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GENERALIZING THE THEOREM OF NAKAGAWA ON BINARY CUBIC FORMS TO NUMBER FIELDS By JORGE DIOSES Bachelor of Science in Mathematics Ponti cal Catholic University of Peru Lima, Lima, Peru 1997 Licentiate in Mathematics Ponti cal Catholic University of Peru Lima, Lima, Peru 2000 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial ful llment of the requirements for the Degree of DOCTOR OF PHILOSOPHY July, 2012 COPYRIGHT c By JORGE DIOSES July, 2012 GENERALIZING THE THEOREM OF NAKAGAWA ON BINARY CUBIC FORMS TO NUMBER FIELDS Dissertation Approved: Dr. David Wright Dissertation Advisor Dr. Heidi Ho er Dr. Anthony Kable Dr. Chris Francisco Dr. Sheryl Tucker Dean of the Graduate College iii TABLE OF CONTENTS Chapter Page 1 Introduction 1 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Basic notation and review of zeta functions of binary cubic forms 13 2.1 Notation for number elds and local elds . . . . . . . . . . . . . . . 13 2.2 Binary cubic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Zeta functions of binary cubic forms and Dirichlet series . . . . . . . 19 2.4 Fourier transforms and the dual Dirichlet series . . . . . . . . . . . . 23 3 Local integrals of a Fourier transform 29 3.1 Statement of the integral to be calculated . . . . . . . . . . . . . . . 29 3.2 Reductions of the local integral . . . . . . . . . . . . . . . . . . . . . 30 3.3 Evaluation of the local integral . . . . . . . . . . . . . . . . . . . . . 33 3.3.1 Type (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.2 Type (2u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.3 Type (2r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3.4 Type (3u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3.5 Type (3r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4 Veri cation of a simple identity . . . . . . . . . . . . . . . . . . . . . 41 4 Residues of the Dirichlet series and generalizing OhnoNakagawa 43 4.1 Filtrations of the Dirichlet series . . . . . . . . . . . . . . . . . . . . . 43 iv 4.2 Poles and residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.3 Residue of the dual Dirichlet series at s = 1 . . . . . . . . . . . . . . 48 4.4 Residue of the dual Dirichlet series at s = 5=6 . . . . . . . . . . . . . 50 4.5 Generalizing Ohno's conjecture . . . . . . . . . . . . . . . . . . . . . 52 5 Decomposing the Dirichlet series according to the resolvent eld 57 5.1 The resolvent eld of an extension k0=k of degree at most 3 . . . . . . 57 5.2 Conductors and discriminants of cubic extensions . . . . . . . . . . . 59 5.3 The resolvent eld identity . . . . . . . . . . . . . . . . . . . . . . . . 62 6 Examples of the resolvent OhnoNakagawa identity 66 6.1 The nite OhnoNakagawa identity . . . . . . . . . . . . . . . . . . . 66 6.2 Resolvent identities over Q . . . . . . . . . . . . . . . . . . . . . . . . 67 6.3 Resolvent identities over Q(i) . . . . . . . . . . . . . . . . . . . . . . 74 7 Expressing the resolvent Dirichlet series as sums of idele class group characters 91 7.1 Simpli cation of the generalized OhnoNakagawa conjecture . . . . . 91 7.2 Shintani's Dirichlet series as sums of idele class group characters . . . 93 BIBLIOGRAPHY 99 v LIST OF TABLES Table Page 6.1 Fields of degree 3 unrami ed outside 2,3. . . . . . . . . . . . . . . 85 6.2 Number of elds (up to conjugacy) of degree 3 unrami ed for primes > p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.3 Polynomials generating quadratic extensions of Q(i) unrami ed outside 1 + i, 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.4 Sextic polynomials generating cubic extensions of Q(i) unrami ed out side 1 + i, 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.5 Cubic polynomials generating cubic extensions of Q(i) unrami ed out side 1 + i, 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.6 Splitting type of 2 in k0=Q, and corresponding type of (1+i)Z[i] in k0=k. 89 6.7 Splitting type of 3 in k0=Q, and corresponding type of 3Z[i] in k0=k. . 90 vi CHAPTER 1 Introduction 1.1 Overview A fundamental problem in number theory is to describe in as precise a manner as possible the collection of algebraic number elds, meaning the extensions of nite degree of the rational number eld Q. For example, all the quadratic extensions may be precisely described as Q( p d) where d ranges over all squarefree integers not equal to 1. One corollary of this is that there is a simple correspondence between elds of positive discriminant Q( p d) and elds of negative discriminant Q( p d). Generalizing these simple statements to even just cubic elds is highly nontrivial. For instance, the smallest negative discriminant of an extension of degree 3 of Q is 23, corresponding to the polynomial x3 x + 1, while the smallest positive discriminant is 49 corresponding to x3 x2 2x + 1. Tables of discriminants of cubic elds, both positive and negative, were calculated in the late nineteenth and early twentieth century, and there was no apparent correlation between the two lists of positive and negative discriminants. In [14] Ohno found a correspondence between these elds, which was most easily explained in terms of class numbers of integral binary cubic forms. This correspondence was stated as a conjecture which was later proved in [13] by Nakagawa using class eld theory. The goal of this thesis is to investigate possible generalizations of this result to the case of quadratic and cubic extensions of an arbitrary number eld. It relies on the work of Datskovsky and Wright in [5] where the original de nitions are extended to global elds of characteristic not equal to 2 or 3. 1 In the second section of Chapter 1, we revise the original statements of the theorem of Nakagawa when the base eld is Q. We also state in simple terms a conjecture that would generalize this result to any number eld. The conjecture is shown as an equation between Dirichlet series, considered as sums over number elds of degree at most 3. We present in Chapter 2 the basic de nitions and notations for number elds, binary cubic forms, zeta functions, and Dirichlet series. We also de ne the appropriate Haar measures that are needed throughout this thesis. In Chapter 3, we evaluate certain local integrals that play an important role in the generalization process. These local integrals de ne the Dirichlet series of one of the sides of our conjecture. We work under the assumption that 3 is unrami ed in the given number eld. The calculation of the residues at its poles of the Dirichlet series mentioned above is given in Chapter 4. We use a technique given by Datskovsky andWright to accomplish this. Based on this, we are able to give a justi cation for our conjecture. In Chapter 5, we introduce additional de nitions that allow us to decompose our conjectured identity into a family of identities between sums over elds. Each side of these identities corresponds to a special eld. We also introduce some terminology and review some basic facts of class eld theory. They will be invoked in the last chapter of this thesis. We test numerically the validity of the conjecture in Chapter 6. In order to do so, we make use of available tables of cubic and quadratic elds. We do this for the quadratic imaginary eld Q(i). This provides strong evidence for the proposed identity. We also verify Nakagawa's theorem. Finally, in Chapter 7, we reduce both sides of our conjectured identity to sums over characters of an idele class group. We proved some results mentioned in the rst chapter. Using the results from Chapter 5, we express these Dirichlet series as sums 2 containing only ideles. 1.2 Statement of results In 1972, Shintani created the theory of zeta functions associated to the space of binary cubic forms in the paper [17], and used that theory to establish the basic analytic properties of two Dirichlet series 1(s) = X1 m=1 h1(m) + 1 3h2(m) ms 2(s) = X1 m=1 h(m) ms (1.1) where h(m) denotes the number of SL2(Z)equivalence classes of integral binary cubic forms of discriminant m, and, for m > 0, h1(m) and h2(m) denote the numbers of classes of discriminant m with isotropy group of order 1 and 3, respectively. Shintani showed that these Dirichlet series converge absolutely for Re(s) > 1, that they have meromorphic continuations to the entire splane, and that they satisfy a functional equation. In addition, he proved that they were holomorphic except for simple poles at s = 1 and s = 5=6, and he gave formulas for the residues of the Dirichlet series at these poles. With this information, Shintani was able to improve a theorem of Davenport [7] on the meanvalue of classnumbers of integral binary cubic forms, by giving a more precise formula for the error term. The functional equation that Shintani discovered actually expresses 1(1s) and 2(1 s) as linear combinations of the dual Dirichlet series ^ 1(s) = X1 m=1 ^h1(m) + 1 3 ^h 2(m) ms ^ 2(s) = X1 m=1 ^h (m) ms where the dual classnumbers are the numbers of SL2(Z)equivalence classes of integral binary cubic forms Fx(u; v) = x1u2 + x2u2v + x3uv2 + x4v3 where the middle coe cients x2, x3 are both divisible by 3. This turns out to be the natural dual lattice to the lattice of integral binary cubic forms. 