0 = X nite ai!0 i: In this fashion one has for an arbitrary element of E a function 2 ChS(E) with (a) = jhaij and (ai) = 0 for all ai 6= a that is constant in its second coordinate as a function of the product hai B. 17 One may write, for a representation ( ; G; V ) induced from ( ;H;W), that (g) = 1 jHj X r2G r1gr2H (r1gr) by the theorem of Frobenius of Section 2.2 noticing now that the sum is over all relevant a set of left coset representatives for H in G. Suppose now that B is a Sylow psubgroup of CG(a). De ne NCG(a)(B) = fr 2 CG(a) j rBr1 = Bg and n = card(fP a pSylow subgroup of CG(a) j b =2 Pg): Note that the induced function IndG E( 0) = X nite aiIndG E(!0 i) satis es IndG E( 0)(ab) = 1 jBj jfr 2 G j r1(ab) = ab0 for some b0 2 Bgj = 1 jBj jfr 2 CG(a)jr1br 2 Bgj = 1 jBj jfr 2 CG(a)jb 2 rBr1gj = n jNCG(a)(B)j jBj [NCG(a)(B) : B] mod p by the rst theorem in this section, and the fact that n = 1 mod p, which holds trivially when b = 1, and in any other case because b 6= 1 permutes by the action of conjugation the Sylow psubgroups of CG(a) which do not contain b in orbits of cardinality equal to some power of p. Of course, p does not divide the index of B in its normalizer in CG(a), as it is a Sylow psubgroup of CG(a), and therefore, there is some integer so that IndG E( 0)(ab) 1 mod p: 18 Then one will have IndG E( 0)(ab) 1 mod p for all b 2 B. One may note that this induced character is zero for every g which is not conjugate to any ab 2 hai B. Also, if g is conjugate to ab for some b 2 B, then one must have IndG E( 0)(g) = IndG E( 0)(ab): Note then that g is conjugate to such an element if and only if the decomposition according to Lemma 2.3 gives that the element g2, of order relatively prime to p and uniquely determined by g, is conjugate to a. One may then consider the conjugacy classes C1; :::; Ck of the elements g2, over all g 2 G. As above, one has for each such conjugacy class a function j in VS(G) where j(g) = 0 if the element g2 in its decomposition does not lie in the conjugacy class Cj , and j(g) 1 mod p if g2 lies in Cj . By taking Xk j=1 j ; one then obtains an element, denoted by p, in VS(g) satisfying (g) 1 mod p for all g 2 G. Suppose then that the order of G is written as p np, where (np; p) = 1. The following analysis will hold for each prime p dividing the order of G. One has p 1 p 1 mod p : 19 Of course, taking an arbitrary element a of G, and considering its conjugacy class, one may notice that the function constructed to lie in VS(hai) induces to a function IndG hai( ) in VS(G) satisfying IndG hai( )(g) = 0 if g is not conjugate to a, and IndG hai( )(g) = jCG(a)j if g lies in the conjugacy class of a. Therefore, a class function whose values are divisible by jGj must lie in VS(G). Thus, in particular, the class function np( p 1 p 1G) must be contained in VS(G). As in the proof that VR(G) is an ideal of UR(G), one must have p 1 p 2 VS(G), and thus np1G 2 VS(G). Noting then that the set of natural numbers np as constructed above, with p ranging over all primes dividing the order of G, has greatest common divisor one, one may nd integers cp satisfying X p cpnp = 1: Therefore 1G = X p cpnp1G 2 VS(G): Writing 1G = X finite ciIndG Ei( i) where i is a character of a representation of Ei and ci 2 S for each i, one may notice that, with ci = P j ci;j"j for integers ci;j , one has 1G = (Xm)1 j=0 "j X i ci;jIndG Ei( i) ! 20 where ( ) denotes Euler's phifunction, as f1; "; :::" (m)1g is a Zbasis for S. Now, one may notice that each of the characters appearing in this sum may be written as a sum of characters of irreducible representations of G, and so one has 1G = X i ci;0IndG Ei( i) 2 VZ(G) as soon as the elements f1; "; :::; " (m)1g are proven to be linearly independent over ChZ(G). For that, a relation (Xm)1 j=0 "j X i ci;j i ! = 0 with the sum over the index i sweeps across characters i of irreducible representations of G requires X i 0 @ (Xm)1 j=1 ci;j"j 1 A i = 0: But as these characters i are orthonormal with respect to the inner product h ; i, so must they be linearly independent, and thus (Xm)1 j=1 ci;j"j = 0 for each i. Therefore ci;j = 0 for each i and each j 2 f1; 2; :::; (m) 1g, and the elements f1; "; :::; " (m)1g are linearly independent over ChZ(G). This establishes Brauer's theorem. This concludes the chapter on representation theory. The next chapter introduces Artin's Lfunction and addresses its basic properties. 21 CHAPTER 3 Artin's Lfunction 3.1 Artin's Lfunction This chapter introduces Artin's Lfunction. One considers now a nite extension K of the rational numbers Q, called an algebraic number eld, and a nite extension F of K. In this work, G(FjK) will denote the Galois group of F over K. Artin's Lfunction hinges upon a representation : G(FjK) ! GL(V ) where, as before, V denotes a vector space over the complex numbers C. An element x 2 K will be called integral over Z if it satis es an equation xn + an1xn1 + + a0 = 0 where ai 2 Z for i 2 f1; 2; :::; n 1g, where here n is understood to be at least one. The set of such elements in K forms a ring, called the integers of K, and is denoted by oK. One may note the following properties of oK: (i) oK is an integral domain; (ii) oK is noetherian as a ring; (iii) an element in K that is integral over oK must be contained in oK; (iv) every nonzero prime ideal in oK is maximal. A ring possessing these four properties is called a Dedekind domain. One then de nes a fractional ideal of K to be an oKsubmodule a of K where ca oK for some c 2 oK. The following result is necessary for the construction of Artin's Lfunction. Theorem 3.1 The nonzero fractional ideals form a group under multiplication of ideals, equal to the free abelian group on the set of prime ideals of oK. 22 Proof. Consider a nonzero ideal a of oK. Since oK is noetherian, some ideal a 6= 0 is maximal with respect to the property that there is not product of prime ideals p1p2 pr a. In particular, this ideal a cannot be a prime ideal, so that there will be b1; b2 2 oK where b1b2 2 a, but neither of b1 and b2 lies in a. Then, if a1 = (a; b1), and a2 = (a; b2), one must have a1a2 a, and each of a1 and a2 is not equal to a. And a is maximal for the property as above, so that there is a product of prime ideals contained in each of a1 and a2. The product of these products of prime ideals is contained in the product of a1 and a2, hence in a, which is a contradiction. Hence every nonzero ideal of a contains a product of prime ideals. Consider then a maximal ideal p, and de ne p1 = fx 2 Kjxp oKg: This will play the role of the multiplicative inverse of p in the group of fractional ideals. One evidently has that p1 oK. Take a nonzero element p 2 p and consider the smallest natural number r for which there exists a product of prime ideals satisfying p1p2 pr (p) p: Such a product exists as above. In this case, p is, of course, prime, and so one of the ideals pi is contained in p, and because pi is maximal, it must equal p. But also, the product of primes contained in (p) as above, with pi removed, is no longer contained in (p), by the minimality of r as selected. Thus there is an element b 2 Y j6=i pj where b =2 (p). However, one must have bp (a), and thus ba1p oK, whence ba1 2 p1. Also, ba1 =2 oK. Therefore in this case p1 6= oK, and p pp1 oK: The ideal p is maximal, so one must have that one of these inclusions must be equality. If cannot, however, be the rst, because then p1 would contain only elements integral 23 over oK, as p is nitely generated, and so would equal oK. Thus the latter inclusion is equality, and p has an inverse. Then one may notice that a nonzero ideal of oK is invertible by a fractional ideal, for if this were not true for every such nonzero ideal of oK, then one would have a maximal noninvertible ideal a 6= 0, because oK is noetherian. By the above analysis, this a could not itself be maximal, and thus one obtains a ap1 aa1 oK: One cannot have the rst inclusion as equality, or else one would again have that p1 = oK, this time because a is nitely generated. But then ap1 would have an inverse by maximality of a in this ordering, and this inverse may be multiplied by p1 to obtain an inverse for a. In this fashion every nonzero ideal of oK has an inverse, and, in fact, this inverse must be a fractional ideal. Considering then a nonzero fractional ideal a, one nds c 2 oK so that ca oK, and so this ideal ca has an inverse b that must be a fractional ideal. Therefore cab = oK, and, as one may easily show, cb = fx 2 K j xa oKg; and so this fractional ideal forms the inverse for a in what is now the group of fractional ideals of K. To establish that this group forms a unique factorization domain, one may notice that if there is a nonzero ideal that is not equal to a product of prime ideals, then there is some maximal ideal a with respect to this property. This ideal a cannot be prime, and so with a p for some prime ideal p, one must have a ( ap1 oK as before. But then by maximality of a according to this ordering, the ideal ap1 must have a prime factorization, which may be multiplied by p to obtain a prime 24 factorization for a. One may then consider two fractional ideals a and b, and say that a divides b if and only if a b. By considering two prime factorizations p1p2 pr = q1q2 qs of an ideal in oK, one notices easily that by maximality of each of the prime ideals in each product, and the fact that each pi is contained in a particular qj , and hence equal to it, one must have r = s and uniqueness of the factorization. Then one may consider fractional ideals a in general, and with c 2 oK satisfying ca oK, one will have with (c) = q1q2 qs and ca = p1p2 pr that q1q2 qsa = (c)a = ca = p1p2 pr; whence a = p1p2 pr q1q2 qs : This establishes the unique factorization, and the proof is complete. With this in mind, one may consider the prime ideals of oK to be the nite primes, for every such prime p de nes a nonarchimedean valuation vp(x) = ep on oK corresponding to the prime factorization (x) = Y p pep : Later, one will deal with in nite primes, which are created by the embeddings of K into the complex numbers. For now, use of the nite primes will su ce. One may observe that for the extension F of K, the analogous ring oF enjoys the above properties mentioned for oK, and thus is also a unique factorization domain. There is a fundamental identity that will be used later, and it is stated here, for its clari cation of a relation between the prime ideals of oK and oF . Before proceeding 25 further, one may notice that the prime ideal p of oK extends via multiplication by oF to an ideal of oF , and therefore, poF has a prime factorization, poF = Pe1 1 Pe2 2 Per r : For any one of the primes Pi appearing in this factorization, one writes Pijp, and calls Pi a prime of oF lying above p. One calls the eld oK=p the residue class eld of p, and similarly in oF for its prime ideals. For the above factorization of p in oF , one calls ei the rami cation index of Pi over p, and de nes fi = [oF =Pi : oK=p]; calling it the inertia degree of Pi over p. For the following theorem, the notion of localization will be used. One may consider for oK what is called its localization at p, equal to the ring na b j a 2 oK; b 2 oKnp o : This will be denoted by oK;p. The ring oF;p will be de ned similarly as na b j a 2 oF ; b 2 oKnp o : The ring oK;p is a principal ideal domain with a unique maximal ideal poK;p and admits a discrete valuation equal to the exponent n 2 Z in the prime factorization x = u n of one of its elements, where is a prime element and u is a unit, where each element in oK;p has such a factorization because the unique maximal ideal is principal. Theorem 3.2 (Fundamental identity) One has the relation [F : K] = Xr i=1 eifi: 26 Proof. The inclusion oK ! oK;p induces an isomorphism oK=poK = oK;p=poK;p; and likewise that oF =poF = oF;p=poF;p: Thus to prove the result, one notes that oF;p=poF;p is a vector space of dimension [F : K] over oF;p=poF;p, and that the factorization of poF;p in oF;p yields poF;p = Pe1;p 1;p Pe2;p 2;p Per;p r;p with fi;p = [oF;p=P1;p : oK;p=poK;p]: One also notices that oF;p is the set of elements in F that are integral over oK;p. It is easy to see that ei;p = ei, and that fi;p = fi for each i 2 f1; 2; :::; rg. Also, oF;p is a Dedekind domain with nitely many primes, and, in particular, its primes are exactly those P1;p;P2;p; :::;Pr;p lying above poK;p. This means that oF;p is a principal ideal domain. For an ideal ap of oF;p factors uniquely as ap = Yr i=1 Pe1 1;pPe2 2;p Per r;p where e1; e2; :::; er are integers, each at least equal to zero. By the Chinese remain der theorem, one may select an element x 2 oF;p so that 27 x = ei i mod Pei+1 i;p where i is an element of Pi;p not contained in P2 i;p. In this way the factorization for the principal ideal (x) is (x) = Yr i=1 Pe1 1;pPe2 2;p Per r;p; and therefore (x) = ap. Thus oF;p is a principal ideal domain. One then notes that the primes P1;p;P2;p; :::;Pr;p are pairwise relatively prime, and thus the Chinese remainder theorem gives an isomorphism oF;p=poF;p ! Yr i=1 oF;p=Pei;p i;p : In view of this isomorphism, and the natural isomorphism oF;p=Pi;p ! Pj i;p=Pj+1 i;p for each i 2 f1; 2; :::; rg, in each case given by multiplication by an element of oF;p by i j , where i is a generator of the principal ideal Pi;p, the identity easily follows. With this machinery in mind, one now considers a nite Galois extension F of K, and de nes for a prime P of oF the decomposition group of P, GP = f 2 G(FjK) j P = Pg; with G intended to denote G(FjK). The homomorphism ! given by x = x mod P for each x 2 oF yields for each 2 GP an associated : oF :=P ! oF =P; and thereby a map from GP to the Galois group of oF =P over oK=P, which has kernel called the inertia group of P, and is denoted by IG;P. The following theorem is of great importance. 28 Theorem 3.3 For any two prime ideals P and P0 of oF lying above p in oK, there exists some 2 G(FjK) so that P = P0, i.e., G(FjK) acts transitively on the set of primes lying above p. Proof. . If the theorem were not true, then the Chinese remainder theorem would yields some x 2 oF where x = 0 mod P0 and x = 1 mod P for all 2 G(FjK). Then one would have NFjK(x) = Y 2G(FjK) x 2 P0 \ oK = p; and yet, as x =2 P for any 2 G(FjK), one must have x =2 P for any 2 G(FjK), whence Y 2G(FjK) x =2 P \ oK = p; which is a contradiction. The following lemma is of importance in constructing Artin's Lfunction. Denote by FP the xed eld of the decomposition group GP. In general, for elds F F0 K with prime ideals PjP0jp where P is a prime in oF , P a prime in oF0 , and p is a prime in oK, one may de ne the inertia degrees f = [oF =P : oK=p], f0 = [oF =P : oF0=P0], and f00 = [oF0=P0 : oK=p]. It follows that f = f0f00. With rami cation indices e, e0, and e00 de ned similarly, one also has e = e0e00. Lemma 3.1 Let P be a prime of oF lying above the prime p of oK. One has oFP=foFP \ Pg = oK=p. Proof. With poF = Yr i=1 Pi ei one has e1 = e2 = = er = e and f1 = f2 = = fr = f from the transitivity of G(FjK) over the primes lying above p, by the previous theorem. In this case, the 29 fundamental identity [F : K] = Pr i=1 eifi reduces to [F : K] = efr where r = (G : GP). Therefore [F : FP] = ef. Considering a particular prime P lying above p, one considers f, f0, and f00 as above, and likewise for e, e0, and e00, taking in this case F0 = FP. Thus one has f00 = e00 = 1, and the result follows. The previous result motivates the following theorem. Theorem 3.4 The map GP ! G(oF =PjoK=p) de ned by mapping 2 GP to 2 G(oF =PjoK=p) satisfying x = x mod P; for every x 2 oK, is surjective. Proof. By the previous lemma, one may assume that K = FP, so that G(FjK) = GP. Consider then a primitive element of oF =P as an extension of oK=p, i.e., an element x satisfying oF =P = foK=pg(x); which exists because the extension oF =P over oK=p is separable. Consider then the minimal polynomial g of x over oK=p, and a lift x of x in oF , and suppose that 2 G(oF =PjoK=p): Then (x) = x0, where x0 is also a root of the polynomial g. Also, the minimal polynomial f of x over K takes all of its roots in oF because F is normal over K, and therefore, f 2 oK[X]. Thus one may consider f, the reduction of f modulo P, and that g must divide f. In this way g must have roots corresponding to the reductions of roots of f taken modulo P, and so x0 also has a lift x0 in oF that is a root of f. In particular, there is a 2 GP so that (x) = x0, and thus (x) = x0, as desired. 30 This proves the surjectivity, because an element in G(oF =PjoK=p) is completely determined by how it acts on the primitive element x. In the cases of these nite primes, one will have that the Galois group of oF =P over oK=p is cyclic, and one may choose as a generator of GP=IG;P an element mapping to what is called the Frobenius element in G(oF =PjoK=p), which is the automorphism given by x ! xq; where q = joK=pj. In this setting, this generator of GP=IG;P is itself called the Frobenius for P on account of the isomorphism GP=IG;P ! G(oF =PjoK=p): Let then ( ; V;G(FjK)) be a representation, and let V IG;P be the subspace of V held xed by IG;P. This V IG;P is called the module of invariants for IG;P, and one may also notice that the Frobenius element for P, and hence all of GP=IG;P, must map V IG;P to itself. Denoting the Frobenius element for P by G;P, one may consider the expression det(I ( G;P)N(p)s j V IG;P): This is intended to denote the determinant of the expression I ( G;P)N(p)s as a matrix acting on V IG;P. Here, N(p) = p[oK=p:Z=pZ] where p lies above p 2 Z. This determinant does not depend upon the prime P chosen, because any two primes P, P0 lying above p have GP and GP0 , IG;P and IG;P0 , and the Frobenius elements in each quotient group GP=IG;P and GP0=IG;P0 as simultaneous conjugates. Also, the above determinant must depend only upon the character of the representation , and thus, so does the product Y p prime p2oK 1 det(I ( G;P)N(p)s j V IG;P) : 31 One de nes this to be the Artin Lfunction, and denotes it by L(FjK; ; s) for a character of a representation of G(FjK). A prime is called unrami ed if IG;P = f1g. One may note that there are nitely many primes that ramify in F, so that all but nitely many p in K have V = V IG;P, so that the expression det(I ( G;P)N(p)sj V IG;P) is a polynomial of degree dim( ) in qs. Theorem 3.5 The Artin Lfunction L(FjK; ; s) converges in the half plane Re(s) > 1. Proof. One may consider the factorization det(I ( G;P)N(p)s j V IG;P) = Ydp i=1 (1 "i;pN(p)s) for a prime p of oK where each "i;p is a root of unity, and dp dim( ). Taking formally the logarithm of the Artin Lfunction L(FjK; ; s) then yields log L(FjK; ; s) = X p X1 m=1 Xdp i=1 "i;p mN(p)ms : If Re(s) > 1, one has X p X1 m=1 Xdp i=1 "i;p mN(p)ms X p X1 m=1 Xdp i=1 "i;p mN(p)ms = X p dp X1 m=1 1 mN(p)mRe(s) dim( ) X p X1 m=1 1 mN(p)mRe(s) [K : Q]dim( ) X p2Q p prime X1 m=1 1 mpmRe(s) = [K : Q]dim( ) log (Re(s)) 32 where denotes Riemann's zeta function, which converges in the halfplane Re(s) > 1. . The following section in this chapter outlines some basic properties of Artin L functions. 3.2 The basic properties of this function There will be a few properties of this function that have use in this work. First, one notes the following theorem. Theorem 3.6 (i) If each of and 0 is a character of a representation of G(FjK), then one has L(FjK; + 0; s) = L(FjK; ; s)L(FjK; 0; s); (ii) if F0 is a Galois extension of K containing F, and is a character of a repre sentation of G(FjK), then with the representation yielding acting on G(F0jK) via the canonical quotient map G(F0jK) ! G(FjK); one has L(F0jK; ; s) = L(FjK; ; s); (iii) if F0 is a sub eld of F containing K, and is a character of a representation of G(FjF0), then L(FjF0; ; s) = L(FjK; IndG H( ); s); where G = G(FjK), and H = G(FjF0). 33 Proof. (i) is trivial. For (ii), one may note that the canonical map G(F0jK) ! G(FjK) yields a surjection G(F0jK)P0=IG(F0jK);P0 ! GP=IG;P where P0jPjp, with P0 a prime of oF0 and P a prime of oF , which maps the Frobenius of P0 to the Frobenius of P. This makes clear (ii). (iii) Suppose that p is a prime ideal of K, and that q1; q2; :::; qr are the prime ideals of oF0 lying above p. Choose then for each i 2 f1; 2; :::; rg a prime ideal Pi of oF lying above qi. One has the equalities GPi \ H = HPi and IG;Pi \ H = IH;Pi : One has N(qi) = N(p)fi ; where fi = jGPi : HPiIG;Pi j: By the previous theorem, one may choose an element i contained in G(FjK) satisfy ing iPi = P1. Then one will have GPi = 1 i GP1 i; and also that IG;Pi = 1 i IG;Pi i. Considering then an element 1 2 GP1 that is mapping to the Frobenius G;P1 2 GP1=IG;P1 , one will also have that i = 1 i 1 i is similarly mapped to the Frobenius G;Pi . Also, the image of fi i in HPi=IH;Pi is the Frobenius H;Pi . Considering then : H ! GL(W) a representation of H yielding as its character, and ( ; G; V ) denoting the induced representation, it will su ce to show that det(I ( 1)t j V IG;P1 ) = Yr i=1 det(I ( fi i )tfi j WIH;Pi ): 34 Henceforth in this proof, the notation ( ) will be written simply as , and likewise for . For each i 2 f1; 2; :::; rg, conjugation by i yields det(I fi i tfi j WIH;Pi ) = det(I fi 1 tfi j iWIG;P1 \ iH 1 i ); where also fi = jGP1 : (GP1 \ iH 1 i )IG;P1 j. For each i 2 f1; 2; :::; rg, one then selects a system of representatives f i;jg of GP1 mod GP1 \ iH 1 i . Then f i;j ig is a system of representatives on the left of G mod H, and one obtains a decomposition for the vector space V corresponding to the induced representation ( ; V;G) of ( ;W;H) as V = M i;j i;j iW: Then by letting Vi = M j i;j iW; one obtains a decomposition V = L i Vi of V as a GP1module. Therefore one must have that det(1 1t j V IG;P1 ) = Yr i=1 det(I 1t j V IG;P1 i ): Now it su ces to show that det(1 1t j V IG;P1 i ) = det(I fi 1 tfi j iWIG;P1 \ iH 1 i ) for each i 2 f1; 2; :::; rg. One has, of course, that ( ; Vi;GP1) is the representation induced from ( i; iW;GP1 \ iH 1 i ). But also, one may notice that V G;IP1 i = Ind GP1 =IG;P1 fGP1 \ iH 1 i g=fIG;P1 \ iH 1 i g ( iWIG;P1 \ iH 1 i ): Taking then a basis fw1; :::;wdg for iWIG;P1 \ iH 1 i and noting the decomposition V IG;P1 i = Mfi1 l=0 l 1 iWIG;P1 \ iH 1 i 35 yields the matrix B = 0 BBBBBBB@ 0 Id d 0 0 0 Id d A 0 0 1 CCCCCCCA for 1 acting on V G;IP1 i , where Id d denotes the d d identity matrix and A denotes the matrix for fi 1 acting on iWIG;P1 \ iH 1 i . Thus, one must have det(I 1t j V IG;P1 Pi ) = det(I fi 1 tfi j iWIG;P1 \ iH 1 i ); a fact which may be seen by multiplication of the rst column of Ifid fid Bt by t and subtraction from its second column, multiplication of the second column of the resulting matrix by t and subtraction from its third column, and so forth. This proves the claim. 36 CHAPTER 4 Class eld theory 4.1 Id eles Class eld theory provides an essential link between nite Galois extensions of an algebraic number eld K and Lfunctions that ultimately leads to the functional equation for Artin's Lfunction, which is addressed in chapter seven of this work. In order to discuss class eld theory, some de nitions are in order. For this, one will deal with the nite primes of an algebraic number eld K, given by p as before, but introduces now the notion of an in nite prime. In this cases, each in nite prime will be given by an embedding : K ! C; with the only restriction on this being that two such embeddings which are complex conjugates of each other are associated with the same prime. The notation p  1 will be used to note that one is dealing with nite primes, and pj1 will imply an in nite prime, given by an embedding as above. Together, these in nite and nite primes comprise what is called the primes of K. Each of the primes determines a valuation. In the case of a nite prime, the valuation, which is nonarchimedean, is given in accordance with that assigned via the localization of K for the prime p. In the case of an in nite prime p, the valuation of K is de ned to be jxjp = j xj where j j denotes the modulus in C, and the embedding yields the in nite prime 37 p, where one may note that taking the complex conjugate of does not alter this valuation, and thus one has motivation for associating complex conjugate embeddings of K with the same in nite prime. Denote the completion of K with respect to the valuation given by the prime p as Kp. One de nes the id ele group IK to be the set of elements ( p), with p ranging across all primes of K, in nite and nite, where p 2 K p is a unit in the ring of integers op of Kp of K with respect to the valuation given by the prime p, for almost all primes p. One equips this product space IK = Y p op with a topology generated by sets of the form Y p2S Wp Y p2S Up where S denotes a nite set of primes containing the in nite primes, and Wp denotes a neighborhood of 1 2 Kp in the topology corresponding to the valuation associated with the prime p. Considering then a nite Galois extension F of K, one may then de ne Fp = Y Pjp FP a norm NFpjKp : F p ! K p for ( P)Pjp by NFpjKp(( P)Pjp) = Y Pjp det( P); with each P here viewed as an automorphism from FP to FP over Kp, and the determinant is taken according to the matrix of this automorphism of FP when viewed as a vector space over Kp. In this way one obtains what is called a global norm NIF jIK : IF ! IK 38 de ned for = ( P)P 2 IF by (NIF jIK( ))p = NFpjKp(( P)Pjp); for each prime p of K. One now de nes for a number eld K the group CK = IK=K ; known as the id ele class group, where each element x of K is viewed as diagonally embedded into IK by de ning x = (xp)p 2 IK to have xp = x for all primes p of K. This is possible because the decomposition of the principal ideal (x) = Y p pep into a product of prime ideals shows that x is a unit in op for almost all primes p. One may then notice that the norm as de ned above yields for x 2 F that (NIF jIK(x))p = NLjK(x); a fact which follows from the canonical isomorphism F K Kp = Y Pjp FPjp; and therefore the norm NIF jIK de nes a homomorphism CF ! CK: For a group G, Suppose now that G0 denotes the commutator subgroup of G. The following theorem is the main theorem of class eld theory, and is known as Artin reciprocity. Theorem 4.1 There is an isomorphism A : CK=NIF jIKCF ! G=G0; 39 and the norm map NIF jIK yields a onetoone correspondence between nite Galois extensions F of K with abelian Galois group over K, and the subgroups of nite index in CK that are closed in the quotient topology induced by the canonical topology on IK. Denoting by NF the group NIF jIKCF , one will then have the following facts for two such extensions F1 and F2 of K: (i) F1 F2 if and only if NF1 NF2 ; (ii) NF1F2 = NF1 \ NF2 ; and (iii) NF1\F2 = NF1NF2 . In particular, this correspondence is explicitly given by the association F $ NF ; and therefore, the eld F corresponding to the closed subgroup N of CK of nite index must therefore have N = NF , and will thus satisfy CK=N = G(FjK): Proof. Construction of the map A is given here to provide context; for the full proof of the theorem, the reader is referred to [Neu, VI]. First, there is a local reciprocity map, which is de ned for nite primes, and also for in nite primes. For a nite prime p and an element 2 G, one chooses an extension ~ of to the maximal unrami ed extension of FP so that, when restricted to the maximal unrami ed extension ~K p of Kp, it is a natural number power of the map of G( ~K pjKp) de ned uniquely as (a) = aq mod ~p for all a in the valuation ring of ~K p, where ~p denotes the maximal ideal of this valuation ring. Then, one considers the xed eld of this extension ~ , and a prime of . One then constructs the map rFPjKp : G(FPjKp) ! K p 40 de ned by rFPjKp( ) = N jKp( ) mod NFPjKpF P: The requisite map for the in nite primes is de ned to be trivial for any such prime corresponding to a real embedding of K. For a nonreal embedding, one de nes the reciprocity map rFPjKp via the natural isomorphism G(CjR) = R =NCjR(C ) obtained by identifying each of these groups with Z=2Z. Local class eld theory gives that the map rFPjKp induces, for any prime p of K, an isomorphism G(FPjKp)=G(FPjKp)0 = K p=NFPjKpFP: It is then established via global class eld theory that the map A(( p)) = Y p r1 FPjKp ( p) for an element ( p) 2 IK is surjective onto G(FjK), trivial on K , and yields the desired isomorphism. The local maps constructed in the previous proof will be of some use in a later chapter, which will yield the functional equation for the Artin Lfunction. That this theory gives information relating to Lfunctions lies in construction of what is called the Artin symbol, which is introduced in the following section of this chapter. 