3 Datskovsky andWright introduced adelic terminology and notation into Shintani's work in the papers [19, 5, 6, 4], and generalized Shintani's results to the space of binary cubic forms over an arbitrary global eld of characteristic di erent from 2 and 3. In [5], it was observed in Proposition 4.1 that Shintani's functional equation has a natural diagonalization. To state this diagonalization, we write (s) = 1(s) 1 p 3 2(s) ^ (s) = ^ 1(s) 1 p 3 ^ 2(s) We de ne the associated gamma factors to be r (s) = 2s 33s 2s (s) s 2 + 1 4 1 6 s 2 + 1 4 1 3 : The diagonalized functional equations are then r (1 s) (1 s) = 3 r (s)^ (s) for either choice of sign . In [14], Ohno observed that these diagonalized functional equations would be especially symmetric if there were an identity between the original Dirichlet series and the dual Dirichlet series. This suggested an identity between classnumbers for the lattice of integral forms and classnumbers for its dual lattice. By extensive computation of classnumbers, Ohno veri ed numerically the conjecture that for the original Shintani series (1.1) ^ 1(s) = 33s 2(s) ^ 2(s) = 313s 1(s) These identities imply that ^ (s) = 3 1 2 3s (s): With that identity, the diagonalized functional equations become " (1 s) (1 s) = " (s) (s) 4 where " (s) = 2s 3 3 2 s 2s (s) s 2 + 1 4 1 6 s 2 + 1 4 1 3 : Ohno also proved by means of Shintani's functional equation that his two conjectured identities above are logically equivalent. Once Ohno's conjecture was stated, work on the truth of it was swift, and Nakagawa proved the conjecture in 1998 in [13]. Nakagawa's proof is based partly on an idea of Scholz [16] relating the 3class group of quadratic elds Q( p d) to the 3class group of Q( p 3d). The research presented here is concerned with the generalization of Ohno's con jecture and Nakagawa's theorem to Shintani Dirichlet series for the space of binary cubic forms over a number eld k. Since the ring of integers o of the number eld k need not have class number 1, the direct generalization of Shintani's Dirichlet series to number elds is more easily expressed in terms of eld extensions k0=k of degree at most 3 than it is in terms of classnumbers of binary cubic forms. To explain this, we rst present an identity proved in [5] for Shintani's original series. Let (s) denote Riemann's zeta function, and k(s) the Dedekind zeta function of the number eld k. Let dk denote the absolute value of the discriminant of k=Q. For a eld k of degree at most 3 over Q, we de ne o(k) = 6 if k = Q and o(k) = [k : Q] otherwise. We also de ne Rk(s) = 8>>>>>>< >>>>>>: (s)3 if k = Q; (s) k(s) if [k : Q] = 2; k(s) if [k : Q] = 3: Then Shintani's Dirichlet series are proved in [5] to be equal to 1(s) = 2 (4s) (6s 1) X k tot. real ds k o(k) Rk(2s) Rk(4s) 2(s) = 2 (4s) (6s 1) X k complex ds k o(k) Rk(2s) Rk(4s) where the rst series ranges over totally real k=Q of degree at most 3, and the sec ond series ranges over k=Q with one complex in nite place and degree at most 3. 5 Nakagawa also provided a proof of this identity in [12]. We can directly use this terminology to state the generalization of Shintani's series to number elds k. First, the series range over the extensions k0=k of degree at most 3. Again we de ne o(k0) = 6 if k0 = k and o(k0) = [k0 : k] otherwise. De ne dk0=k to be the absolute norm of the relative discriminant of the extension k0=k, and put Rk0(s) = 8>>>>>>< >>>>>>: k(s)3 if k0 = k; k(s) k0(s) if [k0 : k] = 2; k0(s) if [k0 : k] = 3: To state the generalized Shintani series over a number eld k, it remains to de ne the notion of signature of an extension k0=k of degree less or equal to 3. For each real place v (or real embedding) of k, either the places of k0 lying over v are either all real, in which case we say v(k0=k) = +, or there is a unique complex place w lying over v, and then we say v(k0=k) = . This situation is unique to extensions of degree at most 3. We call the vector (k0=k) = ( v(k0=k))v real the signature of k0=k. If r1 is the number of real places of k, then there are 2r1 possible signatures; we denote the set of such signatures by A. For each possible signature of k, we denote the set of extensions k0=k of degree at most 3 which have signature by K . At last, we may state the Shintani series for each signature 2 A as (s) = k(4s) k(6s 1) X k02K ds k0=k o(k0) Rk0(2s) Rk0(4s) (1.2) where the sum ranges over all extensions k0 in K . (Note that the factor of 2 has disappeared; in [19, 5] forms are considered equivalent relative to the group GL2 rather than SL2, and that accounts for the factor of 2.) Datskovsky and Wright prove in [5] these series extend to meromorphic functions of s which are holomorphic everywhere except for simple poles at s = 1 and s = 5=6. That paper also provides precise expressions for the residues of these series at 1 and 5=6. Finally, that paper 6 also proves a functional equation expressing (1s) as linear combinations of dual Dirichlet series ^ (s). By utilizing the adelic approach of [5], an expression nearly identical to equation (1.2) may be derived for the dual Dirichlet series ^ (s). (Note: To be consistent with Nakagawa's theorem and Shintani's original notation, we modify the de nition of ^ (s) in DatskovkyWright by a constant factor, to be explained in Chapter 2.) The only di erences are due to new local factors at primes v lying over 3. These new local factors were calculated for Q by Nakagawa in Lemma 3.6, p. 121, of [13]. In order to give the formula for ^ (s), we rst need to de ne the splitting type of the prime v of k in an extension k0 of degree at most 3: Type (1): k0 k kv = k3 v, or k2 v, or kv; Type (2u): k0 k kv = kv F, or F, where F=kv is quadratic unrami ed; Type (2r): k0 k kv = kv F, or F, where F=kv is quadratic rami ed; Type (3u): k0 k kv = F where F=kv is cubic unrami ed; Type (3r): k0 k kv = F where F=kv is cubic rami ed: Then we use the technique of DatskovskyWright to prove the following: Theorem 1.1 Let k be a number eld of degree n in which 3 is unrami ed. Then the dual Dirichlet series for each signature over k may be expressed as ^ (s) = k(4s) k(6s 1) X k02K ds k0=k o(k0) Rk0(2s) Rk0(4s) Y vj3 Tk0;v(s) (1.3) 7 where for each prime v j 3 we have qv = j3j1 v , and we de ne the rational functions Tk0;v(s) = 8>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>: q4s v 1 + q12s v + 2q14s v (1 + q2s v )2 if v is of type (1) in k0; q4s v 1 + q12s v 1 + q4s v if v is of type (2u) in k0; q2s v 1 + q14s v 1 + q2s v if v is of type (2r) in k0; q4s v 1 + q12s v q14s v 1 q2s v + q4s v if v is of type (3u) in k0; 1 if v is of type (3r) in k0: The proof of this theorem appears in Chapter 3. We expect that a similar theorem holds when 3 is rami ed in k, but we have not yet completed the necessary local calculations. The goal of this thesis is to relate the collection of dual Dirichlet series ^ (s) to the collection of original series (s). Just as in Nakagawa's theorem, it will emerge that the proper generalization relates ^ (s) to (s), where the signature is the negative of in the sense that for each real place v of k we have v = + if and only if ( )v = . At the end of this introduction, we shall explain why this involution should occur. Datskovsky and Wright proved that the series (s) and ^ (s) have meromorphic continuations to the entire splane which are holomorphic except for simple poles at s = 1 and s = 5=6, and they gave explicit formulas for the residues of (s) at 1 and 5=6. In order to test the relationship between ^ (s) and (s), in Chapter 4 we use the results of DatskovskyWright to calculate similar formulas for the residues of ^ (s) at 1 and 5=6. Based on those formulas, we prove the following: Theorem 1.2 Let k be a number eld of degree n with r1 real places and r2 complex places. For every signature over k for which we have m real places v with v = +, the identity ^ (s) = 3r2+m3ns (s) 8 is true at s = 1 and 5=6. Moreover, this is the only expression of the form 3A+Bs for which this theorem is true. Notice that this theorem makes no condition on whether 3 is rami ed or not in k. That led us to conjecture this generalization of the OhnoNakagawa identity: Conjecture 1.1 (Generalized Ohno Conjecture) Let k be a number eld of degree n with r1 real places and r2 complex places. For every signature over k for which we have m real places v with v = +, we have: ^ (s) = 3r2+m3ns (s) (1.4) To see how this is consistent with Nakagawa's theorem, in that case we have n = 1, r1 = 1 and r2 = 0. Then in this conjecture + corresponds to 1 in Shintani's notation, and corresponds to 2. For signature = +, we have m = 1 and thus ^ 2(s) = 313s 1(s), while for = we have m = 0 and ^ 1(s) = 33s 2(s). This is precisely Nakagawa's theorem. The remainder of this thesis is dedicated to the reduction of this conjecture to manageable pieces which we hope may be proved by class eld theory and the ideas of Nakagawa. First, by direct substitution of equations (1.2) and (1.3) into conjecture (1.4), we see that a number of factors directly cancel out, leaving only X k02K ds k0=k o(k0) Rk0(2s) Rk0(4s) Y vj3 Tk0;v(s) = 3r2+m3ns X k02K ds k0=k o(k0) Rk0(2s) Rk0(4s) (1.5) In examining this proposed identity, we see that the Euler products multiplying ds k0=k are all of the form X a ca N(a)2s where a ranges over the integral ideals of k, N(a) denotes the absolute norm of a, and the coe cients ca are ordinary rational numbers. The important point here is the factor 2 in the exponent. Since 33ns = N(3o)3s for the ideal generated by 3 in k, the di erence must be made up in the norms of the relative discriminants dk0=k. To be more precise, consider an extension k01 =k counted on the left side of the identity with a corresponding term equal to a constant multiple of 9 dk01 =k N(a1)2 s . This may be `counteracted' by a term dk02 =k N(3o)3 N(a2)2 s on the right side if and only if the ratio of norms of the relative discriminants dk02 =k=dk01 =k is divisible by an odd power of N(3o). This is one suggestion that the conjectured identity (1.5) may be split into a sequence of identities by restricting the elds k0 included in the summations on each side. That idea is also promoted by the one of the key ideas of Nakagawa's proof, which in turn was based on a theorem of Scholz [16]. For an extension k0=k, let L=k be its Galois closure. If k0=k has degree at most 3, then L=k contains a unique subextension F = k( p ) of degree at most 2, generated by the square root of the discriminant of any generating element of k0 over k. We call F the resolvent eld of k0=k. Note that F = k if k0=k is trivial or a cyclic cubic extension. We de ne the dual resolvent eld to F to be ^ F = k( p 3 ). Thus, F and ^ F are sub elds of the extension F( p 3) obtained by adjoining the cube roots of unity to F. The relative discriminants of F=k and ^ F=k are, up to multiplication by the square of an ideal of k, equal to and 3 , respectively. It will turn out that the odd power of 3 in the identity (1.5) will be accounted by restricting the terms on the left to those extensions k0=k with resolvent eld equal to ^ F and the terms on the right to those k0=k with resolvent eld F. This motivates the following re ned conjecture: Conjecture 1.2 (Resolvent Field Identity) Let k be a number eld of degree n with r1 real places and r2 complex places. Let be a nonzero element of k, and de ne C ( ) to be the set of extensions k0=k of degree at most 3 whose resolvent eld is k( p ). Let m be the number of real embeddings of k for which the image of is a positive number. Then, using the notation for Shintani's series which we have previously developed, we have the identity X k02C (3 ) ds k0=k o(k0) Rk0(2s) Rk0(4s) Y vj3 Tk0;v(s) = 3r2+m3ns X k02C ( ) ds k0=k o(k0) Rk0(2s) Rk0(4s) (1.6) Since the generalized Ohno conjecture is just the sum of all these resolvent eld 10 identities for ranging over the nonzero elements of k modulo multiplication by squares (we will denote this set by k =k2), we may at least state: Theorem 1.3 If the resolvent eld identity is true for all nonzero in k, then the generalized Ohno conjecture (1.4) is true for k. For the one known case k = Q, we shall explain in detail why the converse is true, thus proving: Theorem 1.4 For k = Q, the resolvent eld identity is true for all nonzero in Q. The last part of this thesis is devoted to the analysis necessary to eventually complete the proof of the resolvent eld identity in terms of class eld theory. For nonzero 2 k, the elds k0 2 C ( ) have Galois closure L containing F = k( p ). If [k0 : k] 2, then F = k0. If [k0 : k] = 3, then L is a cyclic cubic extension of F. If k0=k is cyclic cubic, then L = k0 and F = k. If k0=k is noncyclic cubic, then L is an S3extension of k containing F, and L contains three cubic extensions of k all conjugate to k0. Class eld theory implies the abelian Galois extensions of a number eld F correspond to the open subgroups (and hence characters) of the idele class group JF of F. Basic notation and statements of class eld theory will be provided in a later section. In this case, the cyclic cubic extensions L=F correspond bijectively to the open subgroups of index 3 in the idele class group JF . These open subgroups in turn correspond in a onetotwo way to nontrivial complex characters of JF satisfying 3 = 1. These observations will allow us to rewrite the resolvent eld series in Theorem 1.3 as a sum over idele class characters of order dividing 3. To state this precisely, for any resolvent eld F let X(F) denote the group of characters of the idele class group JF of F such that 3 = 1 and also NF=k = 1 if [F : k] = 2, where NF=k denotes the relative norm. If F = k, the latter condition is omitted. The conductor of such a character is an integral ideal f in F. Let N(f ) denote the absolute norm of the conductor of . Then for the extension k0=k 11 corresponding to the kernel of , we have dk0=k = dF=k N(f ). The idele class character induces a character on all prime ideals P of F where P  f . By abuse of notation, we will write this character's value as (P). For prime ideal factors of the conductor, we will extend this de nition by setting (P) = 0. Then we will prove: Theorem 1.5 Let k be a number eld, be a nonzero element of k, and F = k( p ). Then, using the notation we have previously developed for the collection C ( ) of ex tensions k0=k and for the character group X(F), we have the identity X k02C ( ) ds k0=k o(k0) Rk0(2s) Rk0(4s) = ds F=k o(F) X 2X(F) N(f )s Y P 1 + (P) N(P)2s (1.7) where o(F) = 3 if F = k and 1 otherwise, and the product is taken over all prime ideals of F. We also will produce a similar expression for the dual discriminant eld series as a sum over cubic characters of the idele class group J^ F , although this is complicated by the presence of the Euler factors Tk0;v for places vj3. The cubic characters of JF and J^ F are related to the cubic characters of the compositum F ^ F. Using the fact that the compositum F ^ F contains the cube roots of unity, Scholz was able to use this relationship to deduce a relationship between the threeclassnumbers of F and ^ F in [16]. The completion of our project and the proof of the generalized Ohno conjecture at least for elds k where 3 is unrami ed would follow from re ning Scholz' ideas to establish the resolvent eld series identity for all nonzero in k. Our future research will be directed toward providing this proof. 12 CHAPTER 2 Basic notation and review of zeta functions of binary cubic forms 2.1 Notation for number elds and local elds Our notation for number elds, local elds, adele rings and idele rings, etc., will be fairly consistent with that presented in [19, 5, 6] and somewhat with Weil [18]. Let k be a number eld of degree n over the rational number eld Q. For any place v of k, we use kv to denote the completion of k at the place v. Let r1 and r2 be the numbers of real and complex, respectively, places of k, i.e. places such that kv = R and C, respectively. Then r1 + 2r2 = n. These together form the set of in nite places of k, and we shall write v j 1 for these places. For all local elds kv, we normalize the absolute value j jv to be the modulus of any additive Haar measure on kv. This means that (aU) = jajv (U) for any open subset U of kv, any Haar measure of kv, and any nonzero element a of kv. For real places v, this means jxjv is the customary absolute value on R, while for complex places v this means jxjv is the square of the usual absolute value on C. For real places v, we choose the Haar measure dvx on kv such that R 1 0 dvx = 1, i.e. the usual Lebesgue measure on R. For complex places v, we choose dvx so that the measure of the unit circle fx 2 C : jxjv 1g is 2 . That means dvx is twice the usual Lebesgue measure on C; the reason for this choice is that it is more convenient for dvx to represent the di erential form jdx ^ dxj in integration formulas. For any nite place v of k, we write v  1, and we denote the maximal compact subring of kv by ov. The unique prime ideal of ov is denoted pv, and we choose a generator of the principal ideal pv and name it v, called a uniformizer of kv. The 13 modulus qv of kv is the order of the nite eld ov=pv. The additive Haar measure dvx on kv is normalized so that R ov dvx = 1. The absolute value jajv on kv is normalized so that dv(ax) = jajvdvx, which implies that j vjv = q1 v . If we work with a nonarchimedean local eld K without reference to a global eld k, we shall denote the maximal compact subring by O, the unique prime ideal by P, a uniformizer by , the modulus by q, the normalized Haar measure by dx, and the normalized absolute value by j j. For any ring R (always commutative with identity), we denote the subgroup of invertible elements by R . Hence, for a eld K, K denotes the nonzero elements of K. Thus, the unit subgroup of ov is denoted by o v , and this is the same as the subgroup of elements x satisfying jxjv = 1. For a number eld k, we denote the ring of adeles by A = Ak = Q0 v kv and the group of ideles by A k = Q0 v k v , where these are restricted direct products in the usual sense. The ideles correspond to units in the ring of adeles, but the restricted product topology is not the same as the subspace topology. The idele norm jajA of an idele a 2 A k is the modulus of multiplication by a relative to any Haar measure on the adeles Ak. This means jajA = Q v javjv, where av denotes the component of a at the place v. For all x 2 k embedded along the diagonal in A k , we have the product formula jxjA = 1. We choose for the Haar measure on the additive group of adeles Ak the restricted tensor product measure dAx = N v dvxv, where x = (xv)v. The number eld k embeds along the diagonal as a discrete subgroup of A such that A=k is compact. With this choice of measure, the induced measure of the quotient is R A=k dAx = d1=2 k , in terms of the absolute value of the discriminant of k (see Prop. V.4.7 in [18]). On the multiplicative group k v of nonzero elements, we choose Haar measures d vx = dvx jxjv if v is an in nite place, and for nite places we choose d v x so that the unit group o v has measure 1. On the ideles A k , we choose the Haar measure 14 d A x = N v d v xv. By Prop. V.4.9 of [18], the measure of the set C(m), the image in A k =k of all ideles x satisfying 1 jxjA m, is k logm with k = 2r1(2 )r2hkRk ek ; where hk is the class number of k, Rk is the regulator of k, and ek is the number of roots of unity in k. Let A1 denote the subgroup of ideles x with idele norm jxjA = 1. Then k is a subgroup of A1, and the quotient group A1=k is compact. (See Theorem IV.4.6 in [18].) There is an embedding of the group of positive real numbers R+ into the ideles A , which we will denote z(t), satisfying jz(t)jA = t for all t 2 R+. One such embedding is de ned as the idele z(t) = (zv(t))v where zv(t) = 1 for all nite places v and zv(t) = t1=n for all in nite places v, with n = [k : Q]. Then we may decompose the Haar measure for A in the following way Z A (x) d A x = Z 1 0 Z A1 (z(t)x) d1 Ax dt t ; for any integrable function . Then the measure of A1=k induced by d1 Ax is k. Finally, we introduce our notation for the Dedekind zeta function k(s). Let o = ok denote the ring of integers of k, and for each integral ideal a of o let N(a) denote the absolute norm of a, i.e. the cardinality of the quotient ring o=a. The Dedekind zeta function is k(s) = X a N(a)s = Y p 1 N(p)s 1 where the sum extends over all integral ideals a and the product extends over all prime ideals p. Equivalently, this may be written as a product over all nite places of v as k(s) = Y v1 (1 qs v )1: This zeta function converges locally uniformly for Re(s) > 1 and has a meromorphic continuation to the entire splane which is holomorphic except for a simple pole at 15 s = 1 with residue Res s=1 k(s) = k d1=2 k : These facts and more may be found in VII.6 of [18]. 2.2 Binary cubic forms This notation is based on [17, 19, 5]. A binary cubic form is an expression Fx(u; v) = x1u3+x2u2v+x3uv2+x4v3 where xj , 1 j 4 are the coe cients. We will generally think of a fourdimensional vector x = (x1; x2; x3; x4) as a binary cubic form. The module of binary cubic forms with coe cients in a ring R is denoted by VR. There is a natural representation of the group G = GL2 on V given by Fg x(u; v) = 1 det g Fx 0 B@ (u; v) 0 B@ a b c d 1 CA 1 CA for x 2 V and g = 0 B@ a b c d 1 CA 2 G. We say two forms x; y are Gequivalent if y = g x for some g 2 G. The discriminant of the form x is a homogeneous polynomial P(x) of degree 4 (see p. 35 in [5]). This polynomial satis es P(g x) = (det g)2P(x) for g 2 G and x 2 V . A binary cubic form x is de ned to be singular if and only if P(x) = 0. We denote the hypersurface of singular forms in V by S, and the subset of nonsingular forms by V 0. Both subsets are Ginvariant. Just as in Shintani [17], we de ne the bilinear form [x; y] = x1y4 1 3 x2y3 + 1 3 x3y2 x4y1 on V . For the involution g = (det g)1g on G, this form satis es [g x; g y] = [x; y] for all x; y 2 V and g 2 G. We next turn to binary cubic forms over a eld K. The splitting eld K(x) of a binary cubic form x 2 VK is the smallest extension of K (in a given algebraic closure 16 of K) for which the form factors into linear factors de ned over K(x). The splitting eld is either K, a quadratic or cubic cyclic extension of K, or an S3extension of K. The key connection between the space of binary cubic forms and eld extensions of K is the following fact: Proposition 2.1 Two nonsingular binary cubic forms x; y 2 VK are GKequivalent if and only if their splitting elds are the same K(x) = K(y). When we wish to refer to the points of G