4.2 The Artin symbol The following theorem has been included for its centrality to the results that are to come in this work. One may notice that the decomposition group GP de ned in the previous chapter applied to nite primes of a nite Galois extension F of K may be easily extended to the case of a valuation w of F extending a particular valuation v of 41 K, once one notes that each such nite prime induces a nonarchimedean valuation on F. In this case, the decomposition group is de ned so that it agrees on the valuations created by nite primes with the previous de nition of decomposition group, and thus there is no inconsistency in writing that the decomposition group for an extension w is de ned to be Gw = f 2 G(FjK) j w = wg: Here, Fw will denote the completion of F with respect to w, and Kv the completion of K with respect to v. One calls two valuations on a eld equivalent if they induce the same topology. Theorem 4.2 Suppose that K is a complete eld with respect to a valuation j j, and that V is an ndimensional normed vector space over K. Then any two norms on V are equivalent. Proof. It su ces to show that, for a given norm j j on V , there exist constants ; 0 > 0 satisfying kxk jxj 0kxk for all x 2 V , where here, with x written in terms of a basis fv1; :::; vng of V as x = x1v1 + xnvn, one has kxk = maxfjx1j; :::; jxnjg: One nds 0 = jv1j + + jvnj as adequate, and is found inductively. For n = 1, a possible choice of is jv1j. Suppose then that the theorem is proven for all such vector spaces of dimension less than n. De ne Vi = Kv1 + + Kvi1 + Kvi+1 + Kvn: 42 Then Vi has the norm j j as induced from V , and this norm is, in particular, equivalent to the maximum norm k k on Vi. Therefore Vi is complete with respect to j j, and thus is closed in V according to this norm. Therefore, the set Vi + vi is also closed. Noticing that 0 =2 [ni =1 Vi + vi and identifying a neighborhood of radius > 0 around zero disjoint from this set, one nds with x = x1v1 + xnvn 6= 0 that x max jxij = x1 max jxij v1 + + vr + + xn max jxij vn ; and thus jxj kxk: This proves the claim. Theorem 4.3 The elements Gw are exactly those which extend uniquely to elements in the Galois group of Fw over Kv, i.e., restriction from Fw to F induces an isomor phism Gal(FwjKv) = Gw. Proof. Since Kv is complete and Fw is a nite extension of Kv, the previous theorem gives that the valuation w on Fw is the unique extension of v from Kv to Fw. Therefore any element 2 G(FwjKv) must lie in Gw when restricted to F. Conversely, an element 2 Gw is, by de nition, continuous with respect to the valuation w. But an arbitrary element 2 G(FjK) continuous with respect to this valuation yields that jxjw < 1 implies jxjw < 1, whence w and w must be equivalent valuations, and therefore, as they must agree on K, are the same, so that 2 Gw. Thus the elements of Gw are the elements of G(FjK) that are continuous with respect to the valuation w, therefore each element of Gw must extend to a continuous automorphism of Fw over Kv via the map fxngn2f1;2;3;:::g = lim n!1 xn for a Cauchy sequence fxngn2f1;2;3;:::g F, as F is dense in Fw. This proves the claim. 43 With this in mind, one may view the group G(FwjKv), for each such extension w of a valuation v as addressed in the theorem, as a subgroup of G(FjK), and one may do so for each valuation v of K. For what follows in this section, one will restrict attention to the nite primes of p. Within this set, one will restrict attention to what are called the unrami ed nite primes. Therefore, in considering the decomposition group G(FPjKp) G(FjK), one may note that this case allows for the Frobenius of P to lie in G(FjK). Assuming that F has abelian Galois group over K, one may then de ne the Frobe nius automorphism p to be the generator of G(FPjKp) associated via isomorphism with G(oF =PjoK=p), as in the previous chapter, with the map x ! xq; and, in fact, this will be the generator for the decomposition group of any P lying above p. A result in class eld theory gives for a prime element p of Kp that rFPjKp( p) = p mod NFPjKpF P; and one de nes p = FjK p : The Artin symbol is de ned on the set of fractional ideals of K which have prime factorization consisting only of unrami ed primes as FjK a = Y p FjK p vp where a = Y p pvp is the prime factorization of the ideal a. Suppose then that m is an ideal of oK whose prime factorization contains every prime ideal that is rami ed. Then one may de ne Jm K to be the group of fractional 44 ideals of K relatively prime to m, and Pm K to be the subset of Jm K consisting of those principal ideals generated by an element x 2 K so that x > 0 for every embedding : K ! R: De ne ClmK = Jm K=Pm K. Also, de ne for any prime p the ring U(0) p = Up, and likewise for vp > 0 U(vp) p = 8>>>>< >>>>: 1 + pvp if p  1; R + if p is real; C if p is complex: 9>>>>= >>>>; One then de nes Im K = Q p Uvp p , and Cm K = ImK =K : With these notions, one notes that a map ( ) may be obtained, induced from sending the id ele = ( p)p to the ideal ( ) = Y p1 pvp( p); where here vp denotes the valuation corresponding to the nite prime p, which yields an isomorphism CK=Cm K = ClmK; proven in this section as a consequence of the following theorem, which relates equiv alent valuations. Note that the valuation on a eld K satisfying jxj = 1 for all x 6= 0 is excluded. Theorem 4.4 (Weak approximation theorem) Consider a eld K, and a set of pair wise inequivalent valuations j j1, j j2,...,j jn on K. Then one has that (i) for a1; a2; :::; an 2 K, and every " > 0, where exists an x 2 K satisfying jx aiji < " for each i 2 f1; 2; :::; ng, and that 45 (ii) for every " > 0 and k 2 f1; 2; 3; :::g, there exists an x 2 K satisfying jxk 1ji < "; for each i 2 f1; 2; :::; ng. Proof. First, one may notice that two valuations j j1 and j j2 are equivalent if and only if jxj1 < 1 implies that jxj2 < 1. This may be seen in the following way. For j j1 = j js 2 with s > 0 implies that j j1 and j j2 are equivalent. Conversely, if two valuations are equivalent, then one must have that jxj1 < 1 implies that jxj2 < 1. Considering then a xed element y 2 K with jyj1 > 1. Considering then x 2 K with x 6= 0, one must have that jxj1 = jyj 1 for some 2 R. Then, supposing that fml nl gl2f1;2;3;:::g is a sequence of rational numbers, with nl > 0 for each l 2 f1; 2; 3; :::g, converging to from above, one must have that jxj1 = jyj 1 < jyj ml nl 1 for each l 2 f1; 2; 3; :::g, whence by the equivalence of j j1 and j j2, it follows that xnl yml 2 < 1; whereby jxj2 jyj ml nl 2 for each l 2 f1; 2; 3; :::g, and thus jxj2 jyj 2 . Considering another such sequence of rational numbers approaching from below yields that jxj2 jyj 2 , and hence that jxj2 = jyj 2 : Therefore one may de ne ln jxj1 ln jxj2 = s; and note that ln jxj1 ln jxj2 = ln jyj1 ln jyj2 ; 46 and therefore that jxj1 = jxjs 2 with s so de ned. Of course, one must have that jyj2 > 1 as jyj1 > 1 and the valuations j j1 and j j2 are equivalent, and therefore that s > 0. Returning now to the pairwise inequivalent valuations j j1; j j2; :::; j jn, one has that because j j1 and j jn are inequivalent, there must be, by the above argument, some 2 K satisfying j j1 < 1 and j jn 1. Likewise, there is some 2 K satisfying j jn < 1 and j j1 1. Therefore the elements y = satis es jyj1 > 1 and jyjn < 1. One now shows inductively that there exists z 2 K satisfying jzj1 > 1 and jzjj < 1 for each j 2 f2; :::; ng. Notice that this has been proven in the rst case n = 2. Therefore, taking an element z 2 K satisfying jzj1 > 1 and jzjj < 1 for each j 2 f2; :::; n1g, one may consider the following cases. First, if jzjn 1, then, with y as before, taking zmy for m 2 f1; 2; 3; :::g chosen to be su ciently large will yield the desired element. If, on the other hand, jzjn > 1, then one must consider the quantity tm = zm 1 + zm and may note that this converges to 1 with respect to j j1 and j jn, and to zero with respect to j j2; :::; j jn. Therefore taking tmy for m 2 f1; 2; 3; :::g su ciently large will yield the desired element. Then, considering the element as desired, and calling it z, one notes that the quantity zm 1 + zm is again of use, and yields a sequence as m ranges over values in f1; 2; 3; :::g that converges to 1 with respect to j j1, and to zero with respect to j j2; :::; j jn. Thus for every i one nds in this fashion an element zi close to 1 with respect to j ji and close to zero with respect to j jj for j 6= i. considering then the element x = a1z1 + + anzn 47 yields (i). For (ii), one may set ai = 1 for each i 2 f1; 2; :::; ng, and note that the element x = z1 + + zn satis es jxk 1ji < ", so long as zi is chosen to be su ciently close to 1 with respect to j ji and to zero with respect to j jj for j 6= i, for each i 2 f1; 2; :::; ng. As promised, one may now establish the following. Theorem 4.5 There is an isomorphism CK=Cm K = ClmK induced by the map sending the id ele = ( p)p to the ideal ( ) = Y p1 pvp( p): Proof. With m decomposed as a product of prime ideals m = Y p pvp as before, one de nes the set I(m) K = f = ( p)p 2 IKj p 2 U(np) p for pjm or pj1g: One has that IK = I(m) K K , as every 2 IK has by the previous theorem that some a 2 K exists satisfying pa 1 mod pnp for pjm, and pa > 0 for every in nite prime corresponding to a real embedding of K into C. Therefore one has the containment = ( pa)p 2 I(m) K ; and that 2 I(m) K K . One notices now that the set I(m) K \ K consists precisely of those elements in K generating principal ideals in the group Pm, and thus that one has a map from I(m) K =(I(m) K \ K ) ! Jm K=Pm K de ned by = ( p)p ( ) ! Y p1 pvp( p); 48 which is clearly surjective. Noting that one has CK = I(m) K K =K = I(m) K =(I(m) K \ K ); one obtains an induced homomorphism from CK to Jm K=Pm K. Under this mapping, as ( ) = 1 for each 2 Im K, the group Cm K = ImK =K is contained in the kernel of the homomorphism induced from CK. Conversely, suppose that the class [ ] in CK corresponding to a representative 2 I(m) K is contained in the kernel of this map. In that case there is some a 2 I(m) K \K with (a) 2 Pm K so that (a) = ( ). But then the id ele ( p)p = = a1 satis es p 2 Up for all nite primes p not dividing m, p 2 U(np) p for all nite primes p diving m, and likewise for all in nite primes. Therefore, in fact, 2 Im K, so that [ ] = [ ] 2 Im KK =K = Cm K: This establishes the result. One may verify, then, the following commutative diagram, with each row exact. 1 ! NIF jIKCF id ! CK A( ) ! G(FjK) ! 1 ( ) ??y ( ) ??y id ??y 1 ! (NIF jIKCF ) id ! ClmK ( FjK ) ! G(FjK) ! 1: This provides the machinery su cient to begin a study of Lfunctions. 49 CHAPTER 5 The functional equation This chapter presents results that, together, yield the functional equation for the Artin Lfunction. 5.1 Hecke's Lfunction This section provides the functional equation for the Hecke Lfunction. Formally, a Hecke Lfunction is given by L ( ; s) = X a2oK (a;m)=1 (a) N (a)s ; where : Jm K ! C is a homomorphism, for which there exists a pair of characters f : (oK=m) ! S1 and 1 : R K ! S1; where here R K corresponds to the set of units in the Minkowski space RK = fx = (x ) 2 Y Cjx = x g; with the product taken over all embeddings of K into C, and denotes the action of complex conjugation, satisfying ((a)) = f (a) 1 (a) 50 for all a 2 oK with ((a) ;m) = 1. Such a homomorphism is called a Gr o encharakter of K. Later, a special case of this will be a Dirichlet Lfunction, where the set Pm K is contained in the kernel of . The function f is called the nite part of , and 1 the in nite part of . De ne now for x = (x ) 2 Q C and y = (y ) 2 Q R the ntuple exponent xy = (xy ) ; where n is the degree of K over Q as a eld. Consider [R R]+ = f(x1; x2) 2 R Rjx1 = x2g: De ne also Y R Y [R R]+ with the rst product over all real embeddings of K into C, and the second over a set of representatives, one for each pair of complex conjugate nonreal embeddings of K into C. A descriptive result for 1 follows. Theorem 5.1 One has the decomposition of 1 as 1 (x) = N xpjxjp+iq where p = (p ) and q = (q ), with (q ) 2 Y R Y [R R]+; where p 2 f0; 1g for all real embeddings of K into C, and p ; p 2 Z is each nonnegative with p p = 0 for each nonreal embedding of K into C. Proof. x 2 R K may be written as x = x jxj jxj; 51 where jxj 2 R K+ = fx 2 R Kj x > 0 for each 2 Hom(K;C)g; and x jxj 2 UK = fx 2 R Kj jxj = 1g: It thus su ces to determine the characters of R K+ and those of UK. The characters of UK are trivially of the form N (xp) where p = (p ) is as stated in the theorem by considering the decomposition UK = Y f0; 1g Y [S1 S1]+: For the characters of R K+, one obtains an isomorphism of R K+ with R = Y R Y [R R]+ via the componentwise natural logarithm, and observes that any character of the latter must be given by (x) = N ((eiq x ) ) with q = (q ) 2 Y R Y [R R]+: Thus a character of R K+ is given by (x) = N eiq ln x = N (xiq) ; and thus 1 (x) = N x jxj p jxjiq = N xpjxjp+iq : Another construction is essential to the desired functional equation. A set X in a Euclidean space is called centrally symmetric if x 2 X implies that x 2 X. A complete lattice in a Euclidean vector space V of dimension n is a topologically discrete set generated by a set of n vectors linearly independent over R. With this generating set denoted by fv1; :::; vng, one de nes the fundamental mesh of to be the set ft1v1 + + tnvn j 0 ti < 1; for each i 2 f1; 2; :::; ngg: The following geometric result is due to Minkowski. 52 Theorem 5.2 Suppose that is a complete lattice in a Euclidean vector space V of dimension n. Suppose also X is a centrally symmetric, convex subset of V , with the volume of X greater than 2n times the volume of the fundamental mesh of . Then X contains at least one nonzero lattice point 2 . Proof. A sketch of this proof is given here. One shows this by demonstrating that the sets 1 2X + cannot be pairwise disjoint as ranges over the elements of , and therefore that there is a point in the intersection of 1 2X + 1 and 1 2X + 2 for some 1; 2 2 with 1 6= 2. Thereby one obtains this point as 1 2 x1 + 1 = 1 2 x2 + 2 for some x1; x2 2 X, whence = 1 2 = 1 2 x2 1 2 x1 is a nonzero element in X, because, by de nition, X is convex and centrally symmet ric. For the next result, one de nes the following terms. The class group of K is the group JK=PK, where JK denotes the group of fractional ideals of K, and PK the group of principal ideals of K. The integer s shall denote the number of in nite primes of K corresponding to nonreal embeddings of K into C. The canonical volume on RK is the volume associated with the metric determined by the Hermitian inner product hx; yi = X 2Hom(K;C) x y Theorem 5.3 The group JK of fractional ideals of K, taken modulo the group of principal ideals PK of K, is nite. Proof. First, it is established that each coset of PK in JK contains an integral ideal a1 so that N (a1) 2 sp jdKj; 53 where dK denotes the discriminant of K. Consider a nonzero integral ideal b of K. Considering c = (c ) satisfying c = c , c 2 R + for each embedding of K into C, and Y c > 2 sp jdKj (oK : b) ; one may notice that the canonical volume of the set X = f(z ) 2 RK j jz j < c for each 2 Hom(K;C)g is su ciently large so as to guarantee by the result of Minkowski a nonzero element b of b contained also in X. Choosing such an element c so that Y c = 2 sp jdKjN (b) + " for " > 0, one obtains a nonzero element in b which is also contained in X, and thus must satisfy jNKjQ ( ) j = Y j j < 2 sp jdKjN (b) + ": This is true for all " > 0, and upon noting that jNKjQ ( ) j is a positive integer, it is clear that some nonzero 2 b satis es jNKjQ ( ) j 2 sp jdKjN (b) : Consider then an arbitrary representative a belonging to a given coset of PK in JK, and 2 oK, not equal to zero, with b = a1 oK: There exists an element 2 b with 6= 0 satisfying jNKjQ ( ) j N (b)1 = N ( ) b1 = N b1 2 sp jdKj; and then a1 = b1 = 1a is then the desired ideal in the coset of a in JK=PK. 54 Considering then a nonzero prime ideal p of oK, with p \ Z = pZ, one has that oK=p is a nite extension of Z=pZ as a eld. Suppose that this extension has degree f. Then, in accordance with the de nition N (a) = (oK : a) ; one must have N (a) = pf . There must be only nitely many prime ideals p lying above p, so that there are only nitely many prime ideals p of oK with N (p) bounded by a given positive real number. One may write a prime factorization for the ideal a, and multiplicativity of N on the ideals of oK implies that there are a nite number of ideals a of oK with N (a) bounded by a given positive real number. But the rst part of this proof established that each coset of PK in JK contains an ideal with norm less than the bound 2 sp jdKj. This proves the claim. For the functional equation for the Hecke Lfunction, two more components are required. Consider a generating set [b1]; [b2]; :::; [br] for JK=PK, and suppose that, for each j 2 f1; 2; :::; rg, hj denotes the order of [bj ] in JK=PK. Then one has for such j that bhj j = (bj) where bj 2 K. One then chooses an element ^bj 2 R K to satisfy ^b j; = ( bi) 1 hi so that ^b j; = ^b j; , for each embedding of K into C, for each j 2 f1; 2; :::; rg. One then constructs the subgroup of Q C generated by K and the elements ^b j for each j 2 f1; 2; :::; rg, and calling this set ^K , one has an isomorphism ( ) : ^K =K = JK=PK de ned by taking the unique representation of an element ^a of ^K as ^a = a^b v1 1 ^b vr r 55 where 0 vj < hj for each j 2 f1; 2; :::; rg, and mapping it to the class of the ideal (^a) = abv1 1 bvr r : The noncanonically de ned set ^K is loosely called the set of ideal numbers. De ning then the set of ideal integers ^ oK as those elements of ^K which, via this map, are sent to ideals in oK, one notices that for the character , which in this case is called a Gr o encharakter modulo m, a unique extension of f is obtained to the set (^o=m) by de ning f = ((a)) 1 (a)1 for each ideal integer a with ((a) ;m) = 1. Viewing f in this way, the Gauss sum ( f ; a) is de ned for an ideal integer a as ( f ; a) = X ^x mod m f (^x) e2 iT r(^xa=md); where here for an element in x = (x ) 2 Q C, Tr (x) = P x , m is an ideal integer with (m) = m, d an ideal integer with (d) = D where D is the di erent of KjQ, and the sum ranges over the classes of (^o=m) mapped to the class 0 for which [(a) 0] = [mD] in JK=PK. With this in hand, and the in nite component of given by 1 (x) = N xpjxjp+iq ; one de nes L1 ( ; s) = N ( s =2) Z R K+ N eyys dy y where s = (s ) and s = s + p iq , for each embedding of K into C. One may then de ne the completed Lfunction as ( ; s) = (jdKjN (m)) s 2L1 ( ; s) L ( ; s) ; with p = (p ) as before, where p = (p ) and p = p . One then sets the Gauss sum of to be ( f ) = ( f ; 1), and W ( ) = " iTr(p)N md jmdj p !#1 ( f ) p N (m) 56 with jmdj = (jm d j). The complex number W( ) is called the root number of . One de nes a Gr o encharakter modulo m to be primitive if it is not the restric tion of a Gr o encharakter modulo m0 for some proper divisor m0 of m. For such a Gr o encharakter modulo m, one has the functional equation for the Hecke Lfunction as ( ; s) = W( ) ( ; 1 s) : The next section develops the notion of the Artin conductor, which will be used to connect Artin's Lfunction to the Hecke Lfunction. 5.2 The Artin conductor Returning to the setting of representation theory, consider a Galois extension F of a eld K, where K is a local eld, i.e., a eld that is complete with respect to a discrete valuation and possesses a nite residue class eld. One de nes the ith rami cation group Gi to be the group Gi (FjK) = f 2 G(FjK) j vF ( a a) i + 1 for all a 2 oF g where here vF denotes the unique extension to F of the additive valuation vK corre sponding to j jK. De ne f ( ) = X i 0 jGij jG0j codim V Gi ; where V Gi denotes the subspace of V that Gi xes via a xed choice of representation ( ; G; V ) with character . De ne FjK (s) = Z s 0 jGxj jG0j dx: One has the following theorem, due to Herbrand. Theorem 5.4 Suppose that F0 is a subextension of F which contains K, and is Galois over K. Let H = G(FjF0). Then one has Gs (FjK)H=H = Gt (F0jK) 57 where t = FjF0 (s) : Proof. With G = G(FjK) and G0 = G(F0jK), one may select for each 0 2 G0 some preimage 0 via the quotient map G(FjK) ! G(F0jK) so that iFjK ( ) = vF ( x x) ; with x chosen to generate oF over oK as an oKmodule, is maximal. Let m = iFjK ( ); one may note that if 2 H belongs to Hm1, then iFjK ( ) m, whence iFjK ( ) m, so that iFjK ( ) = m. If, on the other hand, =2 Hm1, then iFjK ( ) < m, and iFjK ( ) = iFjK ( ). Therefore, one has iFjK ( ) = minfiFjK ( ) ;mg: Therefore, as one must have, with eFjF0 the rami cation index of FjF0, that iF0jK ( 0) = 1 eFjF0 X jF0= 0 iFjK ( ) ; it follows that iF0jK ( 0) = 1 eFjF0 X 2H minfiFjK ( ) ;mg; and because iFjK ( ) = iFjF0 ( ), and eFjF0 = jH0j, one has that iF0jK ( ) 1 = FjF0 iFjK ( ) 1 ; and thus 0 2 GsH=H if and only if iFjK ( ) 1 s by the de nition of the rami  cation groups, which holds if and only if FjF0 iFjK ( ) 1 FjF0 (s) ; 58 which is equivalent to iF0jK ( 0) 1 FjF0 (s) by the above, which in turn is equivalent to 0 2 Gt (F0jK) ; with t = FjF0 (s) : This proves the claim. Similarly to the decomposition groups, one de nes upper numbering as Gt (FjK) = Gs(FjK) where t = FjK(s): The next result is stated, but for brevity is not proven here. It is a theorem due to Hasse and Arf, and employs the theory of LubinTate extensions [Neu, V]. Theorem 5.5 If G(FjK) is abelian, then the points t 1 for which Gt (FjK) 6= Gt+" (FjK) for any " > 0 are contained in Z. Before one proceeds further, some notation is in order. Denote G = G(FjK). Consider the class function aG ( ) de ned on G as equalling aG (1) = fiG ( ) for 6= 1, and aG (1) = f P 6=1 iG ( ) for = 1, where the sum ranges over nontrivial elements of G(FjK). One may alternatively de ne f ( ) according to the decomposition of aG as a class function of G, as aG = X f ( ) ; where f ( ) 2 C. In this setting, one has f ( ) = h ; aGi; where the inner product h 1; 2i is de ned as in Chapter 2 for two complexvalued functions 1 and 2 as 1 jGj X 2G 1( ) 2( ): Thus as aG( ) = aG( 1) for all 2 G, one may de ne f( ) = h ; aGi for an arbitrary class function on G. That the two de nitions given for f( ) agree follows from the following argument. 59 Lemma 5.1 (Frobenius reciprocity) Suppose that H is a subgroup of G. If 1 is a complexvalued class function on H and 2 is a complexvalued class function on G, then one has hIndG H( 1); 2i = h 1; 2jHi: Proof. By de nition, hIndG H( 1); 2i = 1 jGj X 2G IndG H( 1)(g) 2(g) = 1 jGj X 2G X 2R 1( 1 ) 2( ) = 1 jGj X 2R X 2G 1( 1 ) 2( ) = 1 jGj X 2R X 2G 1( ) 2( 1) = 1 jGj X 2R X 2G 1( ) 2( ) = 1 jGj X 2R X 2H 1( ) 2( ) = 1 jHj X 2H 1( ) 2( ) = h 1; 2jHi: Consider then for the rami cation group Gi the character ui corresponding the representation which acts on the subspace W of the algebra C[G] with a basis fv g indexed by the elements of Gi, with W de ned by f X 2Gi x v j X 2Gi x = 0g: This representation is called the augmentation representation for Gi, and ui the aug mentation character for Gi. 60 Theorem 5.6 The two given de nitions for f( ) agree, i.e., h ; aGi = X i 0 jGij jG0j codim(V Gi): Proof. One has aG = X1 i=0 jGij jG0j IndG Gi(ui) and thus f ( ) = h ; aGi = X i 0 jGij jG0j h ; IndG Gi (ui)i; which is equal to = X i 0 jGij jG0j h jGi ; uii by the previous lemma. and one may notice that h jGi ; uii = codim V Gi for i 0, so that the two de nitions of f ( ) agree as desired. Denote now by rG the character of the regular representation of G. The following theorem is of great importance. Theorem 5.7 Suppose that i is a character corresponding to a representation of degree one of a subgroup Hi of G, and that Ki denotes the xed eld of Hi. Then f IndG Hi i = vK dKijK i (1) + fKijKf ( i) ; where dKijK denotes the discriminant ideal of Ki over K, and fKijK the inertia degree. Proof. Considering 2 Hi with 6= 1, one has that aG ( ) = fFjKiG ( ) and aHi ( ) = fFjKiiHi ( ) ; 61 where here Ki denotes the xed eld of Hi, and thus because iG ( ) = iHi ( ), one has aG ( ) = fKijKaHi ( ) = vK dKijK rHi ( ) + fKijKaHi ( ) where the second equality holds because rHi ( ) = 0. In the case that = 1, one may consider DFjK, the di erent of F over K, and supposing that oL = oK[x], with g (X) the minimal polynomial of x over K, one has that DFjK is generated by g0 (x) = Q 6=1 ( x x). Therefore, vL DFjK = vL (g0 (x)) = X 6=1 iG ( ) = 1 fFjK aG (1) : Also, one has dFjK = NFjK DFjK , and thus that, because vK NFjK = fFjKvL, one has aG (1) = vK dFjK : Likewise aHi (1) = vKi dFjKi . One has dFjK = dKijK [F:Ki] NKijK dFjKi : Therefore, aG (1) = [F : Ki]vK dKijK + fKijKvKi dFjKi = vK dKijK rHi (1) + fKijKaHi(1): Therefore f IndG Hi ( i) = hIndG Hi( i); aGi = h i; aGjHii = vK dKijK ( i; rHi) + fKijK ( i; aHi) = vK dKijK i (1) + fKijKf ( i) : This yields the following theorem. 