and V de ned over a ring R, we will write GR and VR, for example, Gk, Vk, Gkv , Vkv , GA, VA, etc. We shall choose the Haar measure dvx on Vkv to be simply the product of the four coordinate measures dvxj and similarly for the measure dAx on VA. We should point out that this choice is di erent from that of DatskovskyWright [5], where the selfdual measure was chosen relative to a standard biinvariant for on VA. We are changing this choice so that the de nition of the dual Dirichlet series will be in agreement with the papers of Shintani, Ohno, and Nakagawa in the case of k = Q. For a local eld K, we choose the invariant measure dg on g 2 GK in the same manner as in [5]. To review, we de ne UK to be the maximal compact subgroups the orthogonal group O(2) if K =R, the unitary group U(2) if K = C, and the group GO if K is nonarchimedean with maximal compact subring O. We de ne BK to be the Borel subgroup of lower triangular matrices in GK. Then by the Iwasawa decomposition GK = UKBK. We choose the invariant measure du on u 2 UK so that UK has measure 1. We de ne a rightinvariant measure db on b 2 Bk satisfying Z BK (b) db = Z K d t1 Z K d t2 Z K dc (n(c)a(t1; t2)) where the Haar measures on K and K are as previously de ned and we use the notation n(c) = 0 B@ 1 0 c 1 1 CA and a(t1; t2) = 0 B@ t1 0 0 t2 1 C A. Here is any integrable function 17 on GK. Then the measure dg on GK is de ned by Z GK (g) dg = Z UK du Z BK db (ub): In the case of a nonarchimedean eld K with maximal compact subring O, if is the characteristic function of the maximal compact subgroup GO, then Z GK (g) dg = Z UK du Z BO db = Z BO db = Z O d t1 Z O d t2 Z O dc = 1; by our choice of Haar measure on K and K . This con rms this measure satis es the condition that GO has measure 1. On the adelizations VA and GA, we take the invariant measures to be the restricted product measures dAx = N v dvxv and dAg = N v dvgv. Then Vk is a discrete cocom pact subgroup of VA such that R VA=Vk dAx = d2 k. On the quotient group GA=Gk, the set F(m) of all points g with 1 j det gjA m has measure k logm, where k = 2 r1 hkRk ek d3=2 k k(2): (This constant comes from the volume calculation after Prop. 6.3, p. 528, in [19], and the volume normalization on p. 66 in [5].) Let G1 A denote the subgroup of all g 2 GA with j det gjA = 1. Then Gk is a discrete subgroup of GA with quotient G1 A=Gk of nite invariant volume. De ne an embedding w(t) of t 2 R+ into GA by w(t) = a(z( p t); z( p t)), using the embedding z : R+ ! A de ned in the previous section. Then we have j detw(t)jA = t, and we may decompose the invariant volume on GA as follow Z GA=Gk (g) dAg = Z 1 0 Z G1 A=Gk (w(t)g) d1 Ag ! dt t ; for integrable functions (g) on GA=Gk. With this de nition, we see that the measure of G1 A=Gk with respect to d1 Ag is k. 18 Finally, we shall describe the orbits of nonsingular binary cubic forms over a local eld kv. By Proposition 2.1, these correspond to the possible splitting elds kv(x) over kv. For a complex place v, there is only one such splitting eld, namely, C, and thus there is only one nonsingular orbit. For a real place v, the splitting eld may be R or C corresponding to whether the discriminant P(x) is positive or negative in kv. For a nite place v, there are nitely many possible splitting elds kv(x) which we group into ve basic types: Type (1): kv(x) = kv, Type (2u): kv(x)=kv is quadratic unrami ed, Type (2r): kv(x)=kv is quadratic rami ed, Type (3u): kv(x)=kv is cubic unrami ed, Type (3r): kv(x)=kv is the Galois closure of a rami ed cubic extension Kv=kv. In the rst four types, we abbreviate Kv = kv(x), while in the last type Kv is a possibly nonGalois cubic extension whose Galois closure is kv(x). There is only one orbit of each type (1), (2u) and (3u), while there may be more than one orbits of types (2r) and (3r). We shall adopt the description of these given on pp. 3536 in [5]. In particular, for each orbit we shall choose orbital representatives x 2 Vkv as described in that paper. Those are arranged so that P(x ) is a relative discriminant (Kv=kv) of Kv=kv. Occasionally, we shall use Av to denote the set of orbits GkvnV 0 kv . 2.3 Zeta functions of binary cubic forms and Dirichlet series The adelic version of Shintani's zeta function associated to the space (G; V ) of binary cubic forms is Z(!; ) = Z GA=Gk !(det g) X x2V 0 k (g x) dAg; where ! is a quasicharacter on A k which is trivial on k and (x) is a SchwartzBruhat function on the adelization VA. Here Gk is a discrete subgroup embedded along the 19 diagonal in GA, and for every nonsingular form x 2 V 0 k the stabilizer subgroup Gk;x is a nite subgroup of order 1, 2, 3 or 6, depending on the splitting eld k(x) as described in [5]. Here we shall simplify the notation because we will not be referring to nonprincipal quasicharacters. We shall put !(x) = jxj2s A , and de ne Z(s; ) = Z GA=Gk j det gj2s A X x2V 0 k (g x) dAg; which is proved in [19] to be absolutely and locally uniformly convergent for Re(s) > 1, and to have a meromorphic continuation to the entire splane which is holomorphic except for simple poles at s = 1 and s = 5=6. By rewriting the inner summation over Gkequivalence classes and then exchanging summation and integration, we obtain Z(s; ) = X x2GknV 0 k 1 jGk;xj Z GA j det gj2s A (g x) dAg; where x ranges over representatives of the Gkequivalence classes of nonsingular forms in V 0 k. Assuming the SchwartzBruhat function = v v is of product form, the integral above has an Euler product Z GA j det gj2s A (g x) dAg = Y v Z Gkv j det gvj2s v v(gv x) dvgv: For each place v, the form x belongs to one v of nitely many orbits in V 0 kv over kv. Thus, there is an element gx;v 2 Gkv such that x v = gx;v x, using the standard orbital representatives mentioned in Section 2.2 (see page 67 of [5]). Note that P(x v ) = P(gx;v x) = (det gx;v)2 P(x). Then the Euler factor may be rewritten by substitution as Z Gkv j det gvj2s v v(gv x) dvgv = Z Gkv j det gvj2s v v(gvg1 x;v x v ) dvgv = j det gx;vj2s v Z Gkv j det gvj2s v v(gv x v ) dvgv = jP(x v )js v jP(x)js v Z Gkv j det gvj2s v v(gv x v ) dvgv 20 Note that for x 2 V 0 k we have jP(x)jA = Q v jP(x)jv = 1 by the product formula. Also, if k0=k is an extension of degree at most 3 whose Galois closure is the splitting eld k(x) of x, then P(x v ) is the vadic component of the adelic relative discriminant (k0=k) as de ned in [8], and so Y v jP(x v )jv = d1 k0=k in terms of the absolute norm of the relative discriminant of k0=k. We introduce the following notation for the local zeta functions of the space of binary cubic forms Z v (s; v) = Z Gkv j det gvj2s v v(gv x v ) dvgv: Then returning to the Euler product, we have Z GA j det gj2s A (g x) dAg = Y v Z Gkv j det gvj2s v v(gv x) dvgv = ds k0=k Y v Z v (s; v): In this notation, v denotes the local orbit corresponding to the form x or its corre sponding extension k0=k of degree at most 3. At this point, we are ready to convert the zeta function from a sum over orbits x 2 GknV 0 k to a sum over extensions k0=k of degree at most 3. We shall put o(k0=k) = jGk;xj in all cases where k(x)=k is a Galois extension of degree at most 3. When the splitting eld k(x)=k is an S3extension, it is the Galois closure of any of three conjugate cubic subextensions k0=k. We shall allow our series to include the same term for each of those cubic subextensions, and to compensate we set o(k0) = o(k0=k) = 3 instead of jGk;xj = 1. With these conventions, we now have the following series expansion of the adelic zeta function Z(s; ) = X k0=k ds k0=k o(k0) Y v Z v (s; v): (2.1) Again the local orbits v depend on the local type of k0=k over v. 21 The Euler products in (2.1) include factors for the in nite places v. Our next step toward producing Shintani's Dirichlet series is to factor out the local zeta functions for the in nite places. This is where the signatures come into play. If v is a complex place, there is only one nonsingular orbit. If v is a real place, there are two nonsingular orbits: one for forms of positive discriminant and one for forms of negative discriminant. We shall denote these choices by v = + and just as in the de nition of signature in Chapter 1.2. We write the signature as a vector = ( v)vj1 of these choices for all in nite places, and we denote the set of signatures by A (or A1 if we wish to emphasize that this is a choice of orbits at the in nite places). For 2 A1, we de ne Z (s; 1) = Y vj1 Z v (s; v): where 1 denotes the part of the SchwartzBruhat function corresponding to the in nite places of k. Thus, 1 is a rapidly decreasing C1function on the real vector space Q vj1 kv of dimension n. As in Chapter 1.2, for each signature , we de ne K to be the collection of all extensions k0=k of degree at most 3 which have signature . Then at last we have the decomposition of the adelic zeta function as follows Z(s; ) = X 2A1 Z (s; 1) (s; 0) (2.2) with (s; 0) = X k02K ds k0=k o(k0) Y v1 Z v (s; v) (2.3) where 0 = v1 v. Again, take note that the orbit v for each nite place is determined by the extension k0=k. For each nite place v, the SchwartzBruhat function v is a locally constant function of compact support on kv. Thus, (s; 0) does turn out to be essentially a Dirichlet series. To obtain Shintani's series in particular, we make a standard choice of these SchwartzBruhat functions at nite places, namely, v = 0;v, the characteristic 22 function of the submodule Vov of all binary cubic forms with coe cients in ov, the maximal compact subring of kv. With this choice, in the case of k = Q (or any eld k of class number 1) the sum over V 0Q in the de nition of the zeta function Z(s; ) reduces to a sum over V 0Z , the set of nonsingular integral binary cubic forms, and we have the identity Z(s; ) = Z1(s; 1) = Z GR=GZ j det gjs X x2V 0 Z 1(g x) dRg; exactly the zeta function de ned by Shintani in [17]. Thus, the natural generalization of Shintani's Dirichlet series is (s) = X k02K ds k0=k o(k0) Y v1 Z v (s; 0;v) DatskovskyWright calculated these local integrals to be Z v (s; 0;v) = (1 q16s v )1(1 q2s v )1 (2.4) 8>>>>>>>>>>>>>>< >>>>>>>>>>>>>>: (1 + q2s v )2; if v is of type (1); 1 + q4s v ; if v is of type (2u); 1 + q2s v ; if v is of type (2r); 1 q2s v + q4s v ; if v is of type (3u); 1; if v is of type (3r); where qv denotes the module of the nite place v. 2.4 Fourier transforms and the dual Dirichlet series The analytic properties of Shintani's Dirichlet series are derived by means of the Poisson Summation Formula and the use of Fourier transforms. We will establish our conventions for Fourier transform in this section and then review the functional equation for the adelic zeta function de ned in the previous section. Then we will ex 23 tract the dual Dirichlet series, which are the other components to the OhnoNakagawa identity. We have to de ne standard nontrivial additive characters h iv on the local elds kv for all places v of k and the adelic additive character hxi = Q vhxviv on adeles x = (xv)v 2 Ak. First, for Q, we choose standard additive characters on Q1 = R and Qp for any prime p as follows hxiR = exp(2 i x) hxip = exp(2 i fxgp) where fxgp is de ned as P1 j=m aj pj 2 Q in terms of the standard padic expansion x = P1 j=m aj pj with coe cients 0 aj p 1. Note that all these characters are trivial on Z. Also, if x 2 Q, then by the theorem of partial fractions we have x P p fxgp 2 Z, where the sum extends over all prime numbers p. That proves that the adelic character hxi = hxiA = Q vhxviv is trivial on Q embedded along the diagonal in AQ. To de ne the additive characters on extensions kv, we use the trace from kv to R or Qp. Thus, for in nite places v, hxiv = hTrkv=R(x)iR, and for nite places v j p we have hxiv = hTrkv=Qp(x)ip. For x 2 k, we have Trk=Q(x) = X vj1 Trkv=R(x) and Trk=Q(x) = X vjp Trkv=Qp(x) for all nite primes p. This implies that the adelic additive character hxi = Q vhxviv on Ak is trivial on k embedded along the diagonal in Ak. As inWeil [18], Defn. VII.2.4, we choose a di erental idele = ( v)v such that, for all v j 1 we have v = 1 and, for 24 all v  1 the largest ideal contained in the kernel of h iv is 1 v ov. Then vov is the di erent of kv=Qp. By Prop. VII.2.6 in Weil [18], we have j jA = d1 k . Using the measure dvx de ned in Section 2.1, we de ne the Fourier transform on SchwartzBruhat functions on kv by ^ (x) = Z kv (y)hyxiv dvy: Then ^ is also a SchwartzBruhat function on kv, and we have the inversion formula (x) = j vj1 v Z kv ^ (y)hyxiv dvy: Similarly, for an adelic SchwartzBruhat function , we de ne ^ (x) = Z Ak (y)hyxiA dAy; with inversion formula (x) = dk Z Ak ^ (y)hyxiA dAy: The selfdual measure on Ak would then be d1=2 k dAx. To extend these concepts to the vector space V of binary cubic forms, we compose the abovede ned additive characters with the bilinear alternating form [x; y] on V de ned in Section 2.2. We use the notation hx; yi = h[x; y]i with subscripts v and A as needed. With Fourier transform for functions on Vkv de ned by ^ (x) = Z Vkv (y) hx; yiv dvy we have the inversion formula (x) = j3j1 v j vj2v Z Vkv ^ (y) hx; yiv dvy; 25 due to the coe cients 1 3 in the bilinear form. A negative sign is not needed due to the bilinear form being alternating. For the adelic Fourier transform ^ , we have (x) = d2 k Z VA ^ (y) hx; yiA dAy; since the product formula implies j3jA = 1. It follows that the selfdual measure on VA is d2 k dAx, where dAx is the measure chosen in Section 2.2. In Theorem 6.1 of [19], Wright generalized Shintani's proof for Q to establish the functional equation Z(s; ^ ) = Z(1 s; ) for the adelic zeta function de ned in Section 2.3. In this functional equation ^ is de ned relative to the selfdual measure on VA. To comply with our choice of measure, we must replace ^ by d2 k ^ , which gives the functional equation Z(s; ^ ) = d2 kZ(1 s; ): We next carry out the same unfolding process for Z(s; ^ ) into Dirichlet series that we did for Z(s; ) in Section 2.3. If = v v has product form, then ^ = v^ v also has product form, and in the end we nd that Z(s; ^ ) = X 2A1 Z (s; ^ 1) (s; ^ 0) with ^ (s; ^ 0) = X k02K ds k0=k o(k0) Y v1 Z v (s; ^ v) with all the same notational conventions as in Section 2.3. We now again make the choice that for all nite places v we have v = 0;v is the characteristic function of Vov . Then the Fourier transform ^ 0;v may be computed to be the characteristic function of ( 1 v ov) (3 1 v ov) (3 1 v ov) ( 1 v ov) Vkv , or more brie y ^ 0;v(x) = 0;v( v(x1; 1 3x2; 1 3x3; x4)), as mentioned on p. 69 in [5] (with a slight typographical 26 error entering 3 instead of 1 3 as it should be). We would like to factor out the di erental element as much as possible. First we de ne 0;v(x) = 0;v(x1; 1 3x2; 1 3x3; x4), so that ^ 0;v(x) = 0;v( vx). Note that 0 B@ v 0 0 v 1 CA x = vx by the de nition of our representation of GL2 on V . Then by changing variables in the integral de ning the local zeta function, we have Z v (s; ^ 0;v) = Z Gkv j det gvj2s v ^ 0;v(gv x v ) dvgv = Z Gkv j det gvj2s v 0;v( 0 B@ v 0 0 v 1 CA gv x v ) dvgv = j vj4s v Z Gkv j det gvj2s v 0;v(gv x ) dvgv = j vj4s v Z v (s; 0;v): Using the fact that Q v j vjv = d1 k , we have, for this choice of v = 0;v for all nite places v, Z(s; ^ ) = d4s k X 2A1 Z (s; ^ 1) ^ (s) with ^ (s) = X k02K ds k0=k o(k0) Y v1 Z v (s; 0;v): This completes our de nition of the dual Dirichlet series ^ (s). It is our major goal now to work out the relationship between ^ (s) and (s) conjectured in Chapter 1.2. As a consequence of the functional equation for the adelic zeta function, we have the relation X 2A1 Z (s; ^ 1) ^ (s) = d24s k X 2A1 Z (1 s; 1) (1 s): (2.5) Later, as needed, we shall review the known facts about the residues of these Dirich let series, and the functional equation satis ed by the local zeta functions at in nite 27 places. In the next chapter, we calculate explicit expressions for Z v (s; 0;v), compa rable to those produced for Z v (s; 0;v) in [5]. Note that if v is a nite place that does not lie over the prime 3, then 3 is a unit in kv and it follows that 0;v = 0;v. Thus, in that case, the evaluation of the local zeta function is given by equation (2.4). In the next chapter, we shall consider exclusively places v j 3. 28 CHAPTER 3 Local integrals of a Fourier transform 3.1 Statement of the integral to be calculated In this chapter, we address the problem of calculating for a nite place v the local zeta function Z v (s; 0;v) as described at the end of Section 2.4. Since our work in this chapter will be exclusively over a local eld, we shall simplify our notation by letting K be a nonarchimedean local eld and using the notation of Sections 2.1 and 2.2. To review, we denote by O, P, , q, respectively, the maximal compact subring of O, the unique prime ideal P of O, a uniformizer or any generator of P so that P = O, and the order of the nite eld O=P, respectively. We normalize the absolute value jaj for a 2 K to be the modulus of multiplication by a with respect to any additive Haar measure on K. Thus, j j = q1. We denote by dy the additive Haar measure on K for which the measure of O is 1, and by dg the invariant measure on GK so that GO has measure 1. Our goal is to evaluate the local integral Z (s; 0) = Z GK j det gj2s 0(g x ) dg where s is a complex number with Re(s) > 1, is an orbit of nonsingular binary cubic forms in VK, x is the standard representative of , and 0(x) = 0(x1; 1 3x2; 1 3x3; x4) where 0 is the characteristic function of the compact subset VO. Hence, 0 is the characteristic function of the subset O 3O 3O O VO. If 3 is a unit in K, then 0 = 0, and this evaluation was completed in [5]. Thus, in this chapter we assume that K is a 3 eld of characteristic not equal to 3. For the de nition of Fourier 29 transform given in Section 2.4 and a di erental element 2 K, we saw in that section that Z (s; ^ 0) = j j4s Z (s; 0) which explains how this integral arises as a Fourier transform. The result we will prove in this chapter is the following: Theorem 3.1 Assuming that 3 is a uniformizer of K, we have Z (s; 0) = (1 q16s)1(1 q2s)1 8>>>>>>>>>>>>>>< >>>>>>>>>>>>>>: q4s(1 + q12s + 2q14s) if is of type (1); q4s(1 + q12s) if is of type (2u); q2s(1 + q14s) if is of type (2r); q4s(1 + q12s q14s) if is of type (3u); 1 if is of type (3r): 3.2 Reductions of the local integral We begin with some simple reductions in this calculation. First, we write GK = UKBK using the Iwasawa decomposition de ned in Section 2.2 of Chapter 2. For any u in UK = GO, we have that u x is in VO if and only if x is in VO. Hence 0(u x) = 0(x) for all x 2 VK. The same holds for 0. Then, considering the measure on GK de ned above, we have Z (s; 0) = Z UK du Z BK db j det(ub)j2s 0(ub x ) = Z UK du j det uj2s Z BK db j det bj2s 0(u (b x )) = Z UK du Z BK db j det bj2s 0(b x ) = Z BK db j det bj2s 0(b x ) : 30 The last integral gives us a motivation to de ne, for any SchwartzBruhat function , the integral transform I (s; ) = Z BK db j det bj2s (b x ) : Since any b in BK can be written as b = n(y)a(t; u) for some y 2 K and t, u in K , we can express this last integral as I (s; ) = Z K d t Z K d u Z K dy jtuj2s (n(y)a(t; u) x ) : Considering the operator D on the space of SchwartzBruhat functions de ned by D = a( 2; ) , or more explicitly, (D )(x) = (x) (a( 2; 1) x), we conclude that I (s;D ) = I (s; ) Z K d t Z K d u Z K dy jtuj2s (a( 2; 1) (n(y)a(t; u) x )) = I (s; ) Z K d t Z K d u Z K dy jtuj2s (n( y)a( 2t; 1u) x ) = I (s; ) j j1+6sI (s; ) = (1 q16s) I (s; ) ; where we have made the substitutions y 7! 1y, t 7! 2t, and u 7! u. In particular, for = 0 and putting 1 = D 0 we obtain Z (s; 0) = (1 q16s)1 I (s; 1) : So in order to calculate the value of this zeta function explicitly, we simply need to evaluate I (s; 1). By de nition, 1(x) = 0(x) 0( 3x1; 2x2; 1x3; x4) = 0(x1; 1 3x2; 1 3x3; x4) 0( 3x1; 1 3 2x2; 1 3 1x3; x4) : In other words, 1 is the characteristic function of (O 3O 3O O) n (P3 3P2 3P O). 31 In cases (1) and (2) we take x = (0; 1; ; ). Case (1) corresponds to = 0 and = 1 and so x = (0; 1; 1; 0). The integral I (s; 1) becomes now I (s; 1) = Z K d t Z K d u Z K dy jtj2s1 juj2s 1 0; t; 2y + u; y(y + u) t : Case (2) corresponds to = and = where O0 = O[ ] is the maximal compact subring of a quadratic extension K0 = K( ) of K and so x = (0; 1; + ; ). If the extension is unrami ed, case (2u), we take to be a unit not congruent to any unit in O module . For the rami ed extension, case (2r), we take to be the uniformizer . Given this choice of x we can write the integral I (s; 1) for case (2), after making an appropriate substitution, as I (s; 1) = Z K d t Z K d u Z K dy jtj2s1 juj2s 1 0; t;Tr(y + u ); N(y + u ) t ; where Tr and N are the relative trace and norm, respectively, for the extension K0 over K. In case (3) we take x = (1; + 0 + 00; 0 + 00 + 0 00; 0 00) where O0 = O[ ] is the maximal compact subring of a cubic extension K0 = K( ) of K. For case (3u) we take to be a unit not congruent to any unit in O module . For the case (3r) we take to be the uniformizer . For this choice of x we can write I (s; 1), following a change of variables, as I (s; 1) = Z K d t Z K d u Z K dy jtj2s1juj6s 1 t;Tr(y + u ); S(y + u ) t ; N(y + u ) t2 ; where Tr, S, and N are the relative trace, second symmetric function, and norm, respectively, for this cubic extension. In the next section we will calculate the value of the integral I (s; 1). By the reductions done in this section we know that this will su ce to prove Theorem 3.1. The calculation will be done assuming that 3 is a uniformizer in K, i.e. 3O = P. 32 When that is the case, 1 becomes the characteristic function of (O P P O) n (P3 P3 P2 O). 