62 Theorem 5.8 Suppose that is a character of G(FjK) corresponding to a represen tation of degree one. Suppose that j is the largest integer so that jGj 6= 1Gj , where when = 1G one sets j = 1. Then one has f ( ) = FjK (j) + 1; and f ( ) 2 Z is nonnegative. Proof. De ne (Gi) = 1 jGij X 2Gi ( ) : If i j, then evidently (Gi) = 0, and thus (1) (Gi) = 1. If i > j, then clearly (Gi) = 1, whence (1) (Gi) = 0. In light of this, one has that f ( ) = X i 0 jGij jG0j ( (1) (Gi)) = Xj i=0 jGij jG0j = FjK (j) + 1; so long as j 0. If j = 1, then one has (1) (Gi) = 0 for all i 0, so that f ( ) = 0 = FjK (1) + 1: Then, considering H the kernel of in G = G(FjK), and F0 the xed eld of H, one has that Gj (FjK)H=H = Gj0 (F0jK) by Herbrand's theorem. In the upper numbering of rami cation groups, this translates as Gt (FjK)H=H = Gt (F0jK) where here t = FjK (j) = F0jK FjF0 (j) = F0jK (j0). However, one must have (Gj (FjK)H=H) 6= 1; 63 and (Gj+ (FjK)H=H) = (Gj+1 (FjK)H=H) = 1 for all > 0, and so Gj (FjK)H=H 6= Gj+ (FjK)H=H for all > 0. Of course, the function FjK is continuous and strictly increasing as de ned, and therefore Gt (F0jK) = Gt (FjK)H=H 6= Gt+" (FjK)H=H = Gt+" (F0jK) for all " > 0, so that by the theorem of Hasse and Arf, one has that t = FjK (j) is an integer. Brauer's theorem on induced characters then yields the following result. Theorem 5.9 An arbitrary character of G(FjK) has f ( ) equal to a nonnegative integer. Proof. This follows from the fact that one may write, by Brauer's theorem, = X nite niIndG Hi ( i) for integers ni and characters i corresponding to representations of degree one of subgroups Hi of G. One does have f ( ) = X nite nif IndG Hi ( i) = X finite ni vK dKijK i (1) + fKijKf ( i) ; with Ki the xed eld of Hi, for each i. This is an integer by the previous theorem, and is nonnegative because jG0jaG is a character of a representation, and is given by X i 0 jGij IndG Gi (ui) : Therefore jG0jf ( ) = ( ; jG0jaG) 0. 64 The following theorem results almost immediately. For this, let K be a completion with respect to a nite prime p of an algebraic number eld, and F a nite extension of K. Theorem 5.10 Suppose that F is Galois over K, and that is a character of G(FjK) corresponding to a representation of degree one. Suppose also that F is the xed eld of the kernel of , and that f is the conductor of F over K, in this case de ned to be the smallest power n 0 of the unique maximal ideal p of K with re spect to its discrete valuation so that U(n) K is contained in the group NFjK (F ), where n 0, and U(n) K = fx 2 Kjx 1 2 png: Then f = f ( ). Proof. One has f ( ) = FjK (j) + 1 as a consequence of a previous theorem, where j is the largest integer such that Gj (FjK) =2 G(FjF ). With t = FjK (j), one has that Gt (F jK) = Gt (FjK)H=H = Gj (FjK)H=H: Also, Gt+" (F jK) Gj+1 (FjK)H=H = 1 for all " > 0. Hence t is an integer by the theorem of Hasse and Arf, and moreover, f ( ) = t + 1 is the smallest integer with Gf( ) (F jK) equal to one. The theory of LubinTate extensions [Neu, V] then implies that the symbol r1 FjK : K ! G(FjK) of local class eld theory maps U(i) K to the rami cation group Gi (FjK) with i = FjK (j), and therefore n is also the smallest nonnegative integer so that Gn (FjK) = 1. Therefore f = f ( ). De ne then the local Artin conductor as fp ( ) = pf( ), with p denoting the unique maximal ideal corresponding to the discrete valuation on K for which K is complete. 65 The above analysis will be applied to the completions of K with respect to its various nite primes. Suppose now that K is an algebraic number eld. Class eld theory gives an object, called a conductor, of a nite Galois extension F of K as the smallest m in oK for which F is contained in the eld Fm, called the ray class eld modulo m, for which G(FmjK) = CK=Cm K; noting that such an Fm must exist by the main theorem of class eld theory. One then de nes f ( ) = Y p1 fp ( ) : This is called the Artin conductor of . Because this conductor does depend upon the choice of underlying elds, it will henceforth be written as f (FjK; ). The following is immediate from the above analysis. Theorem 5.11 Suppose that FjK is a Galois extension of algebraic number elds, and that is a character of G(FjK) corresponding to a representation of degree one. Suppose also that F denotes the xed eld of the kernel of , and f the conductor of F over K. Then one has f = f ( ). Proof. One has by the class eld theory that f = Y p1 fp where fp, for each p  1, is the conductor of FP over Kp. By the previous theorem, one has fp = fp ( ) for each such p, and hence, the result follows. One has the following theorem. The reader is invited to note its similarity to the theorem about the basic properties of the Artin Lfunction appearing in chapter two of this work. Before stating the result, it is worthwhile to note the de nition of norm 66 for ideals of a nite extension F of an algebraic number eld K as NLjK Y P PvP ! = Y p Y Pjp pfPjpvP where here NLjK is mapping from JF ! JK and fPjp is the inertia degree of P over p. Theorem 5.12 (i) f (FjK; + 0) = f (FjK; ) f (FjK; 0); (ii) if F0 is subextension of F that contains K, and is Galois over K, and is a character of G(F0jK), then f (FjK; ) = f (F0jK; ) with acting on G(FjK) via the quotient map G(FjK) ! G(F0jK) ; (iii) if H is a subgroup of F with xed eld K0, and if is a character of H, then one has f FjK; IndG H ( ) = d (1) K0jKNK0jK (f (FjK0; )) : Proof. The proofs of (i) and (ii) are trivial. For (iii), one denotes G = G(FjK), H = G(FjK0), GP = G(FPjKp), where p = P \ K, and one considers the double coset decomposition G = [ GP H: One has naturally that IndG H ( ) jGP = X IndGP GP\ H 1 ( ) where is the character de ned by ( ) = ( 1 ) corresponding to the group GP \ H 1, where one takes as implicit that the representation homomorphism corresponding to the character is modi ed to become ( ) = ( 1 ), and then restricted to GP \ H 1, so as to be a representation to which the character 67 corresponds. Suppose now that dP0 = pvP0 is the discriminant ideal of K0P 0 jKp, where P0 denotes the prime 1P \ K0 of oK0 . Then one has NK0jK (P0 ) = pfP0 according to the de nition of this norm. And fp FjK; IndG H ( ) = pf(IndG H( )jGP) as well as fP0 (FjK0; ) = P0 f jH 1P ; and so, as dK0jK = Q Pjp dK0jKP, it su ces now to show that f IndG H ( ) jGP = X vP0 (1) + fP0 f ( ;H 1P) where H 1P = G 1P \ H. Theorem 5.7 implies that vP0 (1) + fP0 f ( ;H 1P) = f Ind G 1P H 1P jH 1P for each double coset representative , and therefore X vP0 (1) + fP0 f ( ;H 1P) = X f Ind G 1P H 1P jH 1P But one may notice that jH 1P , and thus Ind G 1P H 1P jH 1P , is obtained from conjugation by ; and thus f Ind G 1P H 1P jH 1P = f IndGP GP\ H 1 ( ) because the inertia degrees of the two groups GP and G 1P are the same as in the proof of Lemma 3.1, and because this conjugation by preserves valuations relative to the map 1 : FP ! F 1P. This proves the claim. One is now prepared to address the problem of the functional equation of Artin's Lfunction. 68 5.3 The functional equation De ne now the ideal of Z by c (FjK) = d (1) KjQNKjQ (f (FjK; )) ; where here dKjQ denotes the discriminant ideal of K over Q, with generator jdKj (1)N (f (FjK; )) : Denote this generator by c (FjK; ). One has the following basic result. Theorem 5.13 (i) c (FjK; + 0) = c (FjK; ) c (FjK; 0); (ii) For a subextension F0 of F containing K and Galois over K and a character of G(F0jK), then c (FjK; ) = c (F0jK; ), where gives a character of G(FjK) via the quotient map G(FjK) ! G(F0jK); (iii) If H is a subgroup of G with xed eld K0, then one has c (FjK0; ) = c FjK; IndG H ( ) : Proof. (i) and (ii) follow from the basic properties of the Artin conductor already proven in this work. (iii) does as well, in conjunction with the fact that, with dFjK denoting the discriminant ideal for F over K, one has, for F K0 K, dFjK = NK0jK dFjK0 d[F:K0] K0jK : The following results may be easily proven. De ne the gamma function of a complex variable s with Re(s) > 0 to be (s) = Z 1 0 eyys dy y : Then one has the following functional equations. 69 Theorem 5.14 (i) (s + 1) = s (s); (ii) (s) (1 s) = sin( s) ; (iii) (s) s + 1 2 = 2 p 22s (2s). Proof. These are easily established, and one may refer to [FB, IV] for the proofs. Then, one recalls the Artin Lfunction L (FjK; ; s) = Y p1 1 det I ( P)N (p)s j V IG(FjK);P : De ne L1 (FjK; ; s) = Y pj1 Lp (FjK; ; s) ; where Lp (FjK; ; s) = 2 (2 )s (s) (1) if p is complex, and Lp (FjK; ; s) = s=2 (s=2) n+ (s+1)=2 ((s + 1) =2) n if p is real, with n+ = (1) + ( P) 2 ; n = (1) ( P) 2 ; P is, when pj1, the generator for G(FPjQp), W ( ) 2 C has modulus one, and each p denotes an in nite prime of K, noting that here a prime p is called complex if it corresponds to a pair of complex conjugate nonreal embeddings of K into C, and real if it corresponds to a real embedding of K into C. The following theorem holds naturally. De ne for simplicity the functions LR (s) = s=2 (s=2) 70 and LC (s) = 2 (2 )s (s) ; one applies in the following theorem the identity LR (s) LR (s + 1) = LC (s) : Theorem 5.15 For an in nite prime p, one has (i) Lp (FjK; + 0; s) = Lp (FjK; ; s) Lp (FjK; 0; s); (ii) given a Galois extension F0 of K contained in F, and a character on G(F0jK), one has Lp (FjK; ; s) = Lp (F0jK; ; s) were acts on G(FjK) via the quotient map G(FjK) ! G(F0jK); (iii) if K0 is a eld containing K and contained in F, and is a character of G(FjK0), then with H = G(FjK0) and G = G(FjK) one has Lp (FjK0; ; s) = Lp FjK; IndG H ( ) ; s . Proof. Once again, properties (i) and (ii) are trivial. For (iii), suppose rst that p corresponds to a complex, nonreal prime of K. Then any in nite prime of K0 lying above it is also complex, and one has that the number of such q is equal to [K0 : K]. But also, one has IndG H ( ) (1) = [K0 : K] (1) : This proves the claim for when p is complex. If p is real, one may notice that in the double coset decomposition G = [ H GP where P is a prime of F lying above p, yields a bijection between this decomposition and the set of primes q = P \ K0 of K0 above p. Of course, this element q is real if and only if G P = GP 1 H, which holds if and only if H GP consists of only one coset modulo H. Therefore, the real places among the elements q are obtained 71 by allowing to range through a system of representatives of the right cosets of H in G, and one may notice that, with P the generator for the group GP, one has IndG H ( ) ( P) = X ( P) where the sum ranges over exactly those , representing right cosets of H in G, for which P 1 2 H. One also has that IndG H ( ) (1) = X q complex 2 (1) + X q real (1) ; and therefore the formula LR (s) LR (s + 1) = LC (s) may be used to show that Lp FjK; IndG H ( ) ; s = Y q complex LC (s) (1) Y q real LR (s) (1)+ ( P) 2 Y q real LR (s + 1) (1) ( P) 2 which, in turn, must equal Y qjp Lq (FjK0; ; s) ; by de nition. One may then de ne the completed Artin Lfunction as (FjK; ; s) = c (FjK; )s=2 L1 (FjK; ; s) L (FjK; ; s) ; and the following three properties have been established. Theorem 5.16 (i) (FjK; + 0; s) = (FjK; ; s) (FjK; 0; s); (ii) for a subextension F0 of F that contains K and is Galois over K, and a character of G(F0jK), one has (FjK; ; s) = (F0jK; ; s), where acts on G(FjK) via the quotient map G(FjK) ! G(F0jK); (iii) if K0 is a sub eld of F containing K, and is a character of G(FjK0), then (FjK0; ; s) = FjK; IndG H ( ) ; s , with G = G(FjK) and H = G(FjK0). 72 One has the following important theorem. Theorem 5.17 Consider a character corresponding to a representation of degree one of G(FjK), and denote by F the xed eld of . (i) may be viewed as a primitive Gr o encharakter modulo f(FjK; ); (ii) for every real in nite prime p of K, one has that pp = [F ;P : Kp] where here the notation F ;P indicates the completion with respect to a prime Pjp of the xed eld F of . Proof. (i) The Artin symbol of class eld theory yields a map Jf K=Pf K ! G(F jK) where f denotes the conductor of F over K. In this way, the commutative diagram at the end of chapter four indicates that becomes a Dirichlet character of con ductor f = f (F jK; ) = f(FjK; ). It is trivial that this character then admits a decomposition as a primitive Gr o encharacter modulo f(FjK; ). (ii) Class eld theory gives an isomorphism between IK=If KK and Jf K=Pf K given by the map ( ) from chapter four, and one thus obtains a map from IK=If KK to C by composing ( ) with the Artin symbol and . Considering then p a real in nite prime of K, and = ( p) 2 IK with p = 1 and q = 1 for all q 6= p, it is noted that the class eld theory giving that the image P in G(F jK) of via the map A( ) as de ned in chapter four must be the generator of the decomposition group GP = G(F ;PjKp). The weak approximation theorem yields some a 2 K with a 1 2 f, where a < 0 for the embedding corresponding to the prime p, and 0a > 0 for all other real embeddings 0 of K into C. Therefore a 2 I(f) K ; and as in the proof yielding the result IK=If KK = Jf K=Pf K 73 one may notice that the image of the coset of via the map allowing this isomorphism must be the same as the class of a in Jf K=Pf K, and that a maps to (a), which must then map via the Artin symbol to the generator of GP. Therefore, viewing now as a Gr o encharakter, one has ((a)) = f (a) 1 (a) = ( P) ; and because a = 1 mod f, one has f (a) = 1, and one notes that 1 (a) = N a jaj p = a jajp pp = (1)pp ; and thus P = 1 if and only if pp = 0. Now one may notice that for a character corresponding to a representation of degree one of G(FjK), and that one may write as a character of G(F jK), where F denotes the xed eld of . As a consequence of this, one may compose with the Artin symbol as done above, and, viewing now also as a Gr o encharakter modulo its conductor, one may see the following fundamental theorem due to Artin. Theorem 5.18 For a character corresponding to a representation of G(FjK) of degree one, and with F denoting the xed eld of , one has the equality (FjK; ; s) = ( ; s) where the righthand side of this equality is a Hecke Lfunction with viewed in that case as a character of Jf K=Pf K through composition of : G(F jK) ! C with the Artin symbol, where f denotes the Artin conductor of . Proof. This is a straightforward application of the de nitions at hand. Given this, one may then employ the Brauer theorem, to claim that 74 Theorem 5.19 The completed Artin Lfunction satis es the functional equation (FjK; ; s) = W ( ) (FjK; ; 1 s) where W ( ) is a complex number of modulus equal to one. Proof. This follows by writing as, with G = G(FjK), = X nite niIndG Hi ( i) ; and then applying the basic properties of the Artin Lfunction to decompose it as (FjK; ; s) = Y nite (FjKi; i; s)ni where Ki is the xed eld of i, and this in turn equals, by the previous theorem, Y nite ( i; s)ni ; where, for each i, ( i; s) is the Hecke Lfunction for i when viewed as a Gr o encharakter in the sense of the previous theorem. One has Y nite ( i; s)ni = Y nite (W ( i) ( i; 1 s))ni where each W ( i) is a complex number of modulus equal to one. Also, one has Y nite (W ( i) ( i; 1 s))ni = W ( ) Y nite ( i; 1 s)ni = W ( ) Y nite (FjKi; i; 1 s)ni = W ( ) Y nite (FjK; ; 1 s) ; where W ( ) = Y nite W ( i)ni is a complex number of modulus equal to one. One nal note is necessary that will be used in the sequel. 75 Theorem 5.20 The Artin Lfunction L(FjK; ; s) is nonzero in the halfplane Re(s) > 1. Proof. This follows from the de nition of the Artin Lfunction in the halfplane Re(s) > 1 as a convergent in nite product of analytic functions, each nonzero in the halfplane Re(s) > 1. . This concludes chapter ve. The next chapter establishes the result of Coates and Lichtenbaum on the value of Artin's Lfunction at negative integers. 76 CHAPTER 6 The result of Coates and Lichtenbaum This chapter establishes the result of Coates and Lichtenbaum [CL] on the values of Artin's Lfunction at negative integers. Sections 6.1 and 6.2 are devoted to establish ing the unit theorem of Shintani [Neu, VII]. Sections 6.3 and 6.4 prove a result due to Siegel and Klingen [Neu, VII] on special values of Lfunctions. Section 6.5 establishes the main result of this chapter. 6.1 Polyhedric cones One begins this section by rst considering an ndimensional Rvector space V , a sub eld k of R and Vk a ksubspace of V with V = Vk k R: In practice, the set Vk will simply be an algebraic number eld K. In this setting, a krational simplicial cone of dimension d will be a set of the form C(v1; :::; vd) = ft1v1 + + tdvd j tl 2 R + for each l 2 f1; 2; :::; dgg; where the set fv1; :::; vdg consists of linearly independent elements of Vk. A nite disjoint union of krational simplicial cones will be called a krational polyhedric cone. A linear form L on V will be called krational if its coe cients with respect to a kbasis of Vk lie in k. The following two results are crucial to proving Shintani's theorem, and comprise the body of this section. Theorem 6.1 Every nonempty subset di erent from the set f0g of the form P = fx 2 V j Li(x) 0; 0 < i l;Mj(x) > 0; 0 < j mg 77 for krational linear forms Li and Mj for i 2 f1; 2; :::; lg and j 2 f1; 2; :::;mg, with admittance of the cases where l = 0 or m = 0, is either a krational polyhedric cone or the union of such a cone with the origin. Proof. For the proof, one considers as a rst case P = fx 2 V j Li(x) 0; for each i 2 f1; 2; :::; lgg; for krational linear forms L1; :::;Ll 6= 0. The theorem is trivial for the case of n = 1, and it may be proven by induction. Assuming the theorem true for all vector spaces over R of dimension less than n, one then considers the following argument. If P has no interior point, then there is a linear form Li so that P is contained in the hyperplane L = 0, because V is a vector space, and the induction hypothesis applies to prove the claim. Thus, one may suppose that there is an interior point, and suppose that u 2 P is this point, satisfying L1(u) > 0; :::;Ll(u) > 0: Vk is, by de nition, dense in V , so that one may assume that u 2 Vk. For each i 2 f1; 2; :::; lg, de ne @iP = fx 2 P j Li(x) = 0g: If @iP 6= f0g, then the induction hypothesis again applies to show that @iPnf0g is a krational polyhedric cone, comprised of krational simplicial cones of dimension less than n. If such a cone in @iP has nonempty intersection with @jP, then it is contained in @iP \ @jP, as is obvious from the fact that each @jP is constructed to lie in P. Thus the set @1P [ [ @lPnf0g is a disjoint union of krational simplicial cones of dimension less than n. Denote this disjoint union by [ j2JCj ; 78 where each Cj = C(v1; :::; vdj ) is a krational simplicial cone corresponding to the linearly independent vectors v1; :::; vdj with dj < n, and here, the notation [ will be intended to denote disjoint union. Then, for each j 2 J, de ne Cj(u) = C(v1; :::; vdj ; u): Each of these will be a krational simplicial cone of dimension dj + 1 because u was selected to be an interior point of P. One has Pnf0g = f[ j2J Cjg [ f[ j2J Cj(u)g [ R +u: This may be seen from the following argument. If the point x 2 Pnf0g lies on the boundary of P, then it belongs to some @iP, and thus to some Cj . If, on the other hand, x belongs to the interior of P, then Li(x) > 0 for each i 2 f1; 2; :::; lg, and in this case, if x is a scalar multiple of u, then one has x 2 R +u, but if it is not, then one may consider the following. If s is the minimum of the quantities L1(x) L1(u) ; :::; Ll(x) Ll(u) ; then s > 0, and the element x su lies upon the boundary of P. As by supposition x 6= su, there must be a unique j 2 J for which x su 2 Cj , and therefore a unique j 2 J for which x 2 Cj(u). This proves the rst case of the claim. For the second case, let P be as de ned in the statement of the theorem. Then P = fx 2 V j Li(x) 0;Mj(x) 0; for each i 2 f1; 2; :::; lg and j 2 f1; 2; :::;mgg is a krational polyhedric cone joined with f0g. For each j 2 f1; 2; :::;mg, de ne @jP = fx 2 P j Mj = 0g; similarly to the rst case in this proof. As before, from the de nition of P, one must have that if a simplicial cone in P has nonempty intersection with @jP, then it must be contained in @jP. And P = Pnf[mj =1 @jPg; 79 so that P must also be a krational polyhedric cone. The following result will be used in the proof of the Shintani unit theorem. Theorem 6.2 If C and C0 are krational polyhedric cones, then CnC0 is also a k rational polyhedric cone. Proof. For the proof, one may, of course, suppose that C and C0 are krational sim plicial cones. With d the dimension of C0, one notices that there are n krational linear forms L1; :::;Lnd;M1; :::;Md so that C0 = fx 2 V j L1(x) = = Lnd(x) = 0;M1(x) > 0; :::;Md(x) > 0g: One may then de ne for each i 2 f1; 2; :::; n dg the sets C+ i = fx 2 C j L1(x) = = Li1(x) = 0; Li(x) > 0g and C i = fx 2 C j L1(x) = = Li1(x) = 0:Li(x) < 0g: Also, for each j 2 f1; 2; :::; dg, one may de ne Cj = fx 2 C j L1(x) = = Lnd(x) = 0;M1(x) > 0; :::;Mj1(x) > 0;Mj(x) 0g: One has CnC0 = f[ nd i=1 C+ i g [ f[ nd i=1 C i g [ f [dj =1 Cjg: Each of the sets appearing in this disjoint union is either empty, or is a krational polyhedric cone, from the previous theorem. This proves the result. 6.2 Shintani's unit theorem For completeness, the following theorem has been included, known as Dirichlet's unit theorem. For this part, one considers again the Minkowski space RK: One notices 80 that the eld K naturally embeds into this space via the map jx = Y x: One then takes the logarithm l as (l j)x = Y ln j xj; and one notices that this composite map l j yields an exact sequence 1 ! (K) ! o K ! ! 0 where, in this exact sequence, l j acts on o K, the group of units of the eld K, and maps it into the tracezero hyperplane in H = fx = (x ) 2 Y R j x = x g; and (K), the group of roots of unity contained in K, maps via inclusion into o K. One may now state and prove the Dirichlet unit theorem. Theorem 6.3 The group of units o K of oK is the direct product of the nite cyclic group (K) and a free abelian group of rank r + s 1, where here r denotes the number of real embeddings of K, and s the number of pairs of complex conjugate nonreal embeddings of K into C. Proof. That the group (K) is nite and cyclic is obvious. the group of units o K admits a decomposition into a direct product of (K) with another abelian group will ultimately rely upon the above exact sequence and a decomposition of the image of o K via the embedding l j. To begin, consider a 2 Z with a > 1. One may rst notice that with N(a) = (oK : a) for an ideal a in oK, one has that a principal ideal (a) in oK must satisfy N(a) = jNKjQ(a)j 81 where NKjQ denotes the usual norm from K to Q. Therefore the index of (a) in oK is nite. Therefore, up to multiplication by a unit of oK, there is at most one element in each coset of oK=aoK satisfying jNKjQ( )j = a. This is seen to be true by taking = + a for 2 oK, and noting that = 1 NKjQ( ) 2 oK for NKjQ( ) 2 oK. This holds likewise for , and therefore is equal to up to multiplication by an element of o K. Therefore there are at most (oK : aoK) elements of norm a. Then, one may easily notice that = (l j)o K is a lattice in the set H, because the bounded domain f(x ) 2 Y R j jx j < cg contains nitely many points of , for each c > 0. Now one must show that is a complete lattice in H, in other words, a lattice of dimension equal to the dimension of H, which here is clearly equal to r+s1. To do this, it su ces to nd a bounded set M H so that H = [ 2 M + : In order to nd this set, one constructs a bounded set T in the surface S de ned by S = fy = (y ) 2 RK j jN(y)j = 1g where here N(y) = Q y , so that S = ["2o K Tj": Given x = (x ) 2 T, one then has that the absolute values jx j are bounded away from zero for each embedding of K into C, as Q jx j = 1. In this case, M = l(T) will be bounded as well, and the proof will be nished. Thus, one considers real 82 numbers c > 0 for every embedding of K into C, satisfying c = c for every such embedding of K, and so that Y c > ( 2 )s p jdKj where here dK denotes the discriminant of the number eld K. With (c ) so chosen, one will have, for an element y = (y ) 2 S and the set Xy = f(z ) 2 RK j jz j < c jy j; for all 2 Hom(K;C)g; that the canonical volume of the set Xy is 2s times the Lebesgue volume of the set f(x ) 2 Y R j jx j < c jy j; x2 + x2 < (c jy j)2g; where here , respectively is intended to denote a real, respectively nonreal, em bedding of K into C. Thus the canonical volume of Xy is 2s Y 2c Y ( c2 ) = 2r+s s Y c ; where the product over is taken over a set of representatives, one for each pair of complex conjugate nonreal embeddings of K into C, the product over intends to be over all real embeddings of K into C, and the last product is over all embeddings from K into C. One notes here that these formulations make sense because y so chosen requires jy j = jy j. The canonical volume vol(Xy) satis es vol(Xy) > 2r+s s( 2 )s p jdkj = 2n p jdkj as jN(y)j = 1; and the canonical volume of oK is precisely p jdkj. Thus the result of Minkowski on lattices applies to yield the existence of a point ja 2 Xy for a 2 oK with a 6= 0, for oK does form a complete lattice in the Minkowski space RK According to the rst result in this proof, one may then choose 1; :::; N 2 oK, each nonzero, so that every a 2 oK with 0 < jNKjQ(a)j C is associated in oK to one of the elements 1; :::; N. Therefore the set T = S \ f[Ni =1 X(j i)1g 83 is as desired, for X is bounded, and thus so is X(j i)1 for each i 2 f1; 2; :::;Ng, and thus so must T be bounded. Now one must show that S = ["2o K Tj": This follows from the following argument. If y 2 S, then one may nd a nonzero element of oK as before with ja 2 Xy1, and therefore ja = xy1 for some x 2 X. Of course, one has jNKjQ(a)j = jN(xy1)j = jN(x)j < Y c where the second equality holds because jN(y)j = 1, and therefore a is associated in oK to i, for some i 2 f1; 2; :::;Ng. Therefore, with i = "a for " 2 o K, one has y = x(ja)1 = x(j i)1j"; and therefore, as y and j" are in S, so must x(j i)1 be contained in S, so that x(j i)1 2 T, and therefore y 2 Tj". Then, one notes the exact sequence 1 ! (K) ! o K ! ! 0 as before now has as a free abelian group of rank r + s 1. therefore, considering preimages of a Zbasis for via the map l j, and the subgroup A in o K generated by this basis, one then notes that A = , and that o K = (K) A: This completes the proof. Now, one de nes R KR;+ = fx = (x ) 2 R Kjx > 0 for each embedding : K ! Rg: 84 One then de nes o K;+ = o K \ R KR;+: One is now fully prepared to state and prove Shintani's unit theorem. Theorem 6.4 If E is a subgroup of nite index in o K;+, then there exists a Qrational polyhedric cone P such that R KR;+ = [ "2E "P: Proof. Every element in R KR;+ may be written uniquely as a product of an element contained in the normone surface S, as de ned in the proof of the Dirichlet unit theorem above, and a positive scalar, as x = jN(x)j 1 n x jN(x)j 1 n : Dirichlet's unit theorem gives as above a mapping of E onto a complete lattice of the tracezero space H as de ned before, because E is of nite index in o K. Consider the fundamental mesh of . With the closure of in H being denoted by , one has that l1( ) is closed and bounded, hence compact, in the surface S. Also, one must have S = ["2E "F: Consider then x 2 F, and the set U (x) = fy 2 RK j kx yk < g where > 0 is chosen so that U (x) R KR;+ and the metric k k is any of the set of equivalent metrics on RK extending the usual one on R. One may nd a basis fv1; :::; vng of RK contained in U (x) so that x = t1v1 + tnvn where ti > 0 for each i 2 f1; 2; :::; ng, and K is dense in RK by the weak approximation theorem, so that one may select fv1; :::; vng to lie also in K. 85 One may then notice that C = C(v1; :::; vn) is a Qrational simplicial cone in R KR;+ with x 2 C . Because E is discrete in RK, one may choose su ciently small so that C \ "C = ? for all " 2 E and " 6= 1. For, if not, then one would nd sequences f vzvgv2f1;2;3;:::g; f 0 vz0 vgv2f1;2;3;:::g; f"vgv2f1;2;3;:::g where v; 0 v 2 R +, zv; z0v 2 U1 v (x), "v 2 E, and vzv = "v 0 vzv for each v 2 f1; 2; 3; :::g. The equation v 0 v N(zv) = N(z0 v) for each v 2 f1; 2; 3; :::g, where N denotes the usual componentwise norm on RK, implies that lim v!1 v 0 v = 1: As limv!1 zv = limv!1 z0v = x, one would then have lim v!1 "v = 1; violating the fact that E is discrete in RK. One may notice that each C is open, so that as one considers the set of these as x ranges over elements of F, one nds a nite subcollection C1; :::;Cm in R KR;+ so that F = [mi =1 (Ci \ F): Therefore one has R KR;+ = [mi =1 ["2E "Ci: Now, one denotes C(1) 1 = C1, and C(1) i = Cinf["2E "C1g: 86 As E is discrete, and hence closed, in RK, one may notice that "C1 and Ci are disjoint for almost every " 2 E. For, if not, then one could nd a sequence of elements "vyv = y0 v with yv 2 C1, y0v 2 Ci, and "v 2 E for each v 2 f1; 2; 3; :::g, and thus a sequence f"vgv2f1;2;3;:::g with a limit point, which must be contained in E. This violates the fact that E is discrete. Therefore, by the second theorem in the rst section of this chapter, the set C(1) i is a Qrational polyhedric cone. One then has R KR;+ = [mi =1 ["2E "C(1) i where "C(1) 1 [ C(1) i = ? for each " 2 E and each i 2 f2; :::;mg. In this fashion, one proceeds to de ne C(2) i = C(1) i for i 2, and for i > 2 de nes C(2) i = C(1) i nf["2E "C(1) 2 g; and so forth, and in this fashion inductively arrives at a system C(m1) 1 ; :::;C(m1) m of Qrational polyhedric cones so that R KR;+ = [ mi =1 [ "2E "C(m1) i : This proves the claim. 6.3 Rewriting a particular kind of Lfunction Suppose that K is an algebraic number eld, and m an ideal of oK. Considering then a homomorphism : Jm K=Pm K ! C which, in view of the rst chapter of this work, may be viewed as a onedimensional representation of the group Jm K=Pm K. In this case may also be called a Dirichlet character modulo m. One de nes the Dirichlet Lfunction for by L( ; s) = X a2oK (a;m)=1 (a) N(a)s 87 where N( ) is the usual norm of an ideal as de ned earlier. It is worthy of note here that this is a special case of a Hecke Lfunction. One, of course has the decomposition L( ; s) = X ( ) ( ; s) where the sum varies across the cosets of Pm K in Jm K, and one has ( ; s) = X a2 a2oK 1 N(a) : The cosets are also called classes. Suppose that is such a xed coset and a 2 oK an element belonging to the coset . Suppose that (1 + a1m)+ = (1 + a1m) \ R KR;+ is the set of all totally positive elements in 1 + a1m. The group E de ned as E = om + = f" 2 o K j " = 1 mod m; " 2 R KR;+g acts on (1 + a1m)+ via componentwise multiplication. Two theorems must now be stated to prepare for the result of Siegel and Klingen, which comes in the second section of this chapter. Theorem 6.5 One has a bijection (1 + a1m)+=E ! fa 2 \ oKg mapping the orbit of a to the element aa. Proof. Suppose that a 2 (1 + a1m)+. It follows that a 1 2 m, because a and m are relatively prime. But then aa 2 , because (a) is relatively prime to m, and of course, must be totally positive. Also, one has aa a(1 + a1m) = a + m = oK where the last equality comes from the fact that a and m are relatively prime. 88 To show the surjectivity, consider aa an ideal in \ oK. Then (a 1)a ma m; whence a 2 1 + a1m. Also, as a is totally positive, one thus has a 2 (1 + a1m)+. Considering then a; b 2 (1 + a1m)+, one has naturally that aa = ba if and only if (a) = (b), or in other words, when a = b" for " 2 o K. Of course, such an element " must be contained in R KR;+. Also, " 2 1 + a1m, as is evident from the fact that b(" 1) = a b 2 a1m and thus that " 1 2 m because (b) is relatively prime to m. Of course, if " 2 E and a = b", then clearly (a) = (b), and thus the image under the action stated in the theorem is the same. This proves the claim. Thus, one has for such a representative a the equality ( ; s) = 1 N(a)s X a2 1 jN(a)js where N denotes the usual norm on elements in RK, and a, in line with the previous theorem, ranges across a system of representatives of (1 + a1m)+=E. Now one may notice that Shintani's unit theorem applies to E, for it is of nite index in o K. Therefore with R KR;+ = [ mi =1 [ "2E "Ci; one lets vi1; :::; vidi be a linearly independent set of generators of Ci in K, for each i 2 f1; 2; :::;mg, as in the proof of Shintani's unit theorem. One may multiply these 89 by an appropriate positive element in Z to allow one, without loss of generality, the assumption that vil lies in m for each l 2 f1; 2; :::; dig and i 2 f1; 2; :::;mg. De ne then C1 i = ft1vi1 + + tdividi j 0 < tl 1; for each i 2 f1; 2; :::; digg: De ne also R( ;Ci) = (1 + a1m)+ \ C1 i ; with a chosen as before. One then has the following theorem. Theorem 6.6 The sets R( ;Ci) are nite. Also, ( ; s) = 1 N(a)s Xm i=1 X x2R( ;ci) (Ci; x; s); where (Ci; x; s) = X z jN(x + z1vi1 + zdividi)js; and the latter sum sweeps over all dituples of nonnegative integers z = (z1; :::; zdi). Proof. One knows that the set R( ;Ci) is a bounded subset of a translation by one of the complete lattice a1m, and thus R( ;Ci) is nite. As Ci R KR;+ is the simplicial cone generated by vi1; :::; vidi 2 m, it follows that every element of (1 + a1m) \ Ci may, in fact, be written uniquely as a = Xdi l=1 ylvil where 0 < yl 2 Q for each l 2 f1; 2; :::; lg. With yl = xl + zl; 0 < xl 1, zl 2 Z, for each l 2 f1; 2; :::; dig, one has because Pdi l=1 zlvil 2 m that Xdi l=1 xlvil 2 1 + a1m: Therefore, every a 2 (1 + a1m) \ Ci may be written uniquely as a = x + Xdi l=1 zlvil 90 where x 2 R( ;Ci). And one has, from the Shintani theorem, that (1 + a1m)+ = [mi =1 [ "2E (1 + a1m) \ "Ci; and therefore, as a runs through this set modulo its Eorbits, one has ( ; s) = 1 N(a)s Xm i=1 X x2R( ;Ci) (Ci; x; s): 6.4 Siegel and Klingen's result on special values of Lfunctions This section establishes a result, due to Siegel and Klingen, about the values of certain Dirichlet Lfunctions at negative integers, via the work of Shintani. For the following result, one denotes by Q( ) the eld generated over Q by the values of . Theorem 6.7 If is a Dirichlet character for a totally real algebraic number eld K, then one has L( ;m) 2 Q( ) for m 2 f1; 2; 3; :::g. Proof. First, one notes that, by the rewriting of ( ; s) from the previous section, it su ces to show that the values of (Ci; x; s) lie in Q for each i 2 f1; 2; :::;mg and x 2 R( ;Ci). One may consider only the case where K is totally real, i.e., where every embedding of K into C is real. For if this is not the case, then one automatically has L( ;m) = 0 via the following argument. Denote as before by Fm the ray class eld modulo m. Via the Artin symbol of Chapter 4, one may view as a character of the Galois group G(FmjK). In this way, the function L( ; s) will agree with the Artin Lfunction L(FmjK; ; s) up to nitely many Euler factors at nite primes, and therefore is zero at s = m if and only if the associated Artin Lfunction is zero. It is clear from the functional equation for the Artin Lfunction that L(FjK; ;m) = 0 if K is not totally real. 91 Therefore, one may suppose that K is totally real. Consider the following setup. One de nes Lj(t1; :::; tn) = Xn i=1 ajiti and L i (z1; :::; zr) = Xr j=1 ajizj ; where the elements aji for j 2 f1; 2; :::; rg and i 2 f1; 2; :::; ng form a matrix A, where each aji is real and positive, and (x1; :::; xr) is an rtuple consisting of positive rational numbers. One may then de ne (A; x; s) = X1 z Yn i=1 L i (z + x)s; with the sum over rtuples of nonnegative integers as before. Also, one may let (A; x; s) = (Ci; x; s) as in the rewriting of the Dirichlet Lfunction of the previous section with the matrix A possessing columns equal to the representation of each basis element vil 2 m of Ci as an ntuple with entries consisting of its various embeddings into C because K is totally real. Thus, one proceeds to show that (A; x; s) is a rational number. To do this, one rst looks at (s)n = Z 1 0 Z 1 0 f Yn i=1 etig(t1 tn)s1dt1 dtn; and performs the substitution of ti with L i (z + x)ti, for each i 2 f1; 2; :::; ng. One then easily obtains for Re(s) > r n by summing over all nonnegative rtuples z of integers that (s)n (A; x; s) = Z 1 0 Z 1 0 g(t)(t1 tn)s1dt1 dtn with t = (t1; :::; tn), and g(t) = Yr j=1 exp((1 xj)Lj(t)) exp(Lj(t)) 1 : 92 One then de nes Di = ft 2 Rn j 0 tl ti; for each l 2 f1; 2; :::; ngnfigg; and one notices that (A; x; s) = (s)n Xn i=1 Z Di g(t)(t1 tn)s1dt1 dtn: One then transforms t within Di as t = (t1; :::; tn) = u(y1; :::; yn), where u > 0, 0 yl 1 for l 2 f1; 2; :::; ngnfig, and yi = 1. Therefore one has (A; x; s) = (s)n Xn i=1 Z 1 0 (Z 1 0 Z 1 0 g(uy)( Y l6=i yl)s1 Y l6=i dl ) uns1du; so that by the previous result, one has for Re(s) > r n that and for " su ciently small that (A; x; s) = G(s) Z I"(+1) (Z I"(1)n1 g(uy)uns1( Y l6=i yl)s1 Y l6=i dl ) du where G(s) = (s)n (e2 ins 1)(e2 is 1)n1 ; and I"(a) for a = 1 or1denotes a path in C consisting of the interval [a; "], succeeded by a circle in the counterclockwise direction, and then the interval ["; a], as in the previous theorem. One also notices that this yields a meromorphic continuation of the function (A; x; s) to the complex plane. At s = 1 k, G(s) is equal to (1)n(k1) (k)n n 1 (2 i)n by the basic properties of the function (s) as outlined in section 5.2. The quantity Xn i=1 1 (2 i)n Z 1 0 (Z 1 0 Z 1 0 g(uy)( Y l6=i yl)s1 Y l6=i dl ) uns1du at s = 1 k is the sum over each i 2 f1; 2; :::; ng of the coe cients of un(k1)+r(t1t2 tn)k1 93 in the Taylor expansion of the function ur Yr j=1 exp((1 xj)uLj(t)) exp(uLj(t)) 1 : Denote then di = 1 (2 i)n Z 1 0 (Z 1 0 Z 1 0 g(uy)( Y l6=i yl)s1 Y l6=i dl ) uns1du: One may notice that the elements of A lie in the Galois closure N of K over Q, and that 2 G(NjQ) permutes the elements in each column of the matrix A. Represent ing then as an element of Sn, one then has (di) = d (i) for each i 2 f1; 2; :::; ng. Therefore Pn i=1 di 2 Q. This proves the claim. 6.5 The value of Artin's Lfunction at negative integers In the case of a Galois extension FjK, the functional equation is restated for Artin's Lfunction: L (FjK; ; 1 s) L1 (FjK; ; 1 s) = W ( ) L (FjK; ; s) L1 (FjK; ; s) : With this functional equation, one may note the de nition of a critical point for this Lfunction as s 2 Z where L1 (FjK; ; s) and L1 (FjK; ; 1 s) are nite. One has immediately the following theorem. Theorem 6.8 The point s = m 2 Z with m 2 f1; 2; 3; :::g is critical if and only if L (FjK; ;m) 6= 0: Proof. Passing to the xed eld F of , one has that L (FjK; ; s) = L (F jK; ; s) and the proof reduces to proving the claim for L (F jK; ; s) : Suppose now that s = m is a critical point. Then by de nition the values L1 (FjK; ;m) and L1 (FjK; ; 1 + m) are nite. The lefthand side of Artin's functional equation as written above is nite and nonzero as the Lfunction L (FjK; ; s) must be nonzero 94 at s = 1 + m. Returning to Artin's functional equation, one has that the factor W ( ) is nonzero and nite, as well as the expression L1 (FjK; ;m) by assumption. Thus one must have L (FjK; ;m) 6= 0: Conversely, suppose that this Lfunction is nonzero. Of course, the term L1 (FjK; ; 1 + m) is nonzero and nite by its construc tion, and the term L (FjK; ; 1 + m) is nonzero and nite, as noted before. Thus, the lefthand side of the functional equation is now nonzero and nite, and thus so must the right be nonzero and nite. But W ( ) 6= 0 in any case, and L (FjK; ;m) 6= 0 by assumption. Therefore as the righthand side must be equal to the nite value of the lefthand side, one must have that the value of L1 (FjK; ; 1 + m) is nite. This proves the claim. Suppose now that K = Q, and that : G(FjQ) ! GL(V ) is a complex representation of its Galois group, with character . Passing to the xed eld F of , one observes that Theorem 6.9 One has that s = m with m 2 f1; 2; 3; :::g is a critical point for L (FjQ; ; s) if and only if one of the following is true: (i) m is odd and F is totally real; (ii) F is totally imaginary, the automorphism of complex conjugation is a central element in G F jQ , and ( ) = dim( ). Proof. Suppose that s = m is a critical point. Then by the previous theorem, one has that L (FjQ; ;m) 6= 0: One has that L (FjQ; ;m) = L F jQ; ;m : In that case, the functional equation yields that L1 F jQ; ;m 6= 0; whence, as by de nition, L1 F jQ; ;m = s=2 (s=2) N+ (s+1)=2 ((s + 1) =2) N ; 95 one has poles of this function if and only if m is odd and N = n 2 (1) 1 2 X p real ( P) = 0; or m is even and N+ = n 2 (1) + 1 2 X p real ( P) = 0: In the rst case, one has (1) = ( P) for each in nite prime P, whence acts trivially on each P, and so the generator of any decomposition group of an in nite prime in F is trivial. Thus F is totally real. In the second case, one has (1) = ( P) for each in nite prime P, whence all decomposition groups of in nite primes of F are nontrivial and F is totally imaginary. That the action of complex conjugation is central to the Galois group in this second case follows from Lemma 2.1, as ( ) = (1) = dim( ) and thus = I. For the converse, if (i) holds, then N = 0, whence L F jQ; ;m 6= 0 from the functional equation. If (ii) holds, then one must have N+ = 0, whence L F jQ; ;m 6= 0 from the functional equation. The previous theorem then shows that s = m is a critical point for L (FjQ; ; s) in either case. One then has the following theorem due to Coates and Lichtenbaum [CL]. Theorem 6.10 The values L (FjQ; ;m) for m 2 f1; 2; 3; :::g are algebraic inte gers contained in Q( ). Proof. One has observed that these critical points occur precisely when either the xed eld F of is totally real and m is odd, or the xed eld F is totally imaginary, m is even, and the action of complex conjugation is central to G(F jQ) and has ( ) = dim( ). One may restrict attention to such a point, for if m with 96 m 2 f1; 2; 3; :::g is not critical, then the functional equation implies that the Artin Lfunction L(FjQ; ; s) is zero at s = m. In the rst case, one may use Brauer's theorem to state that = Xl i=1 niIndG Hi ( i) where each Hi is a subgroup of G = G F jQ , each i is a character of degree one of G(FjQ), and each ni 2 Z. One has then L F jQ; ; s = Yl i=1 L F jKi; i; s ni where each Ki denotes the xed eld of the subgroup Hi. One may then observe that L F jKi; i;m 6= 0; for by inducing to F jQ and then restricting via the quotient map G F jQ ! G(FijQ) to the xed eld Fi of that induced character IndG Hi ( i) ; the result of the previous theorem implies, as Fi is totally real and m is odd, that L F jKi; i;m = L F jQ; IndG Hi ( i) ;m = L FijQ; IndG Hi ( i) ;m 6= 0: The work of Siegel and Klingen in the previous chapter then shows that L F jKi; i;m = X i ( ) a where a 2 Q and the sum is over the nitely many cosets of Pm Ki in Jm Ki , for an ap propriate module of de nition m for the xed eld of i. Thus for 2 G QjQ( ) ; one has thus that L F jKi; i;m = L F jKi; ( i) ;m , whence L F jQ; ;m = Yl i=1 L F jKi; i;m ni ! = Yl i=1 L F jKi; ( i) ;m ni 97 = L F jQ; ( ) ;m = L F jQ; ;m ; and thus L F jQ; ;m 2 Q( ), noting that ( i) is viewed for a particular i as a character of G F jKi via the Artin symbol Jm Ki=Pm Ki ! G F ;i jKi with F ;i here denoting the xed eld of the character i of G F jKi , for it has precisely the xed eld that i does. In the second case, it must be that ( ) = dim( ), and one may employ a special case of the Brauer theorem, as given by Serre [CL], yielding = Xl i=1 niIndG Hi ( i) where one may take each Hi to contain the center of G, and therefore the action of complex conjugation, where also i ( ) = 1. On
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Title  On the Galois Structure of Artin's LFunction at its Critical Points 
Date  20090501 
Author  Ward, Kenneth 
Department  Mathematics 
Document Type  
Full Text Type  Open Access 
Abstract  This investigation evaluates a specific relation between an Artin Lfunction of a finite Galois extension F of the rational numbers, and its twist by an even character of degree one, taken here to mean that this character acts trivially on the action of conjugation in the Galois group of F over the rational numbers. First, an integer k is defined to be <italic>critical</italic> for a particular Lfunction if the Euler factors at infinity appearing in the functional equation for the given Lfunction are regular at k and 1k. It is then shown that for such critical points k with k > 1, the twisted Lfunction is equal, up to an element contained in the field generated over the rational numbers by the values of the twisting character and the character appearing in the original Lfunction, to a product of the original Lfunction with a power of the Gauss sum of the twisting character, where the power of this Gauss sum corresponds to the dimension of the representation defining the character of the original Lfunction. 
Note  Thesis 
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Transcript  ON THE GALOIS STRUCTURE OF ARTIN'S LFUNCTION AT ITS CRITICAL POINTS By KENNETH WARD Bachelor of Arts of mathematics The University of Chicago Chicago, Illinois, United States of America 2004 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial ful llment of the requirements for the Degree of Master of Science May 2009 COPYRIGHT c By KENNETH WARD May 2009 ON THE GALOIS STRUCTURE OF ARTIN'S LFUNCTION AT ITS CRITICAL POINTS Thesis Approved: Dr. Anantharam Raghuram Thesis Advisor Dr. Dale Alspach Dr. Alan Noell Dr. A. Gordon Emslie Dean of the Graduate College iii ACKNOWLEDGMENTS I thank Anantharam Raghuram for introducing me to the problem in number theory that is the heart of this thesis. iv TABLE OF CONTENTS Chapter Page 1 Introduction 1 2 Representation theory 3 2.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Brauer's theorem on induction of characters . . . . . . . . . . . . . . 12 3 Artin's Lfunction 22 3.1 Artin's Lfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 The basic properties of this function . . . . . . . . . . . . . . . . . . 33 4 Class eld theory 37 4.1 Id eles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 The Artin symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5 The functional equation 50 5.1 Hecke's Lfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.2 The Artin conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.3 The functional equation . . . . . . . . . . . . . . . . . . . . . . . . . 69 6 The result of Coates and Lichtenbaum 77 6.1 Polyhedric cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.2 Shintani's unit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.3 Rewriting a particular kind of Lfunction . . . . . . . . . . . . . . . . 87 v 6.4 Siegel and Klingen's result on special values of Lfunctions . . . . . . 91 6.5 The value of Artin's Lfunction at negative integers . . . . . . . . . . 94 7 The main result: Behavior of special values under twisting 100 BIBLIOGRAPHY 109 vi CHAPTER 1 Introduction This work provides a relation between an Artin Lfunction for a character corre sponding to a representation of a Galois group of a Galois extension FjQ of number elds, and an Artin Lfunction for a character = of the same group, where is an even Dirichlet character. In particular, it is shown that, with ( ) denoting the Gauss sum of , the value of L(FjQ; ; s) is, up to an element of Q( ; ), equal to ( )dim( )L(FjQ; ; s), so long as s is an integer, at least 2, where the Euler factors at in nity for the Lfunction of are regular at 1 s. This naturally extends a work of Coates and Lichtenbaum [CL], which states that L(FjK; ; s) 2 Q( ) for a character of a Galois extension FjK of algebraic number elds if s is a negative integer. The work of Coates and Lichtenbaum employs a result of Siegel and Klingen [Neu, VII], which states that L(FjQ; ; s) 2 Q( ) if s is a negative integer and is a Dirichlet character. This work, as well as those mentioned above, employ the functional equation for the given Lfunction. It is of some importance that regularity is involved with the main result of this work, since poles emerge in the functional equation at other integer points that limit its use in determining the value of the Lfunction. Preparation is given to the basic topics necessary for understanding the main result. Proofs are provided, with exceptions made for class eld theory and the theory of LubinTate extensions. These omissions are for brevity, and references are provided where omissions have been made. The progression of this work is as follows. Chapter 2 proves Brauer's theorem 1 on induction of characters. Chapter 3 introduces Artin's Lfunction and its basic properties. Chapter 4 surveys the main results of class eld theory. Chapter 5 establishes the functional equation for Artin's Lfunction. Chapter 6 proves the above mentioned result of Coates and Lichtenbaum. Chapter 7 yields the main result. 2 CHAPTER 2 Representation theory 2.1 Basic notions This section introduces the basic notions of representation theory. Here, G will denote a nite group. A representation is a homomorphism : G ! GL(V ) where V denotes a vector space over C of nite dimension. Here, the notation g will be adopted instead of (g). The dimension of V will be called the dimension of the representation . A subspace W V will be called stable if gW W for all g 2 G. De ne a representation to be irreducible if may not be written as 1 M 2 : G ! G(V1) M G(V2) where i : G ! G(Vi) for each i 2 f1; 2g, where V = V1 L V2. Theorem 2.1 Given a representation of a nite group G, there is a decomposition V = Mr i=1 Vi; and likewise = Mr i=1 i; where i : G ! GL(Vi) and is irreducible, for each i 2 f1; 2; :::; rg. 3 Proof. First, consider a subspace W ( V that is stable and proper. Adding elements to a basis for W to obtain a basis for V , one may obtain a subspace W0 satisfying W L W0 = V . Consider the projection p of V onto W via this basis for V , and de ne the average p0 = 1 jGj X g2G gp 1 g : One notes that because W is stable, one must have p0 mapping V into W. Also, because W is stable, one must have gp 1 g x = x; for every x 2 W. Therefore p0x = x for every x 2 W, and so p0 is a projection of V onto W. Its kernel W0 satis es W M W0 = V: But also, one has hp0 1 h = 1 jGj X g2G h gp 1 g 1 h = 1 jGj X g2G hgp 1 hg = p0 for all h 2 G, and therefore x 2 W0 satis es p0x = 0, and thus p0pgx = 0. Thus pgx 2 W0; and W0 is a complement of W in V that is stable. By the de nition of irreducibility, the result then follows by induction on the dimension of V. For a representation , de ne the character as (g) = Tr( g): A few basic properties of this character function are as follows: 4 Theorem 2.2 (i) (1) = dim( ); (ii) is a class function from G to C; (iii) (g1) = (g). Proof. Properties (i) and (ii) are obvious. Property (iii) follows the observation that (g) = dXim( ) i=1 i where each i is a root of unity, and, in fact, an eigenvalue for g. Since g has nite order, one has that g is conjugate to a matrix with nonzero entries only along the diagonal, with each diagonal entry an eigenvalue for g. But then g1 is conjugate to the inverse of this matrix of eigenvalues. Thus ( g1) = dXim( ) i=1 1 i = dXim( ) i=1 i; with the last equality holding because each i is a root of unity, as G has nite order. As dXim( ) i=1 i = (g); the result follows. The following will now demonstrate various aspects of character theory that will be of use to us. In order to do this, one important de nition must be established. Two representations 1 : G ! GL(V1) and 2 : G ! GL(V2) are called isomorphic if there is a isomorphism of vector spaces f : V1 ! V2 such that f 1;g = 2;g f for all g 2 G. This de nition motivates the following. Theorem 2.3 (Schur's lemma) Suppose that 1 : G ! GL(V1) 5 and 2 : G ! GL(V2) are irreducible representations, and that f : V1 ! V2 is a linear map satisfying f 1;g = 2;g f for all g 2 G. (i) If 1 and 2 are not isomorphic, then f 0; (ii) if V1 = V2 and 1 = 2, then f(v) = v for some 2 C. Proof. For the rst part, note that the kernel of f must be stable, as one must have 0 = f 1;gx = 2;g fx = 2;g0 = 0: As V1 is irreducible, this kernel must be zero. Likewise, the image of f is stable. Therefore f is surjective and injective, hence an isomorphism, which contradicts the fact that 1 and 2 are isomorphic. (ii) As a linear map, f has an eigenvalue, say, f(v) = v for some nonzero element v 2 V = V1 = V2. With f = f 1V as a map from V to V , one has that f 1;g = 2;gf for all g 2 G, and therefore the nonzero kernel of this map must equal all of V , as V is irreducible. Therefore f . If 1 and 2 are two complexvalued functions on G, one de nes their inner product as h 1; 2i = 1 jGj X g2G 1(g) 2(g): The following result is of importance here. Theorem 2.4 (Schur's orthogonality of characters) 6 (i) If 1 and 2 correspond to representations 1 : G ! GL(V1) and 2 : G ! GL(V2), respectively, that are irreducible and not isomorphic, then h 1; 2i = 0; (ii) If 1 and 2 are irreducible and isomorphic, then h 1; 2i = 1. Proof. For (i), one considers a mapping, as in the proof of Theorem 2.1, de ned as f0 = 1 jGj X g2G ( 2;g)1f 1;g where f is a linear map from V1 to V2 as before. One may note that 2;gf0 = f0 1;g for all g 2 G. By Schur's lemma, f0 0. In this case, one considers f to be the linear map which, as a matrix, has an element 1 in a particular entry, and zeroes elsewhere. The inner product of 1 and 2, with 1 and 2 written in matrix form as 1;g = 1 ij(g) i;j and 2;g = 2 ij(g) i;j ; satis es h 1; 2i = X i;j h 1 ii; 2j ji: With f as the linear map mapping the jth basis element for V1 onto the ith basis element for V2 and all else to zero, one must have < 1 ii; 1j j >= 0 for all i and j. Therefore the inner product of 1 and 2 is equal to zero. For (ii), if 1 and 2 are irreducible and isomorphic, then one immediately identi es from the de nitions that 1 2. Thus to consider the inner product of these two characters, one may assume that V = V1 = V2, as has been done in the proof of Schur's lemma. With = 1 2, one has Tr(f0) = 1 jGj X g2G Tr( 1 g f g) = Tr(f) as matrices, and as, by the Schur lemma, f0 for some 2 C, it follows that = 1 dim( )Tr(f). With the above matrix notations, by considering f to be the linear map sending the jth basis element in V1 to the ith in V2, and all else to zero, one 7 must have h ii; jji = 1 dim( ) if i = j and zero otherwise. Therefore h 1; 2i = h 1; 1i = dXim( ) i=1 h ii; iii = dXim( ) i=1 1 dim( ) = 1: This yields immediately the uniqueness, up to isomorphism, of the decomposition of a representation into a direct sum of irreducible representations. In the sequel, it will also be useful to note the following fact. Lemma 2.1 (g) = dim( ) or (g) = dim( ) if and only if it holds respectively that g = I or that g = I, where I denotes the identity matrix. Proof. One may represent (g) as the trace of the diagonal matrix with entries along the diagonal consisting of the eigenvalues of g. The matrix g is conjugate to this diagonal matrix of eigenvalues. Also, because g has nite order, each of its eigenval ues is a root of unity, and thus has modulus one as a complex number. Therefore if (g) = dim( ), then g is conjugate, and hence equal, to the identity matrix, as each eigenvalue of g is equal to 1. If (g) = dim( ), then g is conjugate, and hence equal, to the diagonal matrix I, as each eigenvalue of g is equal to 1. The converse holds trivially. The following result will be necessary to the proof of Brauer's theorem on induction of characters. First, one must de ne the regular representation of G to be the C algebra C[G] which has elements of G as its basis, where the action of g on C[G] is de ned as 8 g X g02G cg0g0 = X g02G cg0gg0: Lemma 2.2 The irreducible characters of a nite group G form an orthonormal basis of the space of class functions of G. Proof. Consider a class function f, an irreducible representation ( ; G; V ), and the linear mapping (f) : V ! V de ned by (f) = X g2G f(g) g: One may notice that 1 g1 (f) g1 = X g2G f(g) 1 g1 g g1 = X g2G f(g) g1 1 gg1 = X g2G f(g1gg1 1 ) g = X g2G f(g) g = (f): Therefore, with denoting the character associated with , (f) is equivalent to a scalar = 1 dim(V ) X g2G f(g)Tr( g) = 1 dim(V ) X g2G f(g) (g) = jGj dim(V ) hf; i: Suppose then that f is a class function orthogonal to , for every character that corresponds to an irreducible representation of G. Considering an arbitrary representation ( ; G; V ) of G, one then has, by Schur's lemma and componentwise additivity of the inner product h ; i in each component, that (f) is identically zero. 9 Supposing now that this arbitrary representation is specially chosen to be the regular representation of G, one has (f) 1 = X g2G f(g)g in C[G]. As (f) is identically zero, one has (f) 1 = 0. Thus f(g) = 0 for all g 2 G, and f is identically zero. Therefore the characters from characters corresponding to irreducible repre sentations of G form an orthonormal basis for the set of class functions on G. If is such a character, then is also such a character, a fact which is immediate from the de nition of the inner product h ; i. Therefore the characters of the irreducible representations of G form an orthonormal basis for G. 2.2 Induction In this section, the notion of an induced representation is introduced. As before, suppose that G is a group. Where necessary a representation : G ! V for a group G will be denoted by ( ; V;G). Consider then H a subgroup of G, with a representation ( ;W;H). The induced representation ( ; IndG H(W);G) of ( ;W;H) is de ned as IndG H(W) = C[G] C[H] W; where elements of H act on W via the representation . The following wellknown result is of importance for the proof of Brauer's theorem on induced characters. It is due to Frobenius. In what follows, R will denote a set of left coset representatives for H in G, and G will denote a nite group. A character of H is viewed in the following theorem as a function on G with (g) = 0 for all g =2 H. Theorem 2.5 If ( ;W;H) has character , and ( ; V;G) is the induced representa 10 tion with character , then one has (g) = X r2R r1gr2H (r1gr) . Proof. Consider the direct sum V = M r2R rW; and some g 2 G. One may note that (g) consists of the sums of traces of those indices r 2 R for which grW = rW: Denoting by Rg the set of indices in R so xed by g, one has (g) = X r2Rg Tr rW( g;r) with here g;r denoting the restriction of g to the subspace rW. In fact, the set Rg consists exactly of those elements r 2 R for which r1gr 2 H, and thus, one may note that r is an isomorphism from W onto rW for every r 2 Rg, and therefore, as g r1gr = g;r r if r1gr 2 H, one has (r1gr) = TrW( r1gr) = Tr rW( g;r); and the result holds. This may be used to de ne for any class function f : H ! C the induced function IndG H(f) = X r2R r1gr2H (r1gr); where as with characters of H, f is viewed as a function extended to G by zero. With this machinery, one is now prepared to prove Brauer's theorem, whose nal steps are put forth in the last section of this chapter. 11 2.3 Brauer's theorem on induction of characters This section establishes Brauer's theorem on induction of characters. Lemma 2.3 Let p be a prime number, G a nite group, and g 2 G. Two elements g1 and g2 exist so that g = g1g2, the order of g1 is a power of p, the order of g2 is relatively prime to p, and g1 and g2 commute. An element of G commutes with g if and only if it commutes with both g1 and g2. Also, g1 and g2 are uniquely determined by the element g Proof. All of these facts follow easily from the following construction. Suppose that jgj = p m, where m and p are relatively prime. Then there exist integers a and b satisfying ap + bm = 1: In this case, one may set g1 = gbm and g2 = gap , and one will evidently have g = g1g2. Evidently, also g1 commutes with g2. The order of g1 is a power of p by its construction, and the order of g2 is relatively prime to p by its construction. That an element of G commutes with g if and only if it commutes with both g1 and g2 follows from the construction of g1 and g2 as powers of g. One must now establish uniqueness of g1 and g2. As p and m are relatively prime, one must have that g1 has order p and g2 has order m. Thus, one must have gm 1 = gm, and gp 2 = gp . Therefore, g1 = gbm 1 = gbm; and likewise g2 = gap 2 = gap : This proves the claim. De ne the function 1G(g) = 1; for all g 2 G: 12 Let the symbol p stand for a prime number dividing the order of G, let S denote the ring Z["], where " is a primitive mth root of unity, and m is a natural number where gm = 1 for all g 2 G. For a 2 G, denote the centralizer of a by CG(a) = fg 2 G j ga = agg: Then, suppose that R is a subring of C containing the integers. Denote by E the set of elementary subgroups of G, i.e., the subgroups of G of the form hai B where a 2 G is an element of order relatively prime to p, and B is a psubgroup of CG(a). Denote by Ch(G) the set of characters of representations of G. De ne the following three sets, ascending in the inclusion ordering: VR(G) = f j = X nite riIndG Ei( i);Ei 2 E; ri 2 R; for all ig; ChR(G) = f j = X nite ri i; i 2 Ch(G); ri 2 R; for all ig; UR(G) = f j jE 2 ChR(E) for each E 2 Eg: One may note that VR(G) is naturally an ideal of UR(G), for with 2 VR(G), 2 UR(G), one has, with = X nite riIndG Ei( i); that = X nite ri IndG Ei( i) = X nite riIndG Ei(( i)jEi); with the denoting multiplication. Of course, ( i)jEi = jEi i 2 Ch(Ei); for both jEi and i are characters of representations, and thus their product is the character of the tensor product of the representations for which jEi and i are characters. The following lemmas are necessary precursors to Brauer's theorem on induced characters. 13 Lemma 2.4 Let E be a supersolvable group. Then E is monomial, i.e., an irreducible representation of E is induced from a representation of degree one for a subgroup of E. Proof. This proof is by induction on the order of E. If jEj = 1, then the lemma is obvious. Also, if E is abelian, then an irreducible representation of E is automati cally of dimension one. Suppose then that jEj > 1, and that E is nonabelian. As E is supersolvable, E admits a tower 1 = E0 E1 E2 En = E where each Ei is a normal subgroup of E, and Ei+1=Ei is a cyclic group of prime order. One may then take the quotient of E with its center Z(E), and may thus obtain such a cyclic tower Z(E) = F0 F1 F2 Fn = E where, again, each Fi is a normal subgroup of E, and quotient groups of successive groups in this chain are cyclic of prime order. One may note that F1 must be an abelian, normal subgroup of E, and may suppose that F1 is not equal to Z(E). Suppose that the character corresponds to the irreducible representation of E. By induction, one may assume that is injective. Then, as F1 is not contained in the center of E, there must be some a 2 F1 so that (a) is not identical to a scalar. If the decomposition of the restriction of to F1 into irreducible representations consisted of a collection of pairwise isomorphic representations, then, as F1 is abelian, one would have a decomposition of as ( ; V; F1) = Mr i=1 ( i; Vi; F1) where Vi has dimension one, for each i 2 f1; 2; :::; rg. 14 Of course, as each representation in the direct sum is isomorphic to every other, one would have in particular for i; j 2 f1; 2; :::; rg some Fij : Vj ! Vi satisfying Fij j;f = i;f Fij ; for each f 2 F1. As each representation in the direct sum is also of one dimension, it must be that j;f = i;f for all i; j 2 f1; 2; :::; rg, for each f 2 F1 and thus f would be a scalar for every f 2 F1. This is a contradiction, and so the restriction of to F1 may not be decomposed in this way. Nonetheless, in restricting to F1, one may write ( ; V; F1) = Ms j=1 Mrj i=1 ( i;j ; Vi;j ; F1) where each collection Bj = f( i;j ; Vi;j ; F1)grj i=1 is a maximal collection of irreducible representations of F1 in its decomposition into irreducible representations. The linear transformation permutes the subspaces Vj = Mrj i=1 Vi;j : Considering some Vj , and the subgroup Hj E where hVj = Vj ; one notes that F1 Hj , and that is induced by the representation jHj when viewed as acting on Vj . Also, one must have Hj 6= E. This establishes the result. Theorem 2.6 A character of a representation of G is a Zlinear combination of characters of representations of G as = X nite IndG Hi( i) where, as before, Hi is a subgroup of G, and i is a character of a representation of dimension one for Hi. 15 Proof. One needs only to show that 1G 2 VZ(G). For, once this has been established, one may note that if 1G 2 VZ(G), then as Ch(G) UZ(G) and VZ(G) is an ideal of UZ(G), then so must Ch(G) VZ(G). Consider a character of a representation of G as a nite Zlinear combination of characters of representations G where each character in the linear combination is induced from a character of a representation of a particular element, say, E, of E. Writing = Xn j=1 njIndG Ej ( j) with Ej 2 E and nj 2 Z for each j 2 f1; 2; :::; ng, and with j = Xlj i=1 i;j the expression of j as a sum of characters from irreducible representations of Ej , for each j 2 f1; 2; :::; ng, it follows that = Xn j=1 njIndG Ej ( j) = Xn j=1 njIndG Ej ( Xlj i=1 i;j) = Xn j=1 Xlj i=1 njIndG Ej ( i;j); where the last equality holds because induction is additive on characters of representa tions, from Frobenius' theorem in Section 2.2. And each i;j is induced by a character of a representation of dimension one on a subgroup of Ej , for each i 2 f1; 2; :::; ljg, for each j 2 f1; 2; :::; ng, because each Ej is nilpotent, and thus supersolvable. Therefore will be the desired Zlinear combination. Noting this, let us now must only establish that 1G 2 VZ(G). Consider then an element E 2 E. Suppose, to begin, that E = hai; 16 and let n = jhaij. Suppose then that ! 2 S is a primitive nth root of unity. The class function de ned as (a) = jhaij and (ai) = 0 for all ai 6= a may, by the above, be written as = X nite ai!i where each !i denotes a character of an irreducible representation of E and each ai denotes a complex number. Here, one may, in particular, take !i to be the character uniquely de ned by !i(a) = !i. In this case, the set of these characters !i constitute all of the characters for irreducible representations of E. So written, one has explicitly that ai = h ; !ii = !i 2 S: Therefore one must have that 2 ChS(E), noting here that this de nition does apply because S = Z["] is a subring of C containing Z. Then, considering a general element of E, one may see that this argument extends to these groups of the form hai B where B need not be trivial, by de ning 0 to be equal to the function constructed immediately above on hai, and extended as a function on E that is constant in its second coordinate. Of course, one may do the same with the characters !i, and this yields a set of irreducible characters of E, denoted by !0i for each i, with, by the above, 0 = X nite ai!0 i: In this fashion one has for an arbitrary element of E a function 2 ChS(E) with (a) = jhaij and (ai) = 0 for all ai 6= a that is constant in its second coordinate as a function of the product hai B. 17 One may write, for a representation ( ; G; V ) induced from ( ;H;W), that (g) = 1 jHj X r2G r1gr2H (r1gr) by the theorem of Frobenius of Section 2.2 noticing now that the sum is over all relevant a set of left coset representatives for H in G. Suppose now that B is a Sylow psubgroup of CG(a). De ne NCG(a)(B) = fr 2 CG(a) j rBr1 = Bg and n = card(fP a pSylow subgroup of CG(a) j b =2 Pg): Note that the induced function IndG E( 0) = X nite aiIndG E(!0 i) satis es IndG E( 0)(ab) = 1 jBj jfr 2 G j r1(ab) = ab0 for some b0 2 Bgj = 1 jBj jfr 2 CG(a)jr1br 2 Bgj = 1 jBj jfr 2 CG(a)jb 2 rBr1gj = n jNCG(a)(B)j jBj [NCG(a)(B) : B] mod p by the rst theorem in this section, and the fact that n = 1 mod p, which holds trivially when b = 1, and in any other case because b 6= 1 permutes by the action of conjugation the Sylow psubgroups of CG(a) which do not contain b in orbits of cardinality equal to some power of p. Of course, p does not divide the index of B in its normalizer in CG(a), as it is a Sylow psubgroup of CG(a), and therefore, there is some integer so that IndG E( 0)(ab) 1 mod p: 18 Then one will have IndG E( 0)(ab) 1 mod p for all b 2 B. One may note that this induced character is zero for every g which is not conjugate to any ab 2 hai B. Also, if g is conjugate to ab for some b 2 B, then one must have IndG E( 0)(g) = IndG E( 0)(ab): Note then that g is conjugate to such an element if and only if the decomposition according to Lemma 2.3 gives that the element g2, of order relatively prime to p and uniquely determined by g, is conjugate to a. One may then consider the conjugacy classes C1; :::; Ck of the elements g2, over all g 2 G. As above, one has for each such conjugacy class a function j in VS(G) where j(g) = 0 if the element g2 in its decomposition does not lie in the conjugacy class Cj , and j(g) 1 mod p if g2 lies in Cj . By taking Xk j=1 j ; one then obtains an element, denoted by p, in VS(g) satisfying (g) 1 mod p for all g 2 G. Suppose then that the order of G is written as p np, where (np; p) = 1. The following analysis will hold for each prime p dividing the order of G. One has p 1 p 1 mod p : 19 Of course, taking an arbitrary element a of G, and considering its conjugacy class, one may notice that the function constructed to lie in VS(hai) induces to a function IndG hai( ) in VS(G) satisfying IndG hai( )(g) = 0 if g is not conjugate to a, and IndG hai( )(g) = jCG(a)j if g lies in the conjugacy class of a. Therefore, a class function whose values are divisible by jGj must lie in VS(G). Thus, in particular, the class function np( p 1 p 1G) must be contained in VS(G). As in the proof that VR(G) is an ideal of UR(G), one must have p 1 p 2 VS(G), and thus np1G 2 VS(G). Noting then that the set of natural numbers np as constructed above, with p ranging over all primes dividing the order of G, has greatest common divisor one, one may nd integers cp satisfying X p cpnp = 1: Therefore 1G = X p cpnp1G 2 VS(G): Writing 1G = X finite ciIndG Ei( i) where i is a character of a representation of Ei and ci 2 S for each i, one may notice that, with ci = P j ci;j"j for integers ci;j , one has 1G = (Xm)1 j=0 "j X i ci;jIndG Ei( i) ! 20 where ( ) denotes Euler's phifunction, as f1; "; :::" (m)1g is a Zbasis for S. Now, one may notice that each of the characters appearing in this sum may be written as a sum of characters of irreducible representations of G, and so one has 1G = X i ci;0IndG Ei( i) 2 VZ(G) as soon as the elements f1; "; :::; " (m)1g are proven to be linearly independent over ChZ(G). For that, a relation (Xm)1 j=0 "j X i ci;j i ! = 0 with the sum over the index i sweeps across characters i of irreducible representations of G requires X i 0 @ (Xm)1 j=1 ci;j"j 1 A i = 0: But as these characters i are orthonormal with respect to the inner product h ; i, so must they be linearly independent, and thus (Xm)1 j=1 ci;j"j = 0 for each i. Therefore ci;j = 0 for each i and each j 2 f1; 2; :::; (m) 1g, and the elements f1; "; :::; " (m)1g are linearly independent over ChZ(G). This establishes Brauer's theorem. This concludes the chapter on representation theory. The next chapter introduces Artin's Lfunction and addresses its basic properties. 21 CHAPTER 3 Artin's Lfunction 3.1 Artin's Lfunction This chapter introduces Artin's Lfunction. One considers now a nite extension K of the rational numbers Q, called an algebraic number eld, and a nite extension F of K. In this work, G(FjK) will denote the Galois group of F over K. Artin's Lfunction hinges upon a representation : G(FjK) ! GL(V ) where, as before, V denotes a vector space over the complex numbers C. An element x 2 K will be called integral over Z if it satis es an equation xn + an1xn1 + + a0 = 0 where ai 2 Z for i 2 f1; 2; :::; n 1g, where here n is understood to be at least one. The set of such elements in K forms a ring, called the integers of K, and is denoted by oK. One may note the following properties of oK: (i) oK is an integral domain; (ii) oK is noetherian as a ring; (iii) an element in K that is integral over oK must be contained in oK; (iv) every nonzero prime ideal in oK is maximal. A ring possessing these four properties is called a Dedekind domain. One then de nes a fractional ideal of K to be an oKsubmodule a of K where ca oK for some c 2 oK. The following result is necessary for the construction of Artin's Lfunction. Theorem 3.1 The nonzero fractional ideals form a group under multiplication of ideals, equal to the free abelian group on the set of prime ideals of oK. 22 Proof. Consider a nonzero ideal a of oK. Since oK is noetherian, some ideal a 6= 0 is maximal with respect to the property that there is not product of prime ideals p1p2 pr a. In particular, this ideal a cannot be a prime ideal, so that there will be b1; b2 2 oK where b1b2 2 a, but neither of b1 and b2 lies in a. Then, if a1 = (a; b1), and a2 = (a; b2), one must have a1a2 a, and each of a1 and a2 is not equal to a. And a is maximal for the property as above, so that there is a product of prime ideals contained in each of a1 and a2. The product of these products of prime ideals is contained in the product of a1 and a2, hence in a, which is a contradiction. Hence every nonzero ideal of a contains a product of prime ideals. Consider then a maximal ideal p, and de ne p1 = fx 2 Kjxp oKg: This will play the role of the multiplicative inverse of p in the group of fractional ideals. One evidently has that p1 oK. Take a nonzero element p 2 p and consider the smallest natural number r for which there exists a product of prime ideals satisfying p1p2 pr (p) p: Such a product exists as above. In this case, p is, of course, prime, and so one of the ideals pi is contained in p, and because pi is maximal, it must equal p. But also, the product of primes contained in (p) as above, with pi removed, is no longer contained in (p), by the minimality of r as selected. Thus there is an element b 2 Y j6=i pj where b =2 (p). However, one must have bp (a), and thus ba1p oK, whence ba1 2 p1. Also, ba1 =2 oK. Therefore in this case p1 6= oK, and p pp1 oK: The ideal p is maximal, so one must have that one of these inclusions must be equality. If cannot, however, be the rst, because then p1 would contain only elements integral 23 over oK, as p is nitely generated, and so would equal oK. Thus the latter inclusion is equality, and p has an inverse. Then one may notice that a nonzero ideal of oK is invertible by a fractional ideal, for if this were not true for every such nonzero ideal of oK, then one would have a maximal noninvertible ideal a 6= 0, because oK is noetherian. By the above analysis, this a could not itself be maximal, and thus one obtains a ap1 aa1 oK: One cannot have the rst inclusion as equality, or else one would again have that p1 = oK, this time because a is nitely generated. But then ap1 would have an inverse by maximality of a in this ordering, and this inverse may be multiplied by p1 to obtain an inverse for a. In this fashion every nonzero ideal of oK has an inverse, and, in fact, this inverse must be a fractional ideal. Considering then a nonzero fractional ideal a, one nds c 2 oK so that ca oK, and so this ideal ca has an inverse b that must be a fractional ideal. Therefore cab = oK, and, as one may easily show, cb = fx 2 K j xa oKg; and so this fractional ideal forms the inverse for a in what is now the group of fractional ideals of K. To establish that this group forms a unique factorization domain, one may notice that if there is a nonzero ideal that is not equal to a product of prime ideals, then there is some maximal ideal a with respect to this property. This ideal a cannot be prime, and so with a p for some prime ideal p, one must have a ( ap1 oK as before. But then by maximality of a according to this ordering, the ideal ap1 must have a prime factorization, which may be multiplied by p to obtain a prime 24 factorization for a. One may then consider two fractional ideals a and b, and say that a divides b if and only if a b. By considering two prime factorizations p1p2 pr = q1q2 qs of an ideal in oK, one notices easily that by maximality of each of the prime ideals in each product, and the fact that each pi is contained in a particular qj , and hence equal to it, one must have r = s and uniqueness of the factorization. Then one may consider fractional ideals a in general, and with c 2 oK satisfying ca oK, one will have with (c) = q1q2 qs and ca = p1p2 pr that q1q2 qsa = (c)a = ca = p1p2 pr; whence a = p1p2 pr q1q2 qs : This establishes the unique factorization, and the proof is complete. With this in mind, one may consider the prime ideals of oK to be the nite primes, for every such prime p de nes a nonarchimedean valuation vp(x) = ep on oK corresponding to the prime factorization (x) = Y p pep : Later, one will deal with in nite primes, which are created by the embeddings of K into the complex numbers. For now, use of the nite primes will su ce. One may observe that for the extension F of K, the analogous ring oF enjoys the above properties mentioned for oK, and thus is also a unique factorization domain. There is a fundamental identity that will be used later, and it is stated here, for its clari cation of a relation between the prime ideals of oK and oF . Before proceeding 25 further, one may notice that the prime ideal p of oK extends via multiplication by oF to an ideal of oF , and therefore, poF has a prime factorization, poF = Pe1 1 Pe2 2 Per r : For any one of the primes Pi appearing in this factorization, one writes Pijp, and calls Pi a prime of oF lying above p. One calls the eld oK=p the residue class eld of p, and similarly in oF for its prime ideals. For the above factorization of p in oF , one calls ei the rami cation index of Pi over p, and de nes fi = [oF =Pi : oK=p]; calling it the inertia degree of Pi over p. For the following theorem, the notion of localization will be used. One may consider for oK what is called its localization at p, equal to the ring na b j a 2 oK; b 2 oKnp o : This will be denoted by oK;p. The ring oF;p will be de ned similarly as na b j a 2 oF ; b 2 oKnp o : The ring oK;p is a principal ideal domain with a unique maximal ideal poK;p and admits a discrete valuation equal to the exponent n 2 Z in the prime factorization x = u n of one of its elements, where is a prime element and u is a unit, where each element in oK;p has such a factorization because the unique maximal ideal is principal. Theorem 3.2 (Fundamental identity) One has the relation [F : K] = Xr i=1 eifi: 26 Proof. The inclusion oK ! oK;p induces an isomorphism oK=poK = oK;p=poK;p; and likewise that oF =poF = oF;p=poF;p: Thus to prove the result, one notes that oF;p=poF;p is a vector space of dimension [F : K] over oF;p=poF;p, and that the factorization of poF;p in oF;p yields poF;p = Pe1;p 1;p Pe2;p 2;p Per;p r;p with fi;p = [oF;p=P1;p : oK;p=poK;p]: One also notices that oF;p is the set of elements in F that are integral over oK;p. It is easy to see that ei;p = ei, and that fi;p = fi for each i 2 f1; 2; :::; rg. Also, oF;p is a Dedekind domain with nitely many primes, and, in particular, its primes are exactly those P1;p;P2;p; :::;Pr;p lying above poK;p. This means that oF;p is a principal ideal domain. For an ideal ap of oF;p factors uniquely as ap = Yr i=1 Pe1 1;pPe2 2;p Per r;p where e1; e2; :::; er are integers, each at least equal to zero. By the Chinese remain der theorem, one may select an element x 2 oF;p so that 27 x = ei i mod Pei+1 i;p where i is an element of Pi;p not contained in P2 i;p. In this way the factorization for the principal ideal (x) is (x) = Yr i=1 Pe1 1;pPe2 2;p Per r;p; and therefore (x) = ap. Thus oF;p is a principal ideal domain. One then notes that the primes P1;p;P2;p; :::;Pr;p are pairwise relatively prime, and thus the Chinese remainder theorem gives an isomorphism oF;p=poF;p ! Yr i=1 oF;p=Pei;p i;p : In view of this isomorphism, and the natural isomorphism oF;p=Pi;p ! Pj i;p=Pj+1 i;p for each i 2 f1; 2; :::; rg, in each case given by multiplication by an element of oF;p by i j , where i is a generator of the principal ideal Pi;p, the identity easily follows. With this machinery in mind, one now considers a nite Galois extension F of K, and de nes for a prime P of oF the decomposition group of P, GP = f 2 G(FjK) j P = Pg; with G intended to denote G(FjK). The homomorphism ! given by x = x mod P for each x 2 oF yields for each 2 GP an associated : oF :=P ! oF =P; and thereby a map from GP to the Galois group of oF =P over oK=P, which has kernel called the inertia group of P, and is denoted by IG;P. The following theorem is of great importance. 28 Theorem 3.3 For any two prime ideals P and P0 of oF lying above p in oK, there exists some 2 G(FjK) so that P = P0, i.e., G(FjK) acts transitively on the set of primes lying above p. Proof. . If the theorem were not true, then the Chinese remainder theorem would yields some x 2 oF where x = 0 mod P0 and x = 1 mod P for all 2 G(FjK). Then one would have NFjK(x) = Y 2G(FjK) x 2 P0 \ oK = p; and yet, as x =2 P for any 2 G(FjK), one must have x =2 P for any 2 G(FjK), whence Y 2G(FjK) x =2 P \ oK = p; which is a contradiction. The following lemma is of importance in constructing Artin's Lfunction. Denote by FP the xed eld of the decomposition group GP. In general, for elds F F0 K with prime ideals PjP0jp where P is a prime in oF , P a prime in oF0 , and p is a prime in oK, one may de ne the inertia degrees f = [oF =P : oK=p], f0 = [oF =P : oF0=P0], and f00 = [oF0=P0 : oK=p]. It follows that f = f0f00. With rami cation indices e, e0, and e00 de ned similarly, one also has e = e0e00. Lemma 3.1 Let P be a prime of oF lying above the prime p of oK. One has oFP=foFP \ Pg = oK=p. Proof. With poF = Yr i=1 Pi ei one has e1 = e2 = = er = e and f1 = f2 = = fr = f from the transitivity of G(FjK) over the primes lying above p, by the previous theorem. In this case, the 29 fundamental identity [F : K] = Pr i=1 eifi reduces to [F : K] = efr where r = (G : GP). Therefore [F : FP] = ef. Considering a particular prime P lying above p, one considers f, f0, and f00 as above, and likewise for e, e0, and e00, taking in this case F0 = FP. Thus one has f00 = e00 = 1, and the result follows. The previous result motivates the following theorem. Theorem 3.4 The map GP ! G(oF =PjoK=p) de ned by mapping 2 GP to 2 G(oF =PjoK=p) satisfying x = x mod P; for every x 2 oK, is surjective. Proof. By the previous lemma, one may assume that K = FP, so that G(FjK) = GP. Consider then a primitive element of oF =P as an extension of oK=p, i.e., an element x satisfying oF =P = foK=pg(x); which exists because the extension oF =P over oK=p is separable. Consider then the minimal polynomial g of x over oK=p, and a lift x of x in oF , and suppose that 2 G(oF =PjoK=p): Then (x) = x0, where x0 is also a root of the polynomial g. Also, the minimal polynomial f of x over K takes all of its roots in oF because F is normal over K, and therefore, f 2 oK[X]. Thus one may consider f, the reduction of f modulo P, and that g must divide f. In this way g must have roots corresponding to the reductions of roots of f taken modulo P, and so x0 also has a lift x0 in oF that is a root of f. In particular, there is a 2 GP so that (x) = x0, and thus (x) = x0, as desired. 30 This proves the surjectivity, because an element in G(oF =PjoK=p) is completely determined by how it acts on the primitive element x. In the cases of these nite primes, one will have that the Galois group of oF =P over oK=p is cyclic, and one may choose as a generator of GP=IG;P an element mapping to what is called the Frobenius element in G(oF =PjoK=p), which is the automorphism given by x ! xq; where q = joK=pj. In this setting, this generator of GP=IG;P is itself called the Frobenius for P on account of the isomorphism GP=IG;P ! G(oF =PjoK=p): Let then ( ; V;G(FjK)) be a representation, and let V IG;P be the subspace of V held xed by IG;P. This V IG;P is called the module of invariants for IG;P, and one may also notice that the Frobenius element for P, and hence all of GP=IG;P, must map V IG;P to itself. Denoting the Frobenius element for P by G;P, one may consider the expression det(I ( G;P)N(p)s j V IG;P): This is intended to denote the determinant of the expression I ( G;P)N(p)s as a matrix acting on V IG;P. Here, N(p) = p[oK=p:Z=pZ] where p lies above p 2 Z. This determinant does not depend upon the prime P chosen, because any two primes P, P0 lying above p have GP and GP0 , IG;P and IG;P0 , and the Frobenius elements in each quotient group GP=IG;P and GP0=IG;P0 as simultaneous conjugates. Also, the above determinant must depend only upon the character of the representation , and thus, so does the product Y p prime p2oK 1 det(I ( G;P)N(p)s j V IG;P) : 31 One de nes this to be the Artin Lfunction, and denotes it by L(FjK; ; s) for a character of a representation of G(FjK). A prime is called unrami ed if IG;P = f1g. One may note that there are nitely many primes that ramify in F, so that all but nitely many p in K have V = V IG;P, so that the expression det(I ( G;P)N(p)sj V IG;P) is a polynomial of degree dim( ) in qs. Theorem 3.5 The Artin Lfunction L(FjK; ; s) converges in the half plane Re(s) > 1. Proof. One may consider the factorization det(I ( G;P)N(p)s j V IG;P) = Ydp i=1 (1 "i;pN(p)s) for a prime p of oK where each "i;p is a root of unity, and dp dim( ). Taking formally the logarithm of the Artin Lfunction L(FjK; ; s) then yields log L(FjK; ; s) = X p X1 m=1 Xdp i=1 "i;p mN(p)ms : If Re(s) > 1, one has X p X1 m=1 Xdp i=1 "i;p mN(p)ms X p X1 m=1 Xdp i=1 "i;p mN(p)ms = X p dp X1 m=1 1 mN(p)mRe(s) dim( ) X p X1 m=1 1 mN(p)mRe(s) [K : Q]dim( ) X p2Q p prime X1 m=1 1 mpmRe(s) = [K : Q]dim( ) log (Re(s)) 32 where denotes Riemann's zeta function, which converges in the halfplane Re(s) > 1. . The following section in this chapter outlines some basic properties of Artin L functions. 3.2 The basic properties of this function There will be a few properties of this function that have use in this work. First, one notes the following theorem. Theorem 3.6 (i) If each of and 0 is a character of a representation of G(FjK), then one has L(FjK; + 0; s) = L(FjK; ; s)L(FjK; 0; s); (ii) if F0 is a Galois extension of K containing F, and is a character of a repre sentation of G(FjK), then with the representation yielding acting on G(F0jK) via the canonical quotient map G(F0jK) ! G(FjK); one has L(F0jK; ; s) = L(FjK; ; s); (iii) if F0 is a sub eld of F containing K, and is a character of a representation of G(FjF0), then L(FjF0; ; s) = L(FjK; IndG H( ); s); where G = G(FjK), and H = G(FjF0). 33 Proof. (i) is trivial. For (ii), one may note that the canonical map G(F0jK) ! G(FjK) yields a surjection G(F0jK)P0=IG(F0jK);P0 ! GP=IG;P where P0jPjp, with P0 a prime of oF0 and P a prime of oF , which maps the Frobenius of P0 to the Frobenius of P. This makes clear (ii). (iii) Suppose that p is a prime ideal of K, and that q1; q2; :::; qr are the prime ideals of oF0 lying above p. Choose then for each i 2 f1; 2; :::; rg a prime ideal Pi of oF lying above qi. One has the equalities GPi \ H = HPi and IG;Pi \ H = IH;Pi : One has N(qi) = N(p)fi ; where fi = jGPi : HPiIG;Pi j: By the previous theorem, one may choose an element i contained in G(FjK) satisfy ing iPi = P1. Then one will have GPi = 1 i GP1 i; and also that IG;Pi = 1 i IG;Pi i. Considering then an element 1 2 GP1 that is mapping to the Frobenius G;P1 2 GP1=IG;P1 , one will also have that i = 1 i 1 i is similarly mapped to the Frobenius G;Pi . Also, the image of fi i in HPi=IH;Pi is the Frobenius H;Pi . Considering then : H ! GL(W) a representation of H yielding as its character, and ( ; G; V ) denoting the induced representation, it will su ce to show that det(I ( 1)t j V IG;P1 ) = Yr i=1 det(I ( fi i )tfi j WIH;Pi ): 34 Henceforth in this proof, the notation ( ) will be written simply as , and likewise for . For each i 2 f1; 2; :::; rg, conjugation by i yields det(I fi i tfi j WIH;Pi ) = det(I fi 1 tfi j iWIG;P1 \ iH 1 i ); where also fi = jGP1 : (GP1 \ iH 1 i )IG;P1 j. For each i 2 f1; 2; :::; rg, one then selects a system of representatives f i;jg of GP1 mod GP1 \ iH 1 i . Then f i;j ig is a system of representatives on the left of G mod H, and one obtains a decomposition for the vector space V corresponding to the induced representation ( ; V;G) of ( ;W;H) as V = M i;j i;j iW: Then by letting Vi = M j i;j iW; one obtains a decomposition V = L i Vi of V as a GP1module. Therefore one must have that det(1 1t j V IG;P1 ) = Yr i=1 det(I 1t j V IG;P1 i ): Now it su ces to show that det(1 1t j V IG;P1 i ) = det(I fi 1 tfi j iWIG;P1 \ iH 1 i ) for each i 2 f1; 2; :::; rg. One has, of course, that ( ; Vi;GP1) is the representation induced from ( i; iW;GP1 \ iH 1 i ). But also, one may notice that V G;IP1 i = Ind GP1 =IG;P1 fGP1 \ iH 1 i g=fIG;P1 \ iH 1 i g ( iWIG;P1 \ iH 1 i ): Taking then a basis fw1; :::;wdg for iWIG;P1 \ iH 1 i and noting the decomposition V IG;P1 i = Mfi1 l=0 l 1 iWIG;P1 \ iH 1 i 35 yields the matrix B = 0 BBBBBBB@ 0 Id d 0 0 0 Id d A 0 0 1 CCCCCCCA for 1 acting on V G;IP1 i , where Id d denotes the d d identity matrix and A denotes the matrix for fi 1 acting on iWIG;P1 \ iH 1 i . Thus, one must have det(I 1t j V IG;P1 Pi ) = det(I fi 1 tfi j iWIG;P1 \ iH 1 i ); a fact which may be seen by multiplication of the rst column of Ifid fid Bt by t and subtraction from its second column, multiplication of the second column of the resulting matrix by t and subtraction from its third column, and so forth. This proves the claim. 36 CHAPTER 4 Class eld theory 4.1 Id eles Class eld theory provides an essential link between nite Galois extensions of an algebraic number eld K and Lfunctions that ultimately leads to the functional equation for Artin's Lfunction, which is addressed in chapter seven of this work. In order to discuss class eld theory, some de nitions are in order. For this, one will deal with the nite primes of an algebraic number eld K, given by p as before, but introduces now the notion of an in nite prime. In this cases, each in nite prime will be given by an embedding : K ! C; with the only restriction on this being that two such embeddings which are complex conjugates of each other are associated with the same prime. The notation p  1 will be used to note that one is dealing with nite primes, and pj1 will imply an in nite prime, given by an embedding as above. Together, these in nite and nite primes comprise what is called the primes of K. Each of the primes determines a valuation. In the case of a nite prime, the valuation, which is nonarchimedean, is given in accordance with that assigned via the localization of K for the prime p. In the case of an in nite prime p, the valuation of K is de ned to be jxjp = j xj where j j denotes the modulus in C, and the embedding yields the in nite prime 37 p, where one may note that taking the complex conjugate of does not alter this valuation, and thus one has motivation for associating complex conjugate embeddings of K with the same in nite prime. Denote the completion of K with respect to the valuation given by the prime p as Kp. One de nes the id ele group IK to be the set of elements ( p), with p ranging across all primes of K, in nite and nite, where p 2 K p is a unit in the ring of integers op of Kp of K with respect to the valuation given by the prime p, for almost all primes p. One equips this product space IK = Y p op with a topology generated by sets of the form Y p2S Wp Y p2S Up where S denotes a nite set of primes containing the in nite primes, and Wp denotes a neighborhood of 1 2 Kp in the topology corresponding to the valuation associated with the prime p. Considering then a nite Galois extension F of K, one may then de ne Fp = Y Pjp FP a norm NFpjKp : F p ! K p for ( P)Pjp by NFpjKp(( P)Pjp) = Y Pjp det( P); with each P here viewed as an automorphism from FP to FP over Kp, and the determinant is taken according to the matrix of this automorphism of FP when viewed as a vector space over Kp. In this way one obtains what is called a global norm NIF jIK : IF ! IK 38 de ned for = ( P)P 2 IF by (NIF jIK( ))p = NFpjKp(( P)Pjp); for each prime p of K. One now de nes for a number eld K the group CK = IK=K ; known as the id ele class group, where each element x of K is viewed as diagonally embedded into IK by de ning x = (xp)p 2 IK to have xp = x for all primes p of K. This is possible because the decomposition of the principal ideal (x) = Y p pep into a product of prime ideals shows that x is a unit in op for almost all primes p. One may then notice that the norm as de ned above yields for x 2 F that (NIF jIK(x))p = NLjK(x); a fact which follows from the canonical isomorphism F K Kp = Y Pjp FPjp; and therefore the norm NIF jIK de nes a homomorphism CF ! CK: For a group G, Suppose now that G0 denotes the commutator subgroup of G. The following theorem is the main theorem of class eld theory, and is known as Artin reciprocity. Theorem 4.1 There is an isomorphism A : CK=NIF jIKCF ! G=G0; 39 and the norm map NIF jIK yields a onetoone correspondence between nite Galois extensions F of K with abelian Galois group over K, and the subgroups of nite index in CK that are closed in the quotient topology induced by the canonical topology on IK. Denoting by NF the group NIF jIKCF , one will then have the following facts for two such extensions F1 and F2 of K: (i) F1 F2 if and only if NF1 NF2 ; (ii) NF1F2 = NF1 \ NF2 ; and (iii) NF1\F2 = NF1NF2 . In particular, this correspondence is explicitly given by the association F $ NF ; and therefore, the eld F corresponding to the closed subgroup N of CK of nite index must therefore have N = NF , and will thus satisfy CK=N = G(FjK): Proof. Construction of the map A is given here to provide context; for the full proof of the theorem, the reader is referred to [Neu, VI]. First, there is a local reciprocity map, which is de ned for nite primes, and also for in nite primes. For a nite prime p and an element 2 G, one chooses an extension ~ of to the maximal unrami ed extension of FP so that, when restricted to the maximal unrami ed extension ~K p of Kp, it is a natural number power of the map of G( ~K pjKp) de ned uniquely as (a) = aq mod ~p for all a in the valuation ring of ~K p, where ~p denotes the maximal ideal of this valuation ring. Then, one considers the xed eld of this extension ~ , and a prime of . One then constructs the map rFPjKp : G(FPjKp) ! K p 40 de ned by rFPjKp( ) = N jKp( ) mod NFPjKpF P: The requisite map for the in nite primes is de ned to be trivial for any such prime corresponding to a real embedding of K. For a nonreal embedding, one de nes the reciprocity map rFPjKp via the natural isomorphism G(CjR) = R =NCjR(C ) obtained by identifying each of these groups with Z=2Z. Local class eld theory gives that the map rFPjKp induces, for any prime p of K, an isomorphism G(FPjKp)=G(FPjKp)0 = K p=NFPjKpFP: It is then established via global class eld theory that the map A(( p)) = Y p r1 FPjKp ( p) for an element ( p) 2 IK is surjective onto G(FjK), trivial on K , and yields the desired isomorphism. The local maps constructed in the previous proof will be of some use in a later chapter, which will yield the functional equation for the Artin Lfunction. That this theory gives information relating to Lfunctions lies in construction of what is called the Artin symbol, which is introduced in the following section of this chapter. 4.2 The Artin symbol The following theorem has been included for its centrality to the results that are to come in this work. One may notice that the decomposition group GP de ned in the previous chapter applied to nite primes of a nite Galois extension F of K may be easily extended to the case of a valuation w of F extending a particular valuation v of 41 K, once one notes that each such nite prime induces a nonarchimedean valuation on F. In this case, the decomposition group is de ned so that it agrees on the valuations created by nite primes with the previous de nition of decomposition group, and thus there is no inconsistency in writing that the decomposition group for an extension w is de ned to be Gw = f 2 G(FjK) j w = wg: Here, Fw will denote the completion of F with respect to w, and Kv the completion of K with respect to v. One calls two valuations on a eld equivalent if they induce the same topology. Theorem 4.2 Suppose that K is a complete eld with respect to a valuation j j, and that V is an ndimensional normed vector space over K. Then any two norms on V are equivalent. Proof. It su ces to show that, for a given norm j j on V , there exist constants ; 0 > 0 satisfying kxk jxj 0kxk for all x 2 V , where here, with x written in terms of a basis fv1; :::; vng of V as x = x1v1 + xnvn, one has kxk = maxfjx1j; :::; jxnjg: One nds 0 = jv1j + + jvnj as adequate, and is found inductively. For n = 1, a possible choice of is jv1j. Suppose then that the theorem is proven for all such vector spaces of dimension less than n. De ne Vi = Kv1 + + Kvi1 + Kvi+1 + Kvn: 42 Then Vi has the norm j j as induced from V , and this norm is, in particular, equivalent to the maximum norm k k on Vi. Therefore Vi is complete with respect to j j, and thus is closed in V according to this norm. Therefore, the set Vi + vi is also closed. Noticing that 0 =2 [ni =1 Vi + vi and identifying a neighborhood of radius > 0 around zero disjoint from this set, one nds with x = x1v1 + xnvn 6= 0 that x max jxij = x1 max jxij v1 + + vr + + xn max jxij vn ; and thus jxj kxk: This proves the claim. Theorem 4.3 The elements Gw are exactly those which extend uniquely to elements in the Galois group of Fw over Kv, i.e., restriction from Fw to F induces an isomor phism Gal(FwjKv) = Gw. Proof. Since Kv is complete and Fw is a nite extension of Kv, the previous theorem gives that the valuation w on Fw is the unique extension of v from Kv to Fw. Therefore any element 2 G(FwjKv) must lie in Gw when restricted to F. Conversely, an element 2 Gw is, by de nition, continuous with respect to the valuation w. But an arbitrary element 2 G(FjK) continuous with respect to this valuation yields that jxjw < 1 implies jxjw < 1, whence w and w must be equivalent valuations, and therefore, as they must agree on K, are the same, so that 2 Gw. Thus the elements of Gw are the elements of G(FjK) that are continuous with respect to the valuation w, therefore each element of Gw must extend to a continuous automorphism of Fw over Kv via the map fxngn2f1;2;3;:::g = lim n!1 xn for a Cauchy sequence fxngn2f1;2;3;:::g F, as F is dense in Fw. This proves the claim. 43 With this in mind, one may view the group G(FwjKv), for each such extension w of a valuation v as addressed in the theorem, as a subgroup of G(FjK), and one may do so for each valuation v of K. For what follows in this section, one will restrict attention to the nite primes of p. Within this set, one will restrict attention to what are called the unrami ed nite primes. Therefore, in considering the decomposition group G(FPjKp) G(FjK), one may note that this case allows for the Frobenius of P to lie in G(FjK). Assuming that F has abelian Galois group over K, one may then de ne the Frobe nius automorphism p to be the generator of G(FPjKp) associated via isomorphism with G(oF =PjoK=p), as in the previous chapter, with the map x ! xq; and, in fact, this will be the generator for the decomposition group of any P lying above p. A result in class eld theory gives for a prime element p of Kp that rFPjKp( p) = p mod NFPjKpF P; and one de nes p = FjK p : The Artin symbol is de ned on the set of fractional ideals of K which have prime factorization consisting only of unrami ed primes as FjK a = Y p FjK p vp where a = Y p pvp is the prime factorization of the ideal a. Suppose then that m is an ideal of oK whose prime factorization contains every prime ideal that is rami ed. Then one may de ne Jm K to be the group of fractional 44 ideals of K relatively prime to m, and Pm K to be the subset of Jm K consisting of those principal ideals generated by an element x 2 K so that x > 0 for every embedding : K ! R: De ne ClmK = Jm K=Pm K. Also, de ne for any prime p the ring U(0) p = Up, and likewise for vp > 0 U(vp) p = 8>>>>< >>>>: 1 + pvp if p  1; R + if p is real; C if p is complex: 9>>>>= >>>>; One then de nes Im K = Q p Uvp p , and Cm K = ImK =K : With these notions, one notes that a map ( ) may be obtained, induced from sending the id ele = ( p)p to the ideal ( ) = Y p1 pvp( p); where here vp denotes the valuation corresponding to the nite prime p, which yields an isomorphism CK=Cm K = ClmK; proven in this section as a consequence of the following theorem, which relates equiv alent valuations. Note that the valuation on a eld K satisfying jxj = 1 for all x 6= 0 is excluded. Theorem 4.4 (Weak approximation theorem) Consider a eld K, and a set of pair wise inequivalent valuations j j1, j j2,...,j jn on K. Then one has that (i) for a1; a2; :::; an 2 K, and every " > 0, where exists an x 2 K satisfying jx aiji < " for each i 2 f1; 2; :::; ng, and that 45 (ii) for every " > 0 and k 2 f1; 2; 3; :::g, there exists an x 2 K satisfying jxk 1ji < "; for each i 2 f1; 2; :::; ng. Proof. First, one may notice that two valuations j j1 and j j2 are equivalent if and only if jxj1 < 1 implies that jxj2 < 1. This may be seen in the following way. For j j1 = j js 2 with s > 0 implies that j j1 and j j2 are equivalent. Conversely, if two valuations are equivalent, then one must have that jxj1 < 1 implies that jxj2 < 1. Considering then a xed element y 2 K with jyj1 > 1. Considering then x 2 K with x 6= 0, one must have that jxj1 = jyj 1 for some 2 R. Then, supposing that fml nl gl2f1;2;3;:::g is a sequence of rational numbers, with nl > 0 for each l 2 f1; 2; 3; :::g, converging to from above, one must have that jxj1 = jyj 1 < jyj ml nl 1 for each l 2 f1; 2; 3; :::g, whence by the equivalence of j j1 and j j2, it follows that xnl yml 2 < 1; whereby jxj2 jyj ml nl 2 for each l 2 f1; 2; 3; :::g, and thus jxj2 jyj 2 . Considering another such sequence of rational numbers approaching from below yields that jxj2 jyj 2 , and hence that jxj2 = jyj 2 : Therefore one may de ne ln jxj1 ln jxj2 = s; and note that ln jxj1 ln jxj2 = ln jyj1 ln jyj2 ; 46 and therefore that jxj1 = jxjs 2 with s so de ned. Of course, one must have that jyj2 > 1 as jyj1 > 1 and the valuations j j1 and j j2 are equivalent, and therefore that s > 0. Returning now to the pairwise inequivalent valuations j j1; j j2; :::; j jn, one has that because j j1 and j jn are inequivalent, there must be, by the above argument, some 2 K satisfying j j1 < 1 and j jn 1. Likewise, there is some 2 K satisfying j jn < 1 and j j1 1. Therefore the elements y = satis es jyj1 > 1 and jyjn < 1. One now shows inductively that there exists z 2 K satisfying jzj1 > 1 and jzjj < 1 for each j 2 f2; :::; ng. Notice that this has been proven in the rst case n = 2. Therefore, taking an element z 2 K satisfying jzj1 > 1 and jzjj < 1 for each j 2 f2; :::; n1g, one may consider the following cases. First, if jzjn 1, then, with y as before, taking zmy for m 2 f1; 2; 3; :::g chosen to be su ciently large will yield the desired element. If, on the other hand, jzjn > 1, then one must consider the quantity tm = zm 1 + zm and may note that this converges to 1 with respect to j j1 and j jn, and to zero with respect to j j2; :::; j jn. Therefore taking tmy for m 2 f1; 2; 3; :::g su ciently large will yield the desired element. Then, considering the element as desired, and calling it z, one notes that the quantity zm 1 + zm is again of use, and yields a sequence as m ranges over values in f1; 2; 3; :::g that converges to 1 with respect to j j1, and to zero with respect to j j2; :::; j jn. Thus for every i one nds in this fashion an element zi close to 1 with respect to j ji and close to zero with respect to j jj for j 6= i. considering then the element x = a1z1 + + anzn 47 yields (i). For (ii), one may set ai = 1 for each i 2 f1; 2; :::; ng, and note that the element x = z1 + + zn satis es jxk 1ji < ", so long as zi is chosen to be su ciently close to 1 with respect to j ji and to zero with respect to j jj for j 6= i, for each i 2 f1; 2; :::; ng. As promised, one may now establish the following. Theorem 4.5 There is an isomorphism CK=Cm K = ClmK induced by the map sending the id ele = ( p)p to the ideal ( ) = Y p1 pvp( p): Proof. With m decomposed as a product of prime ideals m = Y p pvp as before, one de nes the set I(m) K = f = ( p)p 2 IKj p 2 U(np) p for pjm or pj1g: One has that IK = I(m) K K , as every 2 IK has by the previous theorem that some a 2 K exists satisfying pa 1 mod pnp for pjm, and pa > 0 for every in nite prime corresponding to a real embedding of K into C. Therefore one has the containment = ( pa)p 2 I(m) K ; and that 2 I(m) K K . One notices now that the set I(m) K \ K consists precisely of those elements in K generating principal ideals in the group Pm, and thus that one has a map from I(m) K =(I(m) K \ K ) ! Jm K=Pm K de ned by = ( p)p ( ) ! Y p1 pvp( p); 48 which is clearly surjective. Noting that one has CK = I(m) K K =K = I(m) K =(I(m) K \ K ); one obtains an induced homomorphism from CK to Jm K=Pm K. Under this mapping, as ( ) = 1 for each 2 Im K, the group Cm K = ImK =K is contained in the kernel of the homomorphism induced from CK. Conversely, suppose that the class [ ] in CK corresponding to a representative 2 I(m) K is contained in the kernel of this map. In that case there is some a 2 I(m) K \K with (a) 2 Pm K so that (a) = ( ). But then the id ele ( p)p = = a1 satis es p 2 Up for all nite primes p not dividing m, p 2 U(np) p for all nite primes p diving m, and likewise for all in nite primes. Therefore, in fact, 2 Im K, so that [ ] = [ ] 2 Im KK =K = Cm K: This establishes the result. One may verify, then, the following commutative diagram, with each row exact. 1 ! NIF jIKCF id ! CK A( ) ! G(FjK) ! 1 ( ) ??y ( ) ??y id ??y 1 ! (NIF jIKCF ) id ! ClmK ( FjK ) ! G(FjK) ! 1: This provides the machinery su cient to begin a study of Lfunctions. 49 CHAPTER 5 The functional equation This chapter presents results that, together, yield the functional equation for the Artin Lfunction. 5.1 Hecke's Lfunction This section provides the functional equation for the Hecke Lfunction. Formally, a Hecke Lfunction is given by L ( ; s) = X a2oK (a;m)=1 (a) N (a)s ; where : Jm K ! C is a homomorphism, for which there exists a pair of characters f : (oK=m) ! S1 and 1 : R K ! S1; where here R K corresponds to the set of units in the Minkowski space RK = fx = (x ) 2 Y Cjx = x g; with the product taken over all embeddings of K into C, and denotes the action of complex conjugation, satisfying ((a)) = f (a) 1 (a) 50 for all a 2 oK with ((a) ;m) = 1. Such a homomorphism is called a Gr o encharakter of K. Later, a special case of this will be a Dirichlet Lfunction, where the set Pm K is contained in the kernel of . The function f is called the nite part of , and 1 the in nite part of . De ne now for x = (x ) 2 Q C and y = (y ) 2 Q R the ntuple exponent xy = (xy ) ; where n is the degree of K over Q as a eld. Consider [R R]+ = f(x1; x2) 2 R Rjx1 = x2g: De ne also Y R Y [R R]+ with the rst product over all real embeddings of K into C, and the second over a set of representatives, one for each pair of complex conjugate nonreal embeddings of K into C. A descriptive result for 1 follows. Theorem 5.1 One has the decomposition of 1 as 1 (x) = N xpjxjp+iq where p = (p ) and q = (q ), with (q ) 2 Y R Y [R R]+; where p 2 f0; 1g for all real embeddings of K into C, and p ; p 2 Z is each nonnegative with p p = 0 for each nonreal embedding of K into C. Proof. x 2 R K may be written as x = x jxj jxj; 51 where jxj 2 R K+ = fx 2 R Kj x > 0 for each 2 Hom(K;C)g; and x jxj 2 UK = fx 2 R Kj jxj = 1g: It thus su ces to determine the characters of R K+ and those of UK. The characters of UK are trivially of the form N (xp) where p = (p ) is as stated in the theorem by considering the decomposition UK = Y f0; 1g Y [S1 S1]+: For the characters of R K+, one obtains an isomorphism of R K+ with R = Y R Y [R R]+ via the componentwise natural logarithm, and observes that any character of the latter must be given by (x) = N ((eiq x ) ) with q = (q ) 2 Y R Y [R R]+: Thus a character of R K+ is given by (x) = N eiq ln x = N (xiq) ; and thus 1 (x) = N x jxj p jxjiq = N xpjxjp+iq : Another construction is essential to the desired functional equation. A set X in a Euclidean space is called centrally symmetric if x 2 X implies that x 2 X. A complete lattice in a Euclidean vector space V of dimension n is a topologically discrete set generated by a set of n vectors linearly independent over R. With this generating set denoted by fv1; :::; vng, one de nes the fundamental mesh of to be the set ft1v1 + + tnvn j 0 ti < 1; for each i 2 f1; 2; :::; ngg: The following geometric result is due to Minkowski. 52 Theorem 5.2 Suppose that is a complete lattice in a Euclidean vector space V of dimension n. Suppose also X is a centrally symmetric, convex subset of V , with the volume of X greater than 2n times the volume of the fundamental mesh of . Then X contains at least one nonzero lattice point 2 . Proof. A sketch of this proof is given here. One shows this by demonstrating that the sets 1 2X + cannot be pairwise disjoint as ranges over the elements of , and therefore that there is a point in the intersection of 1 2X + 1 and 1 2X + 2 for some 1; 2 2 with 1 6= 2. Thereby one obtains this point as 1 2 x1 + 1 = 1 2 x2 + 2 for some x1; x2 2 X, whence = 1 2 = 1 2 x2 1 2 x1 is a nonzero element in X, because, by de nition, X is convex and centrally symmet ric. For the next result, one de nes the following terms. The class group of K is the group JK=PK, where JK denotes the group of fractional ideals of K, and PK the group of principal ideals of K. The integer s shall denote the number of in nite primes of K corresponding to nonreal embeddings of K into C. The canonical volume on RK is the volume associated with the metric determined by the Hermitian inner product hx; yi = X 2Hom(K;C) x y Theorem 5.3 The group JK of fractional ideals of K, taken modulo the group of principal ideals PK of K, is nite. Proof. First, it is established that each coset of PK in JK contains an integral ideal a1 so that N (a1) 2 sp jdKj; 53 where dK denotes the discriminant of K. Consider a nonzero integral ideal b of K. Considering c = (c ) satisfying c = c , c 2 R + for each embedding of K into C, and Y c > 2 sp jdKj (oK : b) ; one may notice that the canonical volume of the set X = f(z ) 2 RK j jz j < c for each 2 Hom(K;C)g is su ciently large so as to guarantee by the result of Minkowski a nonzero element b of b contained also in X. Choosing such an element c so that Y c = 2 sp jdKjN (b) + " for " > 0, one obtains a nonzero element in b which is also contained in X, and thus must satisfy jNKjQ ( ) j = Y j j < 2 sp jdKjN (b) + ": This is true for all " > 0, and upon noting that jNKjQ ( ) j is a positive integer, it is clear that some nonzero 2 b satis es jNKjQ ( ) j 2 sp jdKjN (b) : Consider then an arbitrary representative a belonging to a given coset of PK in JK, and 2 oK, not equal to zero, with b = a1 oK: There exists an element 2 b with 6= 0 satisfying jNKjQ ( ) j N (b)1 = N ( ) b1 = N b1 2 sp jdKj; and then a1 = b1 = 1a is then the desired ideal in the coset of a in JK=PK. 54 Considering then a nonzero prime ideal p of oK, with p \ Z = pZ, one has that oK=p is a nite extension of Z=pZ as a eld. Suppose that this extension has degree f. Then, in accordance with the de nition N (a) = (oK : a) ; one must have N (a) = pf . There must be only nitely many prime ideals p lying above p, so that there are only nitely many prime ideals p of oK with N (p) bounded by a given positive real number. One may write a prime factorization for the ideal a, and multiplicativity of N on the ideals of oK implies that there are a nite number of ideals a of oK with N (a) bounded by a given positive real number. But the rst part of this proof established that each coset of PK in JK contains an ideal with norm less than the bound 2 sp jdKj. This proves the claim. For the functional equation for the Hecke Lfunction, two more components are required. Consider a generating set [b1]; [b2]; :::; [br] for JK=PK, and suppose that, for each j 2 f1; 2; :::; rg, hj denotes the order of [bj ] in JK=PK. Then one has for such j that bhj j = (bj) where bj 2 K. One then chooses an element ^bj 2 R K to satisfy ^b j; = ( bi) 1 hi so that ^b j; = ^b j; , for each embedding of K into C, for each j 2 f1; 2; :::; rg. One then constructs the subgroup of Q C generated by K and the elements ^b j for each j 2 f1; 2; :::; rg, and calling this set ^K , one has an isomorphism ( ) : ^K =K = JK=PK de ned by taking the unique representation of an element ^a of ^K as ^a = a^b v1 1 ^b vr r 55 where 0 vj < hj for each j 2 f1; 2; :::; rg, and mapping it to the class of the ideal (^a) = abv1 1 bvr r : The noncanonically de ned set ^K is loosely called the set of ideal numbers. De ning then the set of ideal integers ^ oK as those elements of ^K which, via this map, are sent to ideals in oK, one notices that for the character , which in this case is called a Gr o encharakter modulo m, a unique extension of f is obtained to the set (^o=m) by de ning f = ((a)) 1 (a)1 for each ideal integer a with ((a) ;m) = 1. Viewing f in this way, the Gauss sum ( f ; a) is de ned for an ideal integer a as ( f ; a) = X ^x mod m f (^x) e2 iT r(^xa=md); where here for an element in x = (x ) 2 Q C, Tr (x) = P x , m is an ideal integer with (m) = m, d an ideal integer with (d) = D where D is the di erent of KjQ, and the sum ranges over the classes of (^o=m) mapped to the class 0 for which [(a) 0] = [mD] in JK=PK. With this in hand, and the in nite component of given by 1 (x) = N xpjxjp+iq ; one de nes L1 ( ; s) = N ( s =2) Z R K+ N eyys dy y where s = (s ) and s = s + p iq , for each embedding of K into C. One may then de ne the completed Lfunction as ( ; s) = (jdKjN (m)) s 2L1 ( ; s) L ( ; s) ; with p = (p ) as before, where p = (p ) and p = p . One then sets the Gauss sum of to be ( f ) = ( f ; 1), and W ( ) = " iTr(p)N md jmdj p !#1 ( f ) p N (m) 56 with jmdj = (jm d j). The complex number W( ) is called the root number of . One de nes a Gr o encharakter modulo m to be primitive if it is not the restric tion of a Gr o encharakter modulo m0 for some proper divisor m0 of m. For such a Gr o encharakter modulo m, one has the functional equation for the Hecke Lfunction as ( ; s) = W( ) ( ; 1 s) : The next section develops the notion of the Artin conductor, which will be used to connect Artin's Lfunction to the Hecke Lfunction. 5.2 The Artin conductor Returning to the setting of representation theory, consider a Galois extension F of a eld K, where K is a local eld, i.e., a eld that is complete with respect to a discrete valuation and possesses a nite residue class eld. One de nes the ith rami cation group Gi to be the group Gi (FjK) = f 2 G(FjK) j vF ( a a) i + 1 for all a 2 oF g where here vF denotes the unique extension to F of the additive valuation vK corre sponding to j jK. De ne f ( ) = X i 0 jGij jG0j codim V Gi ; where V Gi denotes the subspace of V that Gi xes via a xed choice of representation ( ; G; V ) with character . De ne FjK (s) = Z s 0 jGxj jG0j dx: One has the following theorem, due to Herbrand. Theorem 5.4 Suppose that F0 is a subextension of F which contains K, and is Galois over K. Let H = G(FjF0). Then one has Gs (FjK)H=H = Gt (F0jK) 57 where t = FjF0 (s) : Proof. With G = G(FjK) and G0 = G(F0jK), one may select for each 0 2 G0 some preimage 0 via the quotient map G(FjK) ! G(F0jK) so that iFjK ( ) = vF ( x x) ; with x chosen to generate oF over oK as an oKmodule, is maximal. Let m = iFjK ( ); one may note that if 2 H belongs to Hm1, then iFjK ( ) m, whence iFjK ( ) m, so that iFjK ( ) = m. If, on the other hand, =2 Hm1, then iFjK ( ) < m, and iFjK ( ) = iFjK ( ). Therefore, one has iFjK ( ) = minfiFjK ( ) ;mg: Therefore, as one must have, with eFjF0 the rami cation index of FjF0, that iF0jK ( 0) = 1 eFjF0 X jF0= 0 iFjK ( ) ; it follows that iF0jK ( 0) = 1 eFjF0 X 2H minfiFjK ( ) ;mg; and because iFjK ( ) = iFjF0 ( ), and eFjF0 = jH0j, one has that iF0jK ( ) 1 = FjF0 iFjK ( ) 1 ; and thus 0 2 GsH=H if and only if iFjK ( ) 1 s by the de nition of the rami  cation groups, which holds if and only if FjF0 iFjK ( ) 1 FjF0 (s) ; 58 which is equivalent to iF0jK ( 0) 1 FjF0 (s) by the above, which in turn is equivalent to 0 2 Gt (F0jK) ; with t = FjF0 (s) : This proves the claim. Similarly to the decomposition groups, one de nes upper numbering as Gt (FjK) = Gs(FjK) where t = FjK(s): The next result is stated, but for brevity is not proven here. It is a theorem due to Hasse and Arf, and employs the theory of LubinTate extensions [Neu, V]. Theorem 5.5 If G(FjK) is abelian, then the points t 1 for which Gt (FjK) 6= Gt+" (FjK) for any " > 0 are contained in Z. Before one proceeds further, some notation is in order. Denote G = G(FjK). Consider the class function aG ( ) de ned on G as equalling aG (1) = fiG ( ) for 6= 1, and aG (1) = f P 6=1 iG ( ) for = 1, where the sum ranges over nontrivial elements of G(FjK). One may alternatively de ne f ( ) according to the decomposition of aG as a class function of G, as aG = X f ( ) ; where f ( ) 2 C. In this setting, one has f ( ) = h ; aGi; where the inner product h 1; 2i is de ned as in Chapter 2 for two complexvalued functions 1 and 2 as 1 jGj X 2G 1( ) 2( ): Thus as aG( ) = aG( 1) for all 2 G, one may de ne f( ) = h ; aGi for an arbitrary class function on G. That the two de nitions given for f( ) agree follows from the following argument. 59 Lemma 5.1 (Frobenius reciprocity) Suppose that H is a subgroup of G. If 1 is a complexvalued class function on H and 2 is a complexvalued class function on G, then one has hIndG H( 1); 2i = h 1; 2jHi: Proof. By de nition, hIndG H( 1); 2i = 1 jGj X 2G IndG H( 1)(g) 2(g) = 1 jGj X 2G X 2R 1( 1 ) 2( ) = 1 jGj X 2R X 2G 1( 1 ) 2( ) = 1 jGj X 2R X 2G 1( ) 2( 1) = 1 jGj X 2R X 2G 1( ) 2( ) = 1 jGj X 2R X 2H 1( ) 2( ) = 1 jHj X 2H 1( ) 2( ) = h 1; 2jHi: Consider then for the rami cation group Gi the character ui corresponding the representation which acts on the subspace W of the algebra C[G] with a basis fv g indexed by the elements of Gi, with W de ned by f X 2Gi x v j X 2Gi x = 0g: This representation is called the augmentation representation for Gi, and ui the aug mentation character for Gi. 60 Theorem 5.6 The two given de nitions for f( ) agree, i.e., h ; aGi = X i 0 jGij jG0j codim(V Gi): Proof. One has aG = X1 i=0 jGij jG0j IndG Gi(ui) and thus f ( ) = h ; aGi = X i 0 jGij jG0j h ; IndG Gi (ui)i; which is equal to = X i 0 jGij jG0j h jGi ; uii by the previous lemma. and one may notice that h jGi ; uii = codim V Gi for i 0, so that the two de nitions of f ( ) agree as desired. Denote now by rG the character of the regular representation of G. The following theorem is of great importance. Theorem 5.7 Suppose that i is a character corresponding to a representation of degree one of a subgroup Hi of G, and that Ki denotes the xed eld of Hi. Then f IndG Hi i = vK dKijK i (1) + fKijKf ( i) ; where dKijK denotes the discriminant ideal of Ki over K, and fKijK the inertia degree. Proof. Considering 2 Hi with 6= 1, one has that aG ( ) = fFjKiG ( ) and aHi ( ) = fFjKiiHi ( ) ; 61 where here Ki denotes the xed eld of Hi, and thus because iG ( ) = iHi ( ), one has aG ( ) = fKijKaHi ( ) = vK dKijK rHi ( ) + fKijKaHi ( ) where the second equality holds because rHi ( ) = 0. In the case that = 1, one may consider DFjK, the di erent of F over K, and supposing that oL = oK[x], with g (X) the minimal polynomial of x over K, one has that DFjK is generated by g0 (x) = Q 6=1 ( x x). Therefore, vL DFjK = vL (g0 (x)) = X 6=1 iG ( ) = 1 fFjK aG (1) : Also, one has dFjK = NFjK DFjK , and thus that, because vK NFjK = fFjKvL, one has aG (1) = vK dFjK : Likewise aHi (1) = vKi dFjKi . One has dFjK = dKijK [F:Ki] NKijK dFjKi : Therefore, aG (1) = [F : Ki]vK dKijK + fKijKvKi dFjKi = vK dKijK rHi (1) + fKijKaHi(1): Therefore f IndG Hi ( i) = hIndG Hi( i); aGi = h i; aGjHii = vK dKijK ( i; rHi) + fKijK ( i; aHi) = vK dKijK i (1) + fKijKf ( i) : This yields the following theorem. 62 Theorem 5.8 Suppose that is a character of G(FjK) corresponding to a represen tation of degree one. Suppose that j is the largest integer so that jGj 6= 1Gj , where when = 1G one sets j = 1. Then one has f ( ) = FjK (j) + 1; and f ( ) 2 Z is nonnegative. Proof. De ne (Gi) = 1 jGij X 2Gi ( ) : If i j, then evidently (Gi) = 0, and thus (1) (Gi) = 1. If i > j, then clearly (Gi) = 1, whence (1) (Gi) = 0. In light of this, one has that f ( ) = X i 0 jGij jG0j ( (1) (Gi)) = Xj i=0 jGij jG0j = FjK (j) + 1; so long as j 0. If j = 1, then one has (1) (Gi) = 0 for all i 0, so that f ( ) = 0 = FjK (1) + 1: Then, considering H the kernel of in G = G(FjK), and F0 the xed eld of H, one has that Gj (FjK)H=H = Gj0 (F0jK) by Herbrand's theorem. In the upper numbering of rami cation groups, this translates as Gt (FjK)H=H = Gt (F0jK) where here t = FjK (j) = F0jK FjF0 (j) = F0jK (j0). However, one must have (Gj (FjK)H=H) 6= 1; 63 and (Gj+ (FjK)H=H) = (Gj+1 (FjK)H=H) = 1 for all > 0, and so Gj (FjK)H=H 6= Gj+ (FjK)H=H for all > 0. Of course, the function FjK is continuous and strictly increasing as de ned, and therefore Gt (F0jK) = Gt (FjK)H=H 6= Gt+" (FjK)H=H = Gt+" (F0jK) for all " > 0, so that by the theorem of Hasse and Arf, one has that t = FjK (j) is an integer. Brauer's theorem on induced characters then yields the following result. Theorem 5.9 An arbitrary character of G(FjK) has f ( ) equal to a nonnegative integer. Proof. This follows from the fact that one may write, by Brauer's theorem, = X nite niIndG Hi ( i) for integers ni and characters i corresponding to representations of degree one of subgroups Hi of G. One does have f ( ) = X nite nif IndG Hi ( i) = X finite ni vK dKijK i (1) + fKijKf ( i) ; with Ki the xed eld of Hi, for each i. This is an integer by the previous theorem, and is nonnegative because jG0jaG is a character of a representation, and is given by X i 0 jGij IndG Gi (ui) : Therefore jG0jf ( ) = ( ; jG0jaG) 0. 64 The following theorem results almost immediately. For this, let K be a completion with respect to a nite prime p of an algebraic number eld, and F a nite extension of K. Theorem 5.10 Suppose that F is Galois over K, and that is a character of G(FjK) corresponding to a representation of degree one. Suppose also that F is the xed eld of the kernel of , and that f is the conductor of F over K, in this case de ned to be the smallest power n 0 of the unique maximal ideal p of K with re spect to its discrete valuation so that U(n) K is contained in the group NFjK (F ), where n 0, and U(n) K = fx 2 Kjx 1 2 png: Then f = f ( ). Proof. One has f ( ) = FjK (j) + 1 as a consequence of a previous theorem, where j is the largest integer such that Gj (FjK) =2 G(FjF ). With t = FjK (j), one has that Gt (F jK) = Gt (FjK)H=H = Gj (FjK)H=H: Also, Gt+" (F jK) Gj+1 (FjK)H=H = 1 for all " > 0. Hence t is an integer by the theorem of Hasse and Arf, and moreover, f ( ) = t + 1 is the smallest integer with Gf( ) (F jK) equal to one. The theory of LubinTate extensions [Neu, V] then implies that the symbol r1 FjK : K ! G(FjK) of local class eld theory maps U(i) K to the rami cation group Gi (FjK) with i = FjK (j), and therefore n is also the smallest nonnegative integer so that Gn (FjK) = 1. Therefore f = f ( ). De ne then the local Artin conductor as fp ( ) = pf( ), with p denoting the unique maximal ideal corresponding to the discrete valuation on K for which K is complete. 65 The above analysis will be applied to the completions of K with respect to its various nite primes. Suppose now that K is an algebraic number eld. Class eld theory gives an object, called a conductor, of a nite Galois extension F of K as the smallest m in oK for which F is contained in the eld Fm, called the ray class eld modulo m, for which G(FmjK) = CK=Cm K; noting that such an Fm must exist by the main theorem of class eld theory. One then de nes f ( ) = Y p1 fp ( ) : This is called the Artin conductor of . Because this conductor does depend upon the choice of underlying elds, it will henceforth be written as f (FjK; ). The following is immediate from the above analysis. Theorem 5.11 Suppose that FjK is a Galois extension of algebraic number elds, and that is a character of G(FjK) corresponding to a representation of degree one. Suppose also that F denotes the xed eld of the kernel of , and f the conductor of F over K. Then one has f = f ( ). Proof. One has by the class eld theory that f = Y p1 fp where fp, for each p  1, is the conductor of FP over Kp. By the previous theorem, one has fp = fp ( ) for each such p, and hence, the result follows. One has the following theorem. The reader is invited to note its similarity to the theorem about the basic properties of the Artin Lfunction appearing in chapter two of this work. Before stating the result, it is worthwhile to note the de nition of norm 66 for ideals of a nite extension F of an algebraic number eld K as NLjK Y P PvP ! = Y p Y Pjp pfPjpvP where here NLjK is mapping from JF ! JK and fPjp is the inertia degree of P over p. Theorem 5.12 (i) f (FjK; + 0) = f (FjK; ) f (FjK; 0); (ii) if F0 is subextension of F that contains K, and is Galois over K, and is a character of G(F0jK), then f (FjK; ) = f (F0jK; ) with acting on G(FjK) via the quotient map G(FjK) ! G(F0jK) ; (iii) if H is a subgroup of F with xed eld K0, and if is a character of H, then one has f FjK; IndG H ( ) = d (1) K0jKNK0jK (f (FjK0; )) : Proof. The proofs of (i) and (ii) are trivial. For (iii), one denotes G = G(FjK), H = G(FjK0), GP = G(FPjKp), where p = P \ K, and one considers the double coset decomposition G = [ GP H: One has naturally that IndG H ( ) jGP = X IndGP GP\ H 1 ( ) where is the character de ned by ( ) = ( 1 ) corresponding to the group GP \ H 1, where one takes as implicit that the representation homomorphism corresponding to the character is modi ed to become ( ) = ( 1 ), and then restricted to GP \ H 1, so as to be a representation to which the character 67 corresponds. Suppose now that dP0 = pvP0 is the discriminant ideal of K0P 0 jKp, where P0 denotes the prime 1P \ K0 of oK0 . Then one has NK0jK (P0 ) = pfP0 according to the de nition of this norm. And fp FjK; IndG H ( ) = pf(IndG H( )jGP) as well as fP0 (FjK0; ) = P0 f jH 1P ; and so, as dK0jK = Q Pjp dK0jKP, it su ces now to show that f IndG H ( ) jGP = X vP0 (1) + fP0 f ( ;H 1P) where H 1P = G 1P \ H. Theorem 5.7 implies that vP0 (1) + fP0 f ( ;H 1P) = f Ind G 1P H 1P jH 1P for each double coset representative , and therefore X vP0 (1) + fP0 f ( ;H 1P) = X f Ind G 1P H 1P jH 1P But one may notice that jH 1P , and thus Ind G 1P H 1P jH 1P , is obtained from conjugation by ; and thus f Ind G 1P H 1P jH 1P = f IndGP GP\ H 1 ( ) because the inertia degrees of the two groups GP and G 1P are the same as in the proof of Lemma 3.1, and because this conjugation by preserves valuations relative to the map 1 : FP ! F 1P. This proves the claim. One is now prepared to address the problem of the functional equation of Artin's Lfunction. 68 5.3 The functional equation De ne now the ideal of Z by c (FjK) = d (1) KjQNKjQ (f (FjK; )) ; where here dKjQ denotes the discriminant ideal of K over Q, with generator jdKj (1)N (f (FjK; )) : Denote this generator by c (FjK; ). One has the following basic result. Theorem 5.13 (i) c (FjK; + 0) = c (FjK; ) c (FjK; 0); (ii) For a subextension F0 of F containing K and Galois over K and a character of G(F0jK), then c (FjK; ) = c (F0jK; ), where gives a character of G(FjK) via the quotient map G(FjK) ! G(F0jK); (iii) If H is a subgroup of G with xed eld K0, then one has c (FjK0; ) = c FjK; IndG H ( ) : Proof. (i) and (ii) follow from the basic properties of the Artin conductor already proven in this work. (iii) does as well, in conjunction with the fact that, with dFjK denoting the discriminant ideal for F over K, one has, for F K0 K, dFjK = NK0jK dFjK0 d[F:K0] K0jK : The following results may be easily proven. De ne the gamma function of a complex variable s with Re(s) > 0 to be (s) = Z 1 0 eyys dy y : Then one has the following functional equations. 69 Theorem 5.14 (i) (s + 1) = s (s); (ii) (s) (1 s) = sin( s) ; (iii) (s) s + 1 2 = 2 p 22s (2s). Proof. These are easily established, and one may refer to [FB, IV] for the proofs. Then, one recalls the Artin Lfunction L (FjK; ; s) = Y p1 1 det I ( P)N (p)s j V IG(FjK);P : De ne L1 (FjK; ; s) = Y pj1 Lp (FjK; ; s) ; where Lp (FjK; ; s) = 2 (2 )s (s) (1) if p is complex, and Lp (FjK; ; s) = s=2 (s=2) n+ (s+1)=2 ((s + 1) =2) n if p is real, with n+ = (1) + ( P) 2 ; n = (1) ( P) 2 ; P is, when pj1, the generator for G(FPjQp), W ( ) 2 C has modulus one, and each p denotes an in nite prime of K, noting that here a prime p is called complex if it corresponds to a pair of complex conjugate nonreal embeddings of K into C, and real if it corresponds to a real embedding of K into C. The following theorem holds naturally. De ne for simplicity the functions LR (s) = s=2 (s=2) 70 and LC (s) = 2 (2 )s (s) ; one applies in the following theorem the identity LR (s) LR (s + 1) = LC (s) : Theorem 5.15 For an in nite prime p, one has (i) Lp (FjK; + 0; s) = Lp (FjK; ; s) Lp (FjK; 0; s); (ii) given a Galois extension F0 of K contained in F, and a character on G(F0jK), one has Lp (FjK; ; s) = Lp (F0jK; ; s) were acts on G(FjK) via the quotient map G(FjK) ! G(F0jK); (iii) if K0 is a eld containing K and contained in F, and is a character of G(FjK0), then with H = G(FjK0) and G = G(FjK) one has Lp (FjK0; ; s) = Lp FjK; IndG H ( ) ; s . Proof. Once again, properties (i) and (ii) are trivial. For (iii), suppose rst that p corresponds to a complex, nonreal prime of K. Then any in nite prime of K0 lying above it is also complex, and one has that the number of such q is equal to [K0 : K]. But also, one has IndG H ( ) (1) = [K0 : K] (1) : This proves the claim for when p is complex. If p is real, one may notice that in the double coset decomposition G = [ H GP where P is a prime of F lying above p, yields a bijection between this decomposition and the set of primes q = P \ K0 of K0 above p. Of course, this element q is real if and only if G P = GP 1 H, which holds if and only if H GP consists of only one coset modulo H. Therefore, the real places among the elements q are obtained 71 by allowing to range through a system of representatives of the right cosets of H in G, and one may notice that, with P the generator for the group GP, one has IndG H ( ) ( P) = X ( P) where the sum ranges over exactly those , representing right cosets of H in G, for which P 1 2 H. One also has that IndG H ( ) (1) = X q complex 2 (1) + X q real (1) ; and therefore the formula LR (s) LR (s + 1) = LC (s) may be used to show that Lp FjK; IndG H ( ) ; s = Y q complex LC (s) (1) Y q real LR (s) (1)+ ( P) 2 Y q real LR (s + 1) (1) ( P) 2 which, in turn, must equal Y qjp Lq (FjK0; ; s) ; by de nition. One may then de ne the completed Artin Lfunction as (FjK; ; s) = c (FjK; )s=2 L1 (FjK; ; s) L (FjK; ; s) ; and the following three properties have been established. Theorem 5.16 (i) (FjK; + 0; s) = (FjK; ; s) (FjK; 0; s); (ii) for a subextension F0 of F that contains K and is Galois over K, and a character of G(F0jK), one has (FjK; ; s) = (F0jK; ; s), where acts on G(FjK) via the quotient map G(FjK) ! G(F0jK); (iii) if K0 is a sub eld of F containing K, and is a character of G(FjK0), then (FjK0; ; s) = FjK; IndG H ( ) ; s , with G = G(FjK) and H = G(FjK0). 72 One has the following important theorem. Theorem 5.17 Consider a character corresponding to a representation of degree one of G(FjK), and denote by F the xed eld of . (i) may be viewed as a primitive Gr o encharakter modulo f(FjK; ); (ii) for every real in nite prime p of K, one has that pp = [F ;P : Kp] where here the notation F ;P indicates the completion with respect to a prime Pjp of the xed eld F of . Proof. (i) The Artin symbol of class eld theory yields a map Jf K=Pf K ! G(F jK) where f denotes the conductor of F over K. In this way, the commutative diagram at the end of chapter four indicates that becomes a Dirichlet character of con ductor f = f (F jK; ) = f(FjK; ). It is trivial that this character then admits a decomposition as a primitive Gr o encharacter modulo f(FjK; ). (ii) Class eld theory gives an isomorphism between IK=If KK and Jf K=Pf K given by the map ( ) from chapter four, and one thus obtains a map from IK=If KK to C by composing ( ) with the Artin symbol and . Considering then p a real in nite prime of K, and = ( p) 2 IK with p = 1 and q = 1 for all q 6= p, it is noted that the class eld theory giving that the image P in G(F jK) of via the map A( ) as de ned in chapter four must be the generator of the decomposition group GP = G(F ;PjKp). The weak approximation theorem yields some a 2 K with a 1 2 f, where a < 0 for the embedding corresponding to the prime p, and 0a > 0 for all other real embeddings 0 of K into C. Therefore a 2 I(f) K ; and as in the proof yielding the result IK=If KK = Jf K=Pf K 73 one may notice that the image of the coset of via the map allowing this isomorphism must be the same as the class of a in Jf K=Pf K, and that a maps to (a), which must then map via the Artin symbol to the generator of GP. Therefore, viewing now as a Gr o encharakter, one has ((a)) = f (a) 1 (a) = ( P) ; and because a = 1 mod f, one has f (a) = 1, and one notes that 1 (a) = N a jaj p = a jajp pp = (1)pp ; and thus P = 1 if and only if pp = 0. Now one may notice that for a character corresponding to a representation of degree one of G(FjK), and that one may write as a character of G(F jK), where F denotes the xed eld of . As a consequence of this, one may compose with the Artin symbol as done above, and, viewing now also as a Gr o encharakter modulo its conductor, one may see the following fundamental theorem due to Artin. Theorem 5.18 For a character corresponding to a representation of G(FjK) of degree one, and with F denoting the xed eld of , one has the equality (FjK; ; s) = ( ; s) where the righthand side of this equality is a Hecke Lfunction with viewed in that case as a character of Jf K=Pf K through composition of : G(F jK) ! C with the Artin symbol, where f denotes the Artin conductor of . Proof. This is a straightforward application of the de nitions at hand. Given this, one may then employ the Brauer theorem, to claim that 74 Theorem 5.19 The completed Artin Lfunction satis es the functional equation (FjK; ; s) = W ( ) (FjK; ; 1 s) where W ( ) is a complex number of modulus equal to one. Proof. This follows by writing as, with G = G(FjK), = X nite niIndG Hi ( i) ; and then applying the basic properties of the Artin Lfunction to decompose it as (FjK; ; s) = Y nite (FjKi; i; s)ni where Ki is the xed eld of i, and this in turn equals, by the previous theorem, Y nite ( i; s)ni ; where, for each i, ( i; s) is the Hecke Lfunction for i when viewed as a Gr o encharakter in the sense of the previous theorem. One has Y nite ( i; s)ni = Y nite (W ( i) ( i; 1 s))ni where each W ( i) is a complex number of modulus equal to one. Also, one has Y nite (W ( i) ( i; 1 s))ni = W ( ) Y nite ( i; 1 s)ni = W ( ) Y nite (FjKi; i; 1 s)ni = W ( ) Y nite (FjK; ; 1 s) ; where W ( ) = Y nite W ( i)ni is a complex number of modulus equal to one. One nal note is necessary that will be used in the sequel. 75 Theorem 5.20 The Artin Lfunction L(FjK; ; s) is nonzero in the halfplane Re(s) > 1. Proof. This follows from the de nition of the Artin Lfunction in the halfplane Re(s) > 1 as a convergent in nite product of analytic functions, each nonzero in the halfplane Re(s) > 1. . This concludes chapter ve. The next chapter establishes the result of Coates and Lichtenbaum on the value of Artin's Lfunction at negative integers. 76 CHAPTER 6 The result of Coates and Lichtenbaum This chapter establishes the result of Coates and Lichtenbaum [CL] on the values of Artin's Lfunction at negative integers. Sections 6.1 and 6.2 are devoted to establish ing the unit theorem of Shintani [Neu, VII]. Sections 6.3 and 6.4 prove a result due to Siegel and Klingen [Neu, VII] on special values of Lfunctions. Section 6.5 establishes the main result of this chapter. 6.1 Polyhedric cones One begins this section by rst considering an ndimensional Rvector space V , a sub eld k of R and Vk a ksubspace of V with V = Vk k R: In practice, the set Vk will simply be an algebraic number eld K. In this setting, a krational simplicial cone of dimension d will be a set of the form C(v1; :::; vd) = ft1v1 + + tdvd j tl 2 R + for each l 2 f1; 2; :::; dgg; where the set fv1; :::; vdg consists of linearly independent elements of Vk. A nite disjoint union of krational simplicial cones will be called a krational polyhedric cone. A linear form L on V will be called krational if its coe cients with respect to a kbasis of Vk lie in k. The following two results are crucial to proving Shintani's theorem, and comprise the body of this section. Theorem 6.1 Every nonempty subset di erent from the set f0g of the form P = fx 2 V j Li(x) 0; 0 < i l;Mj(x) > 0; 0 < j mg 77 for krational linear forms Li and Mj for i 2 f1; 2; :::; lg and j 2 f1; 2; :::;mg, with admittance of the cases where l = 0 or m = 0, is either a krational polyhedric cone or the union of such a cone with the origin. Proof. For the proof, one considers as a rst case P = fx 2 V j Li(x) 0; for each i 2 f1; 2; :::; lgg; for krational linear forms L1; :::;Ll 6= 0. The theorem is trivial for the case of n = 1, and it may be proven by induction. Assuming the theorem true for all vector spaces over R of dimension less than n, one then considers the following argument. If P has no interior point, then there is a linear form Li so that P is contained in the hyperplane L = 0, because V is a vector space, and the induction hypothesis applies to prove the claim. Thus, one may suppose that there is an interior point, and suppose that u 2 P is this point, satisfying L1(u) > 0; :::;Ll(u) > 0: Vk is, by de nition, dense in V , so that one may assume that u 2 Vk. For each i 2 f1; 2; :::; lg, de ne @iP = fx 2 P j Li(x) = 0g: If @iP 6= f0g, then the induction hypothesis again applies to show that @iPnf0g is a krational polyhedric cone, comprised of krational simplicial cones of dimension less than n. If such a cone in @iP has nonempty intersection with @jP, then it is contained in @iP \ @jP, as is obvious from the fact that each @jP is constructed to lie in P. Thus the set @1P [ [ @lPnf0g is a disjoint union of krational simplicial cones of dimension less than n. Denote this disjoint union by [ j2JCj ; 78 where each Cj = C(v1; :::; vdj ) is a krational simplicial cone corresponding to the linearly independent vectors v1; :::; vdj with dj < n, and here, the notation [ will be intended to denote disjoint union. Then, for each j 2 J, de ne Cj(u) = C(v1; :::; vdj ; u): Each of these will be a krational simplicial cone of dimension dj + 1 because u was selected to be an interior point of P. One has Pnf0g = f[ j2J Cjg [ f[ j2J Cj(u)g [ R +u: This may be seen from the following argument. If the point x 2 Pnf0g lies on the boundary of P, then it belongs to some @iP, and thus to some Cj . If, on the other hand, x belongs to the interior of P, then Li(x) > 0 for each i 2 f1; 2; :::; lg, and in this case, if x is a scalar multiple of u, then one has x 2 R +u, but if it is not, then one may consider the following. If s is the minimum of the quantities L1(x) L1(u) ; :::; Ll(x) Ll(u) ; then s > 0, and the element x su lies upon the boundary of P. As by supposition x 6= su, there must be a unique j 2 J for which x su 2 Cj , and therefore a unique j 2 J for which x 2 Cj(u). This proves the rst case of the claim. For the second case, let P be as de ned in the statement of the theorem. Then P = fx 2 V j Li(x) 0;Mj(x) 0; for each i 2 f1; 2; :::; lg and j 2 f1; 2; :::;mgg is a krational polyhedric cone joined with f0g. For each j 2 f1; 2; :::;mg, de ne @jP = fx 2 P j Mj = 0g; similarly to the rst case in this proof. As before, from the de nition of P, one must have that if a simplicial cone in P has nonempty intersection with @jP, then it must be contained in @jP. And P = Pnf[mj =1 @jPg; 79 so that P must also be a krational polyhedric cone. The following result will be used in the proof of the Shintani unit theorem. Theorem 6.2 If C and C0 are krational polyhedric cones, then CnC0 is also a k rational polyhedric cone. Proof. For the proof, one may, of course, suppose that C and C0 are krational sim plicial cones. With d the dimension of C0, one notices that there are n krational linear forms L1; :::;Lnd;M1; :::;Md so that C0 = fx 2 V j L1(x) = = Lnd(x) = 0;M1(x) > 0; :::;Md(x) > 0g: One may then de ne for each i 2 f1; 2; :::; n dg the sets C+ i = fx 2 C j L1(x) = = Li1(x) = 0; Li(x) > 0g and C i = fx 2 C j L1(x) = = Li1(x) = 0:Li(x) < 0g: Also, for each j 2 f1; 2; :::; dg, one may de ne Cj = fx 2 C j L1(x) = = Lnd(x) = 0;M1(x) > 0; :::;Mj1(x) > 0;Mj(x) 0g: One has CnC0 = f[ nd i=1 C+ i g [ f[ nd i=1 C i g [ f [dj =1 Cjg: Each of the sets appearing in this disjoint union is either empty, or is a krational polyhedric cone, from the previous theorem. This proves the result. 6.2 Shintani's unit theorem For completeness, the following theorem has been included, known as Dirichlet's unit theorem. For this part, one considers again the Minkowski space RK: One notices 80 that the eld K naturally embeds into this space via the map jx = Y x: One then takes the logarithm l as (l j)x = Y ln j xj; and one notices that this composite map l j yields an exact sequence 1 ! (K) ! o K ! ! 0 where, in this exact sequence, l j acts on o K, the group of units of the eld K, and maps it into the tracezero hyperplane in H = fx = (x ) 2 Y R j x = x g; and (K), the group of roots of unity contained in K, maps via inclusion into o K. One may now state and prove the Dirichlet unit theorem. Theorem 6.3 The group of units o K of oK is the direct product of the nite cyclic group (K) and a free abelian group of rank r + s 1, where here r denotes the number of real embeddings of K, and s the number of pairs of complex conjugate nonreal embeddings of K into C. Proof. That the group (K) is nite and cyclic is obvious. the group of units o K admits a decomposition into a direct product of (K) with another abelian group will ultimately rely upon the above exact sequence and a decomposition of the image of o K via the embedding l j. To begin, consider a 2 Z with a > 1. One may rst notice that with N(a) = (oK : a) for an ideal a in oK, one has that a principal ideal (a) in oK must satisfy N(a) = jNKjQ(a)j 81 where NKjQ denotes the usual norm from K to Q. Therefore the index of (a) in oK is nite. Therefore, up to multiplication by a unit of oK, there is at most one element in each coset of oK=aoK satisfying jNKjQ( )j = a. This is seen to be true by taking = + a for 2 oK, and noting that = 1 NKjQ( ) 2 oK for NKjQ( ) 2 oK. This holds likewise for , and therefore is equal to up to multiplication by an element of o K. Therefore there are at most (oK : aoK) elements of norm a. Then, one may easily notice that = (l j)o K is a lattice in the set H, because the bounded domain f(x ) 2 Y R j jx j < cg contains nitely many points of , for each c > 0. Now one must show that is a complete lattice in H, in other words, a lattice of dimension equal to the dimension of H, which here is clearly equal to r+s1. To do this, it su ces to nd a bounded set M H so that H = [ 2 M + : In order to nd this set, one constructs a bounded set T in the surface S de ned by S = fy = (y ) 2 RK j jN(y)j = 1g where here N(y) = Q y , so that S = ["2o K Tj": Given x = (x ) 2 T, one then has that the absolute values jx j are bounded away from zero for each embedding of K into C, as Q jx j = 1. In this case, M = l(T) will be bounded as well, and the proof will be nished. Thus, one considers real 82 numbers c > 0 for every embedding of K into C, satisfying c = c for every such embedding of K, and so that Y c > ( 2 )s p jdKj where here dK denotes the discriminant of the number eld K. With (c ) so chosen, one will have, for an element y = (y ) 2 S and the set Xy = f(z ) 2 RK j jz j < c jy j; for all 2 Hom(K;C)g; that the canonical volume of the set Xy is 2s times the Lebesgue volume of the set f(x ) 2 Y R j jx j < c jy j; x2 + x2 < (c jy j)2g; where here , respectively is intended to denote a real, respectively nonreal, em bedding of K into C. Thus the canonical volume of Xy is 2s Y 2c Y ( c2 ) = 2r+s s Y c ; where the product over is taken over a set of representatives, one for each pair of complex conjugate nonreal embeddings of K into C, the product over intends to be over all real embeddings of K into C, and the last product is over all embeddings from K into C. One notes here that these formulations make sense because y so chosen requires jy j = jy j. The canonical volume vol(Xy) satis es vol(Xy) > 2r+s s( 2 )s p jdkj = 2n p jdkj as jN(y)j = 1; and the canonical volume of oK is precisely p jdkj. Thus the result of Minkowski on lattices applies to yield the existence of a point ja 2 Xy for a 2 oK with a 6= 0, for oK does form a complete lattice in the Minkowski space RK According to the rst result in this proof, one may then choose 1; :::; N 2 oK, each nonzero, so that every a 2 oK with 0 < jNKjQ(a)j C is associated in oK to one of the elements 1; :::; N. Therefore the set T = S \ f[Ni =1 X(j i)1g 83 is as desired, for X is bounded, and thus so is X(j i)1 for each i 2 f1; 2; :::;Ng, and thus so must T be bounded. Now one must show that S = ["2o K Tj": This follows from the following argument. If y 2 S, then one may nd a nonzero element of oK as before with ja 2 Xy1, and therefore ja = xy1 for some x 2 X. Of course, one has jNKjQ(a)j = jN(xy1)j = jN(x)j < Y c where the second equality holds because jN(y)j = 1, and therefore a is associated in oK to i, for some i 2 f1; 2; :::;Ng. Therefore, with i = "a for " 2 o K, one has y = x(ja)1 = x(j i)1j"; and therefore, as y and j" are in S, so must x(j i)1 be contained in S, so that x(j i)1 2 T, and therefore y 2 Tj". Then, one notes the exact sequence 1 ! (K) ! o K ! ! 0 as before now has as a free abelian group of rank r + s 1. therefore, considering preimages of a Zbasis for via the map l j, and the subgroup A in o K generated by this basis, one then notes that A = , and that o K = (K) A: This completes the proof. Now, one de nes R KR;+ = fx = (x ) 2 R Kjx > 0 for each embedding : K ! Rg: 84 One then de nes o K;+ = o K \ R KR;+: One is now fully prepared to state and prove Shintani's unit theorem. Theorem 6.4 If E is a subgroup of nite index in o K;+, then there exists a Qrational polyhedric cone P such that R KR;+ = [ "2E "P: Proof. Every element in R KR;+ may be written uniquely as a product of an element contained in the normone surface S, as de ned in the proof of the Dirichlet unit theorem above, and a positive scalar, as x = jN(x)j 1 n x jN(x)j 1 n : Dirichlet's unit theorem gives as above a mapping of E onto a complete lattice of the tracezero space H as de ned before, because E is of nite index in o K. Consider the fundamental mesh of . With the closure of in H being denoted by , one has that l1( ) is closed and bounded, hence compact, in the surface S. Also, one must have S = ["2E "F: Consider then x 2 F, and the set U (x) = fy 2 RK j kx yk < g where > 0 is chosen so that U (x) R KR;+ and the metric k k is any of the set of equivalent metrics on RK extending the usual one on R. One may nd a basis fv1; :::; vng of RK contained in U (x) so that x = t1v1 + tnvn where ti > 0 for each i 2 f1; 2; :::; ng, and K is dense in RK by the weak approximation theorem, so that one may select fv1; :::; vng to lie also in K. 85 One may then notice that C = C(v1; :::; vn) is a Qrational simplicial cone in R KR;+ with x 2 C . Because E is discrete in RK, one may choose su ciently small so that C \ "C = ? for all " 2 E and " 6= 1. For, if not, then one would nd sequences f vzvgv2f1;2;3;:::g; f 0 vz0 vgv2f1;2;3;:::g; f"vgv2f1;2;3;:::g where v; 0 v 2 R +, zv; z0v 2 U1 v (x), "v 2 E, and vzv = "v 0 vzv for each v 2 f1; 2; 3; :::g. The equation v 0 v N(zv) = N(z0 v) for each v 2 f1; 2; 3; :::g, where N denotes the usual componentwise norm on RK, implies that lim v!1 v 0 v = 1: As limv!1 zv = limv!1 z0v = x, one would then have lim v!1 "v = 1; violating the fact that E is discrete in RK. One may notice that each C is open, so that as one considers the set of these as x ranges over elements of F, one nds a nite subcollection C1; :::;Cm in R KR;+ so that F = [mi =1 (Ci \ F): Therefore one has R KR;+ = [mi =1 ["2E "Ci: Now, one denotes C(1) 1 = C1, and C(1) i = Cinf["2E "C1g: 86 As E is discrete, and hence closed, in RK, one may notice that "C1 and Ci are disjoint for almost every " 2 E. For, if not, then one could nd a sequence of elements "vyv = y0 v with yv 2 C1, y0v 2 Ci, and "v 2 E for each v 2 f1; 2; 3; :::g, and thus a sequence f"vgv2f1;2;3;:::g with a limit point, which must be contained in E. This violates the fact that E is discrete. Therefore, by the second theorem in the rst section of this chapter, the set C(1) i is a Qrational polyhedric cone. One then has R KR;+ = [mi =1 ["2E "C(1) i where "C(1) 1 [ C(1) i = ? for each " 2 E and each i 2 f2; :::;mg. In this fashion, one proceeds to de ne C(2) i = C(1) i for i 2, and for i > 2 de nes C(2) i = C(1) i nf["2E "C(1) 2 g; and so forth, and in this fashion inductively arrives at a system C(m1) 1 ; :::;C(m1) m of Qrational polyhedric cones so that R KR;+ = [ mi =1 [ "2E "C(m1) i : This proves the claim. 6.3 Rewriting a particular kind of Lfunction Suppose that K is an algebraic number eld, and m an ideal of oK. Considering then a homomorphism : Jm K=Pm K ! C which, in view of the rst chapter of this work, may be viewed as a onedimensional representation of the group Jm K=Pm K. In this case may also be called a Dirichlet character modulo m. One de nes the Dirichlet Lfunction for by L( ; s) = X a2oK (a;m)=1 (a) N(a)s 87 where N( ) is the usual norm of an ideal as de ned earlier. It is worthy of note here that this is a special case of a Hecke Lfunction. One, of course has the decomposition L( ; s) = X ( ) ( ; s) where the sum varies across the cosets of Pm K in Jm K, and one has ( ; s) = X a2 a2oK 1 N(a) : The cosets are also called classes. Suppose that is such a xed coset and a 2 oK an element belonging to the coset . Suppose that (1 + a1m)+ = (1 + a1m) \ R KR;+ is the set of all totally positive elements in 1 + a1m. The group E de ned as E = om + = f" 2 o K j " = 1 mod m; " 2 R KR;+g acts on (1 + a1m)+ via componentwise multiplication. Two theorems must now be stated to prepare for the result of Siegel and Klingen, which comes in the second section of this chapter. Theorem 6.5 One has a bijection (1 + a1m)+=E ! fa 2 \ oKg mapping the orbit of a to the element aa. Proof. Suppose that a 2 (1 + a1m)+. It follows that a 1 2 m, because a and m are relatively prime. But then aa 2 , because (a) is relatively prime to m, and of course, must be totally positive. Also, one has aa a(1 + a1m) = a + m = oK where the last equality comes from the fact that a and m are relatively prime. 88 To show the surjectivity, consider aa an ideal in \ oK. Then (a 1)a ma m; whence a 2 1 + a1m. Also, as a is totally positive, one thus has a 2 (1 + a1m)+. Considering then a; b 2 (1 + a1m)+, one has naturally that aa = ba if and only if (a) = (b), or in other words, when a = b" for " 2 o K. Of course, such an element " must be contained in R KR;+. Also, " 2 1 + a1m, as is evident from the fact that b(" 1) = a b 2 a1m and thus that " 1 2 m because (b) is relatively prime to m. Of course, if " 2 E and a = b", then clearly (a) = (b), and thus the image under the action stated in the theorem is the same. This proves the claim. Thus, one has for such a representative a the equality ( ; s) = 1 N(a)s X a2 1 jN(a)js where N denotes the usual norm on elements in RK, and a, in line with the previous theorem, ranges across a system of representatives of (1 + a1m)+=E. Now one may notice that Shintani's unit theorem applies to E, for it is of nite index in o K. Therefore with R KR;+ = [ mi =1 [ "2E "Ci; one lets vi1; :::; vidi be a linearly independent set of generators of Ci in K, for each i 2 f1; 2; :::;mg, as in the proof of Shintani's unit theorem. One may multiply these 89 by an appropriate positive element in Z to allow one, without loss of generality, the assumption that vil lies in m for each l 2 f1; 2; :::; dig and i 2 f1; 2; :::;mg. De ne then C1 i = ft1vi1 + + tdividi j 0 < tl 1; for each i 2 f1; 2; :::; digg: De ne also R( ;Ci) = (1 + a1m)+ \ C1 i ; with a chosen as before. One then has the following theorem. Theorem 6.6 The sets R( ;Ci) are nite. Also, ( ; s) = 1 N(a)s Xm i=1 X x2R( ;ci) (Ci; x; s); where (Ci; x; s) = X z jN(x + z1vi1 + zdividi)js; and the latter sum sweeps over all dituples of nonnegative integers z = (z1; :::; zdi). Proof. One knows that the set R( ;Ci) is a bounded subset of a translation by one of the complete lattice a1m, and thus R( ;Ci) is nite. As Ci R KR;+ is the simplicial cone generated by vi1; :::; vidi 2 m, it follows that every element of (1 + a1m) \ Ci may, in fact, be written uniquely as a = Xdi l=1 ylvil where 0 < yl 2 Q for each l 2 f1; 2; :::; lg. With yl = xl + zl; 0 < xl 1, zl 2 Z, for each l 2 f1; 2; :::; dig, one has because Pdi l=1 zlvil 2 m that Xdi l=1 xlvil 2 1 + a1m: Therefore, every a 2 (1 + a1m) \ Ci may be written uniquely as a = x + Xdi l=1 zlvil 90 where x 2 R( ;Ci). And one has, from the Shintani theorem, that (1 + a1m)+ = [mi =1 [ "2E (1 + a1m) \ "Ci; and therefore, as a runs through this set modulo its Eorbits, one has ( ; s) = 1 N(a)s Xm i=1 X x2R( ;Ci) (Ci; x; s): 6.4 Siegel and Klingen's result on special values of Lfunctions This section establishes a result, due to Siegel and Klingen, about the values of certain Dirichlet Lfunctions at negative integers, via the work of Shintani. For the following result, one denotes by Q( ) the eld generated over Q by the values of . Theorem 6.7 If is a Dirichlet character for a totally real algebraic number eld K, then one has L( ;m) 2 Q( ) for m 2 f1; 2; 3; :::g. Proof. First, one notes that, by the rewriting of ( ; s) from the previous section, it su ces to show that the values of (Ci; x; s) lie in Q for each i 2 f1; 2; :::;mg and x 2 R( ;Ci). One may consider only the case where K is totally real, i.e., where every embedding of K into C is real. For if this is not the case, then one automatically has L( ;m) = 0 via the following argument. Denote as before by Fm the ray class eld modulo m. Via the Artin symbol of Chapter 4, one may view as a character of the Galois group G(FmjK). In this way, the function L( ; s) will agree with the Artin Lfunction L(FmjK; ; s) up to nitely many Euler factors at nite primes, and therefore is zero at s = m if and only if the associated Artin Lfunction is zero. It is clear from the functional equation for the Artin Lfunction that L(FjK; ;m) = 0 if K is not totally real. 91 Therefore, one may suppose that K is totally real. Consider the following setup. One de nes Lj(t1; :::; tn) = Xn i=1 ajiti and L i (z1; :::; zr) = Xr j=1 ajizj ; where the elements aji for j 2 f1; 2; :::; rg and i 2 f1; 2; :::; ng form a matrix A, where each aji is real and positive, and (x1; :::; xr) is an rtuple consisting of positive rational numbers. One may then de ne (A; x; s) = X1 z Yn i=1 L i (z + x)s; with the sum over rtuples of nonnegative integers as before. Also, one may let (A; x; s) = (Ci; x; s) as in the rewriting of the Dirichlet Lfunction of the previous section with the matrix A possessing columns equal to the representation of each basis element vil 2 m of Ci as an ntuple with entries consisting of its various embeddings into C because K is totally real. Thus, one proceeds to show that (A; x; s) is a rational number. To do this, one rst looks at (s)n = Z 1 0 Z 1 0 f Yn i=1 etig(t1 tn)s1dt1 dtn; and performs the substitution of ti with L i (z + x)ti, for each i 2 f1; 2; :::; ng. One then easily obtains for Re(s) > r n by summing over all nonnegative rtuples z of integers that (s)n (A; x; s) = Z 1 0 Z 1 0 g(t)(t1 tn)s1dt1 dtn with t = (t1; :::; tn), and g(t) = Yr j=1 exp((1 xj)Lj(t)) exp(Lj(t)) 1 : 92 One then de nes Di = ft 2 Rn j 0 tl ti; for each l 2 f1; 2; :::; ngnfigg; and one notices that (A; x; s) = (s)n Xn i=1 Z Di g(t)(t1 tn)s1dt1 dtn: One then transforms t within Di as t = (t1; :::; tn) = u(y1; :::; yn), where u > 0, 0 yl 1 for l 2 f1; 2; :::; ngnfig, and yi = 1. Therefore one has (A; x; s) = (s)n Xn i=1 Z 1 0 (Z 1 0 Z 1 0 g(uy)( Y l6=i yl)s1 Y l6=i dl ) uns1du; so that by the previous result, one has for Re(s) > r n that and for " su ciently small that (A; x; s) = G(s) Z I"(+1) (Z I"(1)n1 g(uy)uns1( Y l6=i yl)s1 Y l6=i dl ) du where G(s) = (s)n (e2 ins 1)(e2 is 1)n1 ; and I"(a) for a = 1 or1denotes a path in C consisting of the interval [a; "], succeeded by a circle in the counterclockwise direction, and then the interval ["; a], as in the previous theorem. One also notices that this yields a meromorphic continuation of the function (A; x; s) to the complex plane. At s = 1 k, G(s) is equal to (1)n(k1) (k)n n 1 (2 i)n by the basic properties of the function (s) as outlined in section 5.2. The quantity Xn i=1 1 (2 i)n Z 1 0 (Z 1 0 Z 1 0 g(uy)( Y l6=i yl)s1 Y l6=i dl ) uns1du at s = 1 k is the sum over each i 2 f1; 2; :::; ng of the coe cients of un(k1)+r(t1t2 tn)k1 93 in the Taylor expansion of the function ur Yr j=1 exp((1 xj)uLj(t)) exp(uLj(t)) 1 : Denote then di = 1 (2 i)n Z 1 0 (Z 1 0 Z 1 0 g(uy)( Y l6=i yl)s1 Y l6=i dl ) uns1du: One may notice that the elements of A lie in the Galois closure N of K over Q, and that 2 G(NjQ) permutes the elements in each column of the matrix A. Represent ing then as an element of Sn, one then has (di) = d (i) for each i 2 f1; 2; :::; ng. Therefore Pn i=1 di 2 Q. This proves the claim. 6.5 The value of Artin's Lfunction at negative integers In the case of a Galois extension FjK, the functional equation is restated for Artin's Lfunction: L (FjK; ; 1 s) L1 (FjK; ; 1 s) = W ( ) L (FjK; ; s) L1 (FjK; ; s) : With this functional equation, one may note the de nition of a critical point for this Lfunction as s 2 Z where L1 (FjK; ; s) and L1 (FjK; ; 1 s) are nite. One has immediately the following theorem. Theorem 6.8 The point s = m 2 Z with m 2 f1; 2; 3; :::g is critical if and only if L (FjK; ;m) 6= 0: Proof. Passing to the xed eld F of , one has that L (FjK; ; s) = L (F jK; ; s) and the proof reduces to proving the claim for L (F jK; ; s) : Suppose now that s = m is a critical point. Then by de nition the values L1 (FjK; ;m) and L1 (FjK; ; 1 + m) are nite. The lefthand side of Artin's functional equation as written above is nite and nonzero as the Lfunction L (FjK; ; s) must be nonzero 94 at s = 1 + m. Returning to Artin's functional equation, one has that the factor W ( ) is nonzero and nite, as well as the expression L1 (FjK; ;m) by assumption. Thus one must have L (FjK; ;m) 6= 0: Conversely, suppose that this Lfunction is nonzero. Of course, the term L1 (FjK; ; 1 + m) is nonzero and nite by its construc tion, and the term L (FjK; ; 1 + m) is nonzero and nite, as noted before. Thus, the lefthand side of the functional equation is now nonzero and nite, and thus so must the right be nonzero and nite. But W ( ) 6= 0 in any case, and L (FjK; ;m) 6= 0 by assumption. Therefore as the righthand side must be equal to the nite value of the lefthand side, one must have that the value of L1 (FjK; ; 1 + m) is nite. This proves the claim. Suppose now that K = Q, and that : G(FjQ) ! GL(V ) is a complex representation of its Galois group, with character . Passing to the xed eld F of , one observes that Theorem 6.9 One has that s = m with m 2 f1; 2; 3; :::g is a critical point for L (FjQ; ; s) if and only if one of the following is true: (i) m is odd and F is totally real; (ii) F is totally imaginary, the automorphism of complex conjugation is a central element in G F jQ , and ( ) = dim( ). Proof. Suppose that s = m is a critical point. Then by the previous theorem, one has that L (FjQ; ;m) 6= 0: One has that L (FjQ; ;m) = L F jQ; ;m : In that case, the functional equation yields that L1 F jQ; ;m 6= 0; whence, as by de nition, L1 F jQ; ;m = s=2 (s=2) N+ (s+1)=2 ((s + 1) =2) N ; 95 one has poles of this function if and only if m is odd and N = n 2 (1) 1 2 X p real ( P) = 0; or m is even and N+ = n 2 (1) + 1 2 X p real ( P) = 0: In the rst case, one has (1) = ( P) for each in nite prime P, whence acts trivially on each P, and so the generator of any decomposition group of an in nite prime in F is trivial. Thus F is totally real. In the second case, one has (1) = ( P) for each in nite prime P, whence all decomposition groups of in nite primes of F are nontrivial and F is totally imaginary. That the action of complex conjugation is central to the Galois group in this second case follows from Lemma 2.1, as ( ) = (1) = dim( ) and thus = I. For the converse, if (i) holds, then N = 0, whence L F jQ; ;m 6= 0 from the functional equation. If (ii) holds, then one must have N+ = 0, whence L F jQ; ;m 6= 0 from the functional equation. The previous theorem then shows that s = m is a critical point for L (FjQ; ; s) in either case. One then has the following theorem due to Coates and Lichtenbaum [CL]. Theorem 6.10 The values L (FjQ; ;m) for m 2 f1; 2; 3; :::g are algebraic inte gers contained in Q( ). Proof. One has observed that these critical points occur precisely when either the xed eld F of is totally real and m is odd, or the xed eld F is totally imaginary, m is even, and the action of complex conjugation is central to G(F jQ) and has ( ) = dim( ). One may restrict attention to such a point, for if m with 96 m 2 f1; 2; 3; :::g is not critical, then the functional equation implies that the Artin Lfunction L(FjQ; ; s) is zero at s = m. In the rst case, one may use Brauer's theorem to state that = Xl i=1 niIndG Hi ( i) where each Hi is a subgroup of G = G F jQ , each i is a character of degree one of G(FjQ), and each ni 2 Z. One has then L F jQ; ; s = Yl i=1 L F jKi; i; s ni where each Ki denotes the xed eld of the subgroup Hi. One may then observe that L F jKi; i;m 6= 0; for by inducing to F jQ and then restricting via the quotient map G F jQ ! G(FijQ) to the xed eld Fi of that induced character IndG Hi ( i) ; the result of the previous theorem implies, as Fi is totally real and m is odd, that L F jKi; i;m = L F jQ; IndG Hi ( i) ;m = L FijQ; IndG Hi ( i) ;m 6= 0: The work of Siegel and Klingen in the previous chapter then shows that L F jKi; i;m = X i ( ) a where a 2 Q and the sum is over the nitely many cosets of Pm Ki in Jm Ki , for an ap propriate module of de nition m for the xed eld of i. Thus for 2 G QjQ( ) ; one has thus that L F jKi; i;m = L F jKi; ( i) ;m , whence L F jQ; ;m = Yl i=1 L F jKi; i;m ni ! = Yl i=1 L F jKi; ( i) ;m ni 97 = L F jQ; ( ) ;m = L F jQ; ;m ; and thus L F jQ; ;m 2 Q( ), noting that ( i) is viewed for a particular i as a character of G F jKi via the Artin symbol Jm Ki=Pm Ki ! G F ;i jKi with F ;i here denoting the xed eld of the character i of G F jKi , for it has precisely the xed eld that i does. In the second case, it must be that ( ) = dim( ), and one may employ a special case of the Brauer theorem, as given by Serre [CL], yielding = Xl i=1 niIndG Hi ( i) where one may take each Hi to contain the center of G, and therefore the action of complex conjugation, where also i ( ) = 1. On 



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