3.3 Evaluation of the local integral Let us complete the task of evaluating I (s; 0) for each of the possible types of . 3.3.1 Type (1) In the corresponding integral, since 2 is a unit, we can make the following change of variables (t; u; y) 7! (t=4; u; (y u)=2) to obtain I (s; 1) = Z K d t Z K d u Z K dy jtj2s1 juj2s 1 0; t; y; y2 u2 t : The integral is nonzero only if t; y; y2 u2 t belongs to (P P O)n(P3 P2 O) and in this case its value is I (s; 1) = Z K jtj2s1 d t Z K juj2s d u Z K dy : The integral is therefore nonzero in the following cases: (a) t 2 P , y 2 P, y2 u2 2 tO. These conditions imply that u2 2 P and so u 2 P = O. Thus, the integral in this case becomes I(a) = Z O jtj2s1 d t Z P juj2s d u Z P dy =q12s X1 l=1 Z lO juj2s d u q1 =q2s X1 l=1 q2sl = q4s 1 q2s : 33 (b) t 2 2O , y 2 P, y2 u2 2 tO. Again as before, u2 2 P and so u 2 P = O. Hence, the integral is now I(b) = Z 2O jtj2s1 d t Z P juj2s d u Z P dy =q24s X1 l=1 Z lO juj2s d u q1 =q14s X1 l=1 q2sl = q16s 1 q2s : (c) t 2 lO , y 2 O , y2 u2 2 tO, l 3. From these conditions it follows that (y u)(y + u) 2 Pl P. But P is a prime ideal so either y u 2 P or y + u 2 P. In the former case, since y + u = 2y (y u) 2 O , we conclude that y u 2 Pl1 and so u 2 y + Pl1 = y(1 + Pl2). In the latter case, since y u = 2y (y + u) 2 O , we obtain that y + u 2 Pl1 and so u 2 y+Pl1 = y(1+Pl2). This means that u 2 y(1+Pl2)ty(1+Pl2) for a xed y. Before we continue we observe that according to the de nition of the measures given in Chapter 2, we have d u = 1 1 q1 du juj : Moreover, since O =(1 + P) = (O=P) and (1 + Pm1)=(1 + Pm) = O=P for m 2, we have that the measure d u satis es Z 1+Pm d u = 1 (q 1)qm1 m 1 : 34 We can nally calculate the integral for this case. I(c) = X1 l=3 Z lO jtj2s1 d t Z O dy Z y(1+Pl2) juj2s d u + Z y(1+Pl2) juj2s d u = X1 l=3 Z lO jtj2s1 d t Z O dy 2q2s Z 1+Pl2 juj2s d u = X1 l=3 q(12s)l (1 q1)q1 2q2s (q 1)ql3 =2q12s X1 l=3 q2sl = 2q18s 1 q2s : Now, combining these results we obtain the desired result for type (1) I (s; 1) = I(a) + I(b) + I(c) = q4s(1 + q12s + 2q14s) 1 q2s : 3.3.2 Type (2u) When K0 = K( ) is a unrami ed quadratic extension of K with maximal compact subring O0 = O[ ], we have that is a unit and is also a uniformizer of K0. Hence, y +u is in (P0)m if and only if both y and u are in P. The corresponding integral is nonzero only if t;Tr(y + u ); N(y + u ) t is in (P P O) n (P3 P2 O) and its value is given by I (s; 1) = Z K jtj2s1 d t Z K juj2s d u Z K dy : The integral is nonzero for the following cases: (a) t 2 O , Tr(y +u ) 2 P, N(y +u ) 2 tO. Under these conditions, y +u 2 P0 and so y 2 P and u 2 P. This integral was calculated above, type (1) case (a), and its value is I(a) = Z O jtj2s1 d t Z P juj2s d u Z P dy = q4s 1 q2s : 35 (b) t 2 2O , Tr(y + u ) 2 P, N(y + u ) 2 tO. With these conditions, as before, y 2 P and u 2 P. We have already calculated this integral, type (1) case (b), and we found that I(b) = Z 2O jtj2s1 d t Z P juj2s d u Z P dy = q16s 1 q2s : Hence, we have found the value of the integral for type (2u) I (s; 1) = I(a) + I(b) = q4s(1 + q12s) 1 q2s : 3.3.3 Type (2r) When K0 = K( ) is a rami ed quadratic extension of K with maximal compact subring O0 = O[ ], we have that is a uniformizer of K0 and N( ) is a uniformizer of K. Then N(y + u ) is in Pm if and only jyj qm=2 and juj q(m1)=2. The associated integral is nonzero only if t;Tr(y + u ); N(y + u ) t is in (P P O)n(P3 P2 O) and this case it reduces to I (s; 1) = Z K jtj2s1 d t Z K juj2s d u Z K dy : The integral is nonzero for the following cases: (a) t 2 O , Tr(y + u ) 2 P, N(y + u ) 2 tO. These conditions imply that jyj q1=2 and juj 1 and so y 2 P and u 2 O. Then the integral is I(a) = Z O jtj2s1 d t Z O juj2s d u Z P dy =q12s X1 l=0 Z lO juj2s d u q1 =q2s X1 l=0 q2sl = q2s 1 q2s : 36 (b) t 2 2O , Tr(y + u ) 2 P, N(y + u ) 2 tO. These conditions imply that jyj q1 and juj q1=2 and so y 2 P and u 2 P. Then the value of the integral is already known, see type (1) case (b). I(b) = Z 2O jtj2s1 d t Z P juj2s d u Z P dy = q16s 1 q2s : Therefore, we have calculated the value of the integral for type (2r) I (s; 1) = I(a) + I(b) = q2s(1 + q4s) 1 q2s : 3.3.4 Type (3u) If K0 = K( ) is a unrami ed cubic extension of K with maximal compact subring O0 = O[ ], then is a unit and is a uniformizer of K0. Thus, y + u is in (P0)m if and only if both y and u are in Pm. Additionally, we can assume that Tr( ) = 0 and S( ) and N( ) are units of K. And for any y and u in K, we have Tr(y + u ) = 3y S(y + u ) = 3y2 + u2 S( ) N(y + u ) = y3 + yu2 S( ) + u3 N( ) : The corresponding integral is nonzero only if t;Tr(y + u ); S(y + u ) t ; N(y + u ) t2 is in (O P P O) n (P3 P3 P2 O) and its value is I (s; 1) = Z K jtj2s1 d t Z K juj6s d u Z K dy : Therefore, the integral is nonzero in the following cases: (a) t 2 O , Tr(y + u ) 2 P, S(y + u ) 2 tP , N(y + u ) 2 t2O. The rst and last conditions implies that y +u 2 O0 and so y 2 O and u 2 O. But to satisfy the 37 third condition, we actually require u 2 P. So the integral becomes I(a) = Z O jtj2s1 d t Z P juj6s d u Z O dy =1 X1 l=1 Z lO juj6s d u 1 = X1 l=1 q6sl = q6s 1 q6s : (b) t 2 O , Tr(y + u ) 2 P, S(y + u ) 2 tP , N(y + u ) 2 t2O. These conditions imply that y + u 2 P0 and so y 2 P and u 2 P. With this the integral is now I(b) = Z O jtj2s1 d t Z P juj6s d u Z P dy =q2s+1 X1 l=1 Z lO juj6s d u q1 =q2s X1 l=1 q6sl = q4s 1 q6s : (c) t 2 2O , Tr(y + u ) 2 P, S(y + u ) 2 tP , N(y + u ) 2 t2O. With these conditions we have y +u 2 P2 and so y 2 P2 and u 2 P2. And the integral in this case is given by I(c) = Z 2O jtj2s1 d t Z P2 juj6s d u Z P2 dy =q4s+2 X1 l=2 Z lO juj6s d u q2 =q4s X1 l=2 q6sl = q8s 1 q6s : (d) t 2 3O , (Tr(y + u ); S(y + u )) 2 (P tP ) n (P3 tP 2), N(y + u ) 2 t2O. The rst and last conditions imply that y + u 2 (P0)2 and so y 2 P2 and 38 u 2 P2. However, the remaining condition will be valid only if u 2 2O . So the integral becomes I(d) = Z 3O jtj2s1 d t Z 2O juj6s d u Z P2 dy =q6s+3 q12s q2 =q16s : Putting together these partial results gives us the value of the integral for type (3u) I (s; 1) = I(a) + I(b) + I(c) + I(d) = q4s(1 + q12s q14s) 1 q2s : 3.3.5 Type (3r) If K0 = K( ) is a rami ed cubic extension of K with maximal compact subring O0 = O[ ], then is a uniformizer of K0 and N( ) is a uniformizer of K. Hence, N(y + u ) is in Pm if and only if jyj qm=3 and juj q(m1)=3. Moreover, we assume that Tr( ) 2 P, S( ) 2 P, and N( ) 2 O . For any y and u in K we have Tr(y + u ) = 3y + uTr( ) S(y + u ) = 3y2 + 2yuTr( ) + t2 S( ) N(y + u ) = y3 + y2uTr( ) + yu2 S( ) + u3 N( ) : As before, the integral is nonzero only if t;Tr(y + u ); S(y + u ) t ; N(y + u ) t2 is in (O P P O) n (P3 P3 P2 O) and its value is I (s; 1) = Z K jtj2s1 d t Z K juj6s d u Z K dy : Therefore, to have a nonzero integral we have to consider following cases: (a) t 2 O , Tr(y + u ) 2 P, S(y + u ) 2 tP , N(y + u ) 2 t2O. From these assumptions, we conclude that jyj 1 and juj q1=3, i.e. y 2 O and u 2 O. 39 The integral reduces in this case to I(a) = Z O jtj2s1 d t Z O juj6s d u Z O dy =1 X1 l=0 Z lO juj6s d u 1 = X1 l=0 q6sl = 1 1 q6s : (b) t 2 O , Tr(y + u ) 2 P, S(y + u ) 2 tP , N(y + u ) 2 t2O. Using these assumptions, we obtain jyj q2=3 and juj q1=3, i.e. y 2 P and u 2 P. The integral, which was calculated for type (3u) case (b), is I(b) = Z O jtj2s1 d t Z P juj6s d u Z P dy = q4s 1 q6s : (c) t 2 2O , Tr(y+u ) 2 P, S(y+u ) 2 tP , N(y+u ) 2 t2O. These assumptions imply that jyj q4=3 and juj q1, i.e. y 2 P2 and u 2 P. So our integral is I(c) = Z 2O jtj2s1 d t Z P juj6s d u Z P2 dy =q4s+2 X l=1 Z lO juj6s d u q2 =q4s X1 l=1 q6sl = q2s 1 q6s : These results can be combined to get the integral for type (3r) I (s; 1) = I(a) + I(b) + I(c) = 1 1 q2s : This concludes the evaluation of I (s; 0) in all cases, and therefore completes the proof of Theorem 3.1. 40 3.4 Veri cation of a simple identity In this section we use an identity which was shown to be true by Datskovsky and Wright in order to verify the validity of Theorem 3.1. Consider any locally integrable function on VK. By formula (2.4) on page 38 of [5] we have the following identity Z VK (x) dx = X bKc Z GK j det gj2 (g x ) dg ; where the sum is taken over all the GK{orbits in VK, bK = q12e(1q1)(1q2), e is the order of in K, and c = jP(x )j o( ) where P(x ) is the discriminant of x and o( ) is the order of the stabilizer in GK of any x 2 . In particular, we can take to be ^ 0. The left hand side is simply Z VK ^ 0(x) dx = j3j j j2 0(0) = j3j j j2 by the Fourier inversion formula and our choice of the measure dx on VK. The right hand side becomes X bKc Z (1; ^ 0) = bKj j4 X c Z (1; 0) by the de nition of the orbital zeta function. These formulas are valid for any p{ eld whose characteristic is not 2 or 3. We want to very the above identity for the kind of elds we have been considering in this chapter, that is for 3{ elds for which 3 is a uniformizer. Therefore, under these assumptions using Theorem 3.1 and after canceling out common terms, the identity we are trying to show reduces to X c Z (1; 0) = b1 K j3j j j2 = q2(1 q1)1(1 q2)1 : For convenience, we write (j) to indicate that is an orbit of type (j), where j take the values 1, 2u, 3u, 2r, 3r. We recall that c (1) = 1=6, c (2u) = 1=2, and c (3u) = 1=3. Moreover, X (2r) c (2r) = 1 q and X (3r) c (3r) = 1 q2 ; 41 where the sums are taken over all the orbits of type (2r) and (3r), respectively. Since j j = q1, the orbital zeta function of 0 at 1 can be written by Theorem 3.1 and the de nition of , as Z (1; 0) = (1 q5)1(1 q2)1 8>>>>>>>>>>>>>>< >>>>>>>>>>>>>>: q4(1 + q1 + 2q3) if is of type (1); q4(1 + q1) if is of type (2u); q2(1 + q3) if is of type (2r); q4(1 + q1 q3) if is of type (3u); 1 if is of type (3r): We can now use all this information to calculate X c Z (1; 0) = 1 6 Z (1)(1; 0) + 1 2 Z (2u)(1; 0) + 1 3 Z (3u)(1; 0) + 1 q Z (2r)(1; 0) + 1 q2 Z (3r)(1; 0) =(1 q5)1(1 q2)1 1 6 q4(1 + q1 + 2q3) + 1 2 q4(1 + q1) + 1 3 q4(1 + q1 q3) + 1 q q2(1 + q3) + 1 q2 1 =(1 q5)1(1 q2)1(q2 + q3 + q4 + q5 + q6) =q2(1 q5)1(1 q2)1(1 + q1 + q2 + q3 + q4) =q2(1 q1)1(1 q2)1 which concludes the veri cation of the identity. 42 CHAPTER 4 Residues of the Dirichlet series and generalizing OhnoNakagawa In [5], formulas were calculated for the residues of the Dirichlet series (s) at its poles s = 1 and s = 5=6, but the calculation of the residue formulas for the dual Dirichlet series ^ (s) was left incomplete. The method of that paper required the calculation of local integrals of a SchwartzBruhat function, which was completed for nonarchimedean local elds for the characteristic functions 0, but not for its Fourier transform ^ 0. As we saw in the previous chapter, these new local integrals are more di cult to calculate. We shall use instead the ltration method of [6] to calculate the residues of the dual Dirichlet series, and thus our results in this chapter will be valid for all number elds k, and not simply those where 3 is unrami ed. At the end of this chapter, we shall use the complete set of residues of both (s) and ^ (s) to deduce what the correct generalization of Ohno's conjecture should be. Our method will verify this conjecture at least at s = 1 and s = 5=6. 4.1 Filtrations of the Dirichlet series It will be necessary to generalize the decomposition of the adelic zeta function given in equation (2.2) on page 22. Instead of singling out just the in nite places, we shall now allow an arbitrary nite set S of places of k to be distinguished. We shall assume that S contains all in nite places as well as possibly some nite places. For a place v of k, let Av denote the nite set of Gkv orbits of nonsingular forms in Vkv . Let AS = Y v2S Av. Then = ( v) will denote a choice from AS, meaning a choice of an orbit at each place in S. We will call these choices orbit vectors over S. 43 Then just as in Section 2.3 of Chapter 2, assuming the SchwartzBruhat function = v v has product type and that v = 0;v for all nite places v =2 S, the adelic zeta function may be decomposed as Z(s; ) = X 2AS Z ;S(s; ) ;S(s) where now Z ;S(s; ) = Y v2S Z v (s; v); (4.1) Z v (s; v) = Z Gkv j det gvj2s v v(gv x v ) dvgv ;S(s) = X k02K ds k0=k o(k0) Y v =2S Z v (s; 0;v): Here x v is the standard choice of representative for the chosen orbit v of binary cubic forms over kv (described in Prop. 2.1, p. 35 of [5]). Also, K denotes the set of extensions k0=k of degree at most 3 so that k0 k kv corresponds to the choice of orbit v for each place v 2 S. The choice of S is built into the choice of the orbit vector 2 AS, but we will indicate S explicitly in the notation ;S because the technique we wish to use in this chapter is to extend the distinguished set S of places until relations between the various Dirichlet series are easier to detect. In particular, suppose T is a nite set of places of k that contains S. We denote the set of places in T which do not belong to S as T n S. Then there is a natural restriction mapping from orbit vectors in AT to orbit vectors in AS. We will generally use to denote a choice of orbits in AS and to denote a choice of orbits in AT . If v = v for all v 2 S, we say jS = , meaning that restricted to S agrees with . We can decompose the Dirichlet series ;S(s) in terms of the series ;T (s) with 44 jS = as follows ;S(s) = X k02K ds k0=k o(k0) Y v =2S Z v (s; 0;v) = X jS= X k02K ds k0=k o(k0) Y v2TnS Z v (s; 0;v) Y v =2T Z v (s; 0;v) where the sum ranges over those 2 AT whose restriction to S is , and we have explicitly written the Euler product in terms of the local zeta integral factors. Recall that for v =2 T, v is determined as the orbit corresponding to k0=k at v. Thus, it should be kept in mind that for v =2 T the orbit v is a function of the extension k0=k. Rearranging the above sum and product gives ;S(s) = X jS= 2 4 Y v2TnS Z v (s; 0;v) 3 5 ;T (s) (4.2) This is the main ltration formula for the original Dirichlet series. We can extend these ideas to the dual Dirichlet series as well and obtain the formulas ^ ;S(s) = X k02K ds k0=k o(k0) Y v =2S Z v (s; 0;v) (4.3) = X jS= 2 4 Y v2TnS Z v (s; 0;v) 3 5 ^ ;T (s) (4.4) The main idea exploited in this chapter is that if the set of places T contains all places lying over 3, then 3 is a unit for all v =2 T, and therefore 0;v = 0;v for all v =2 T. Consequently ^ ;T (s) = ;T (s) for all 2 AT . Thus, the component series in the two decompositions of ;S and ^ ;S(s) are the same, and the sole di erence lies in the nitely many local zeta function factors Z v (s; 0;v) for v 2 T n S. 45 4.2 Poles and residues First, let us review the slightly more general framework of the earlier papers of DatskovskyWright. In [19, 4], it is proved that the adelic zeta function Z(!; ) = Z GA=Gk !(det g) X x2V 0 k (g x) dAg; has a meromorphic continuation to the entire complex manifold k of quasicharacters ! on A =k . This continuation is holomorphic except for simple poles at ! = !0, !2, !1=3 and !5=3 where is any character satisfying 3 = 1. In this thesis, we are restricting the quasicharacters to principal ones ! = !2s = j j2s A for complex s. Thus, Z(s; ) = Z GA=Gk j det gj2s A X x2V 0 k (g x) dAg; is holomorphic in the entire splane with the exception of simple poles at s = 0, 1 6 , 5 6 , and 1. The decomposition (2.2) of the zeta function Z(s; ) in terms of the Dirichlet series (s) allows us to prove that the Dirichlet series have meromorphic continuations to the entire splane which are holomorphic except for simple poles at s = 1 and s = 5=6. Very general residue formulas are stated in Theorem 6.2, p. 71, in [5]. The measure on GA used in that paper is not the tensor product measure de ned in Section 2 of Chapter 2. Thus, after tracing through the notation presented in [19, 5], we nd the following residue formulas Res s=1 ;S(s) = k 2 p dk S k;S(2) c [1 + b ] (4.5) Res s=5=6 ;S(s) = k 6dk S k;S 1 3 c a where k and dk are de ned in Chapter 2, S = Y v2S v1 (1 q1 v ); k;S(s) = Y v =2S (1 qs v )1; (partial Dedekind zeta function, analytically continued) 46 and the a = Q v2S a v , b = Q v2S b v , and c = Q v2S c v are constants describing the structure of the orbit v de ned on pages 58, 61 and 38 of [5]. To describe a v , b v and c v , suppose that the orbit v corresponds to the local extension k0w =kv of degree at most 3 (up to conjugacy) or equivalently the simple algebra k0 k kv which has dimension 3 over kv. In [5], these extensions are classi ed into ve types: (1), (2u), (2r), (3u) and (3r), where the number is the degree of the extension, and the letter indicates whether that extension is unrami ed or ram i ed. Let v = (k0w =kv) be the relative discriminant, which is an element of k v determined uniquely modulo multiplication by squares of units, and let o( v) be the number of automorphisms of k0 kv over kv. For types (1), (2), and (3), respectively, we have o( v) = 6, 2, and 3 or 1, respectively, the last depending on whether the local extension is Galois or not. Then from [5], page 38 and top of page 36, we have c v = j v jv o( v) : (4.6) On page 61 of [5], b v is simply de ned as 3, 1, 0, resp. for types (1), (2), (3), resp. The de nition of a v is the most involved and is described in a chart on page 58 of [5]. To obtain the residue formulas for the dual Dirichlet series ^ ;S(s), we choose a set T of places that contains S and all nite places v j 3. Then, as we mentioned before, for all orbit vectors 2 AT , we have ^ ;T (s) = ;T (s). Our ltration formulas now give the following relations among the residues of all these series, for r = 1 and 5=6: Res s=r ;S(s) = X jS= 2 4 Y v2TnS Z v (r; 0;v) 3 5Res s=r ;T (s) (4.7) Res s=r ^ ;S(s) = X jS= 2 4 Y v2TnS Z v (r; 0;v) 3 5Res s=r ;T (s) In the next sections, we shall use properties of the local zeta functions together with these ltrations to calculate formulas for the residues of the dual Dirichlet series. 47 4.3 Residue of the dual Dirichlet series at s = 1 In this section, we abbreviate k = k 2 p dk , all the factors in the residue at s = 1 that depend on k but on nothing else. The ltration formula (4.7) together with the residue formulas (4.5) yield Res s=1 ^ ;S(s) = X jS= 2 4 Y v2TnS Z v(1; 0;v) 3 5Res s=1 ;T (s) = X jS= 2 4 Y v2TnS Z v(1; 0;v) 3 5 k T k;T (2) c [1 + b ] = k T k;T (2) X jS= 2 4 Y v2TnS Z v(1; 0;v) 3 5 c [1 + b ]: Note that the local zeta function was shown in [5] to be holomorphic for Re(s) > 1=6, and thus we can simply substitute s = 1 in the residue calculation. By the organizing principle that a sum of products may be rearranged as a product of sums, we will now manipulate the above residue formulas. Both c and b factor as products over the places v 2 T; however, before factoring out the terms corresponding to places in S, we must split the residue formula and then factor, using the facts that k;T (s) = k;S(s) Y v2TnS (1 qs v ) T = S Y v2TnS (1 q1 v ) c = c Y v2TnS c v b = b Y v2TnS b v 48 This leads to Res s=1 ^ ;S(s) = k T k;T (2) X jS= 2 4 Y v2TnS Z v(1; 0;v) 3 5 c + X jS= 2 4 Y v2TnS Z v(1; 0;v) 3 5 c b = k S k;S(2) c X jS= 2 4 Y v2TnS (1 q1 v )(1 q2 v )c v Z v(1; 0;v) 3 5+ c b X jS= 2 4 Y v2TnS (1 q1 v )(1 q2 v )c vb v Z v(1; 0;v) 3 5 = k S k;S(2) c Y v2TnS " (1 q1 v )(1 q2 v ) X v2Av c v Z v(1; 0;v) # + c b Y v2TnS " (1 q1 v )(1 q2 v ) X v2Av c vb v Z v(1; 0;v) # The sums that appear can be simpli ed by means of Fourier transform formulas proved in [5]. Keeping in mind that we are using the measure such that Vov has measure 1, the formula (2.4) on page 38 in [5], restated as Proposition 5.1 on page 52, implies that, for nite places v, (1 q1 v )(1 q2 v ) X v2Av c vZ v(1; 0;v) = Z Vkv 0;v(x) dvx : Since 0;v is the characteristic function of ov 3ov 3ov ov Vkv , the integral is j3j2v . Thus, we have the rst sum in the residue formula equal to (1 q1 v )(1 q2 v ) X v2Av c vZ v(1; 0;v) = j3j2v : The other part of the residue at 1 involves Theorem 5.2, Proposition 5.2 and the de nition of the singular invariant distributions 3 from [5]. Assume throughout that v is a nite place and kv is a p eld. Theorem 5.2 on page 61 says that (1 q1 v ) X v2Av c vb v Z v(1; 0;v) = 3( 0;v) 49 and the distribution 3 is de ned on page 54 as 3( v) = 3(2; v) = Z k v d v t Z kv dvx Z kv dvy jtj2v v(0; t; x; y); for Govsymmetric functions v, where ov is the maximal compact subring in kv. Then it is straightforward to evaluate 3( 0;v) = Z k v d v t Z kv dvx Z kv dvy jtj2v 0;v(0; t; x; y) = Z 3ov jtj2v d v t j3jv = j3j3v (1 q2 v )1 Combining this with our earlier equation for 3( 0;v) produces (1 q1 v ) X v2Av c v b v Z v(1; 0;v) = j3j3v (1 q2 v )1: This gives the second sum in our residue formula at 1 as (1 q1 v )(1 q2 v ) X v2Av c v b v Z v(1; 0;v) = j3j3v : This leads to the full residue formula at s = 1: Res s=1 ^ ;S(s) = k S k;S(2) c Y v2TnS j3j2v 2 41 + b Y v2TnS j3jv 3 5 : Our last task is to account for our assumption that T contains all the places v j 3. De ne jxjS = Q v2S jxjv for any x 2 k. By our assumptions on T, we have j3jT = j3jA = 1, by the idele product formula. Then Q v2TnS j3jv = j3jT =j3jS = j3j1 S . Then our nal residue formula at 1 is Res s=1 ^ ;S(s) = k 2 p dk S k;S(2) c j3j2 S 1 + b j3j1 S : (4.8) 4.4 Residue of the dual Dirichlet series at s = 5=6 Next, we turn to the residue formula at 5=6. Here we abbreviate k = k 6dk , all the factors in the residue at s = 5=6 that depend on k but on nothing else. We use the 50 same notation and arguments at the beginning of Section 4.3 along with the formula a = a Y v2TnS a v , but start with the residue formula for ;S at 5=6. This leads to Res s=5=6 ^ ;S(s) = X jS= 2 4 Y v2TnS Z v(5=6; 0;v) 3 5 Res s=5=6 ;T (s) = X jS= 2 4 Y v2TnS Z v(5=6; 0;v) 3 5 k T k;T 1 3 c a = k T k;T 1 3 X jS= 2 4 Y v2TnS Z v(5=6; 0;v) 3 5 c a = k S k;S 1 3 c a X jS= 2 4 Y v2TnS (1 q1 v )(1 q1=3 v ) c v a v Z v(5=6; 0;v) 3 5 = k S k;S 1 3 c a Y v2TnS " (1 q1 v )(1 q1=3 v ) X v2Av c v a v Z v(5=6; 0;v) # after exchanging the sum and product in exactly the same way as in the preceding section. The sum in the above formula corresponds to a second Fourier inversion formula. For nite places v, Theorem 5.1 in [5] states that (1 q1 v ) X v2Av c v a v Z v(5=6; 0;v) = 4( 0;v); for the distribution 4 de ned on pp. 33 and 34 of [5] by the integral 4( v) = 4(1=3; v) = Z k v d v t Z kv dvx Z kv dvy Z kv dvz jtj1=3 v v(t; x; y; z) if v is any Govsymmetric function. Replacing v by 0;v, this is easy to calculate as 4( 0;v) = Z k v d v t Z kv dvx Z kv dvy Z kv dvz jtj1=3 v 0;v(t; x; y; z) = Z ov jtj1=3 v d v t j3j2v = j3j2v (1 q1=3 v )1 51 Therefore, the sum in the residue formula becomes (1 q1)(1 q1=3) X v2Av c v a v Z v(5=6; 0;v) = j3j2v : Putting this altogether gives Res s=5=6 ^ ;S(s) = k S k;S 1 3 c a Y v2TnS j3j2v : Assuming again that T contains all places lying over 3, we can nally give the formula for the residue at 5=6: Res s=5=6 ^ ;S(s) = k 6dk S k;S 1 3 c a j3j2 S : (4.9) 4.5 Generalizing Ohno's conjecture Ohno's conjecture (see [14, 13]) takes the form ^ 1(s) = 33s 2(s) ^ 2(s) = 313s 1(s) where 1(s), 2(s) are Shintani's Dirichlet series (see [17]) corresponding to integral binary cubic forms of positive and negative discriminant, respectively, and ^ 1(s), ^ 2(s) are the analogous series for the dual lattice. This is the case k = Q with the set of places S limited to just the one in nite place v = 1. The completion of Q at the place 1 is simply the real numbers R, and there are two GRorbits of nonsingular real binary cubic forms, namely, the totally real forms of positive discriminant, and the complex forms of negative discriminant. We will attempt to generalize this pattern to any number eld k by taking S to be the set of in nite places v j 1 of k. For any choice of orbits 2 A1, we will de ne as another choice in the following way. First, we decompose A1 as a direct product A1 = Q vj1 Av and = ( v)v. If kv = C, there is only one GCorbit, and we de ne v = v, the lone orbit. If kv = R, there are two GRorbits, and we simply 52 de ne v to be the other orbit besides v. Then for 2 A1, we de ne in the natural componentwise fashion. Our goal is to establish a formula of the form ^ (s) = 3A+Bs (s) for any 2 A1, for some constants A and B dependent on k and . The reason for the comparison of series for the orbit types and lies in the theorem of Scholz in [16] about the relationship between the 3class numbers of quadratic eld of positive and negative discriminant. There the key tool is to adjoin the cube roots of unity to the Galois closure of a noncyclic cubic eld, with quadratic resolvent eld of discriminant D. The extended eld now contains another family of conjugate cubic elds with quadratic resolvent eld of discriminant 3D. That correspondence changes the sign of the discriminants of the cubic elds. Hopefully, this mechanism will be made more precise in the course of our research. First, we collect the residue calculations of this chapter as well as the original calculations of DatskovskyWright into a convenient reference theorem: Theorem 4.1 For a nite set S of places of the number eld k containing all in nite places, the residues of the Shintani Dirichlet series ;S(s) and the dual series ^ ;S(s), as de ned in Sections 2.3 and 2.4, are given by the following formulas: Res s=1 ;S(s) = k 2 p dk S k;S(2) c [1 + b ] Res s=1 ^ ;S(s) = k 2 p dk S k;S(2) c j3j2 S 1 + b j3j1 S Res s=5=6 ;S(s) = k 6dk S k;S 1 3 c a Res s=5=6 ^ ;S(s) = k 6dk S k;S 1 3 c a j3j2 S When S is the set of in nite places, we have j3jS = j3j1 = 3n, where n = [k : Q]. 53 Then, using the preceding theorem, we have ^ (1) (1) = 32n c c 1 + b 3n 1 + b ^ (5=6) (5=6) = 32n c c a a : The next step in simplifying these formulas is to manipulate the formulas for a , b and c when corresponds to an orbit type over R or C. Since we are only considering places v j 1, there are only three local orbit types v to consider, which we will denote as v = 0 if kv = C, v = + if kv = R and v corresponds to the binary cubic forms of positive discriminant, and v = if kv = R and v corresponds to the binary cubic forms of negative discriminant. Recall that in Section 4.2, equation (4.6), we reviewed the de nitions of b and c established in [5], and we give these again strictly for the archimedean places c0 = 1 6 ; c+ = 1 6 ; c = 1 2 ; b0 = 3; b+ = 3; b = 1: Considering ratios, we have c0 c0 = 1; c+ c = 1 3 ; c c+ = 3: For = ( v)v 2 A1, suppose that for m of the real places v we have v = + and then for the other r1 m places we have v = . Then by taking products of the formulas from the previous paragraph we get c c = 3m 1 3 r1m = 32mr1 ; b = 3r2+m; b = 3r2+r1m: Then 1 + b 3n 1 + b = 1 + 3r1+r2mn 1 + 3r2+m = 1 + 3r2m 1 + 3r2+m = 3r2m; since n = r1+2r2. It is noteworthy that under our choices this ratio simpli es to just a power of 3. Then combining these results we obtain ^ (1) (1) = 32n 32mr1 3r2m = 3m2nr1r2 (4.10) 54 For the value of the ratio at 5=6, we need the formulas for a given on page 58 in [5]. a0 = 3 p 3 4 2 1 3 6 ; a+ = 3 p 3 2 1 3 3 ; a = 3 2 1 3 3 : Then the ratios are a0 a0 = 1; a+ a = p 3; a a+ = 1 p 3 : For 2 A1 with m real places v such that v = +, as before, we have a a = 1 p 3 m ( p 3)r1m = 3 1 2 r1m: Then ^ (5=6) (5=6) = 32n 32mr1 3 1 2 r1m = 3m2n1 2 r1 : (4.11) Both values are consistent with ^ (s)= (s) being powers of 3. We may now solve for the constants A;B in this conjectural form: ^ (1) (1) = 3A+B = 3m2nr1r2 ^ (5=6) (5=6) = 3A+5 6B = 3m2n1 2 r1 : This leads to the linear equations: A + B = m 2n r1 r2 A + 5 6 B = m 2n 1 2 r1; which have the unique solution A = m + r2 B = 3n: Then the proposed generalization of Ohno's Conjecture (and Nakagawa's theorem) is ^ (s) (s) = 3m+r23ns : (4.12) The calculations presented here establish Theorem 1.2 and motivate Conjecture 1.1. 55 Finally, let us compare this conjecture to Nakagawa's theorem for k = Q. In that case we have n = [Q : Q] = 1, r1 = 1, and r2 = 0. Then our conjecture would say ^ 2(s) 1(s) = 31+03s = 313s; ^ 1(s) 2(s) = 30+03s = 33s: This is exactly the theorem of Nakagawa mentioned at the beginning of this section. 56 CHAPTER 5 Decomposing the Dirichlet series according to the resolvent eld Datskovsky and Wright established the expression of Shintani Dirichlet (s) series as a sum over extensions k0=k of degree at most 3, as mentioned in the introduction, and in Chapter 3 of this thesis we established the analogous formula for the dual Dirichlet series ^ (s). Then after cancelling common factors, as described in Chapter 1.2 at equation (1.5), our generalization of Ohno's conjecture becomes X k02K ds k0=k o(k0) Rk0(2s) Rk0(4s) Y vj3 Tk0;v(s) = 3r2+m3ns X k02K ds k0=k o(k0) Rk0(2s) Rk0(4s) In this chapter, we shall decompose this identity according to the resolvent elds of the extensions k0=k, and give the proofs of Theorems 1.3 and 1.4 in Chapter 1.2. 5.1 The resolvent eld of an extension k0=k of degree at most 3 If k0=k has degree strictly less than 3, we simply de ne the resolvent eld to be F = k0. If k0=k is a cubic extension, it is either cyclic, in which case we de ne the resolvent eld to be F = k, or it is noncyclic and its Galois closure over k contains a unique quadratic eld F, which is called the resolvent eld in that case. Each resolvent eld F has degree at most 2 over k, and thus can be expressed in the form F = k( p ) for some nonzero element of k. By Kummer theory, k( p 1) = k( p 2) if and only if 1= 2 2 k2, the subgroup of squares in k . Thus, the possible resolvent elds F of k0=k bijectively correspond to the cosets in k =k2. For each 2 k , de ne C ( ) to be the set of all extensions k0=k of degree at most 3 which have resolvent eld equal to F = k( p ). 57 For each real embedding : k ! R of k, and for any 2 k , either > 0 or < 0. Thus, we can de ne a signature of by setting v = + if > 0 and v = otherwise. This signature is the same as the signature of the extension k( p )=k as de ned in Chapter 1.2. If v = +, then k( p ) k kv = R R, and if v = , then k( p ) k kv = C. For any extension k0=k 2 C ( ), since the Galois closure of k0 over k contains k( p ), this shows that k0 k kv must be a direct sum of three copies of R. Similar reasoning in case v = proves that the signature of any k0 2 C ( ) is the same as the signature of . Thus, for any signature , the set of extensions K is the disjoint union of C ( ) over representatives of all cosets 2 k =k2 with signature . Finally, for any resolvent eld F = k( p ), we de ne the dual resolvent eld to be ^ F = k( p 3 ). The reason for this choice of dual is that the compositum F ^ F must contain the eld F0 = k( p 3) generated by the cube roots of unity over k. Kummer theory says that any cyclic cubic extension F0=F for which F contains the cube roots of unity must be of the form F0 = F( 3 p ) for some 2 F , and that fact plays a special role in Scholz' re ection theorem in [16] and Nakagawa's proof. Note that duality is symmetric in that the dual eld of ^ F is just F. If the signature of is , then clearly the signature of 3 is . We summarize these observations about the resolvent elds in the following propo sition: Proposition 5.1 For any signature of the eld k and its negative , the sets of extensions K and K , respectively, are the disjoint unions of the subsets C ( ) and C (3 ) as ranges over representatives of each coset in k =k2 which has signature . By summing over the coset representatives 2 k =k2 with signature , this proposition directly proves what we stated as Theorem 1.3 in Chapter 1.2. 58 Theorem 5.1 If for every 2 k , we have X k02C (3 ) ds k0=k o(k0) Rk0(2s) Rk0(4s) Y vj3 Tk0;v(s) = 3r2+m3ns X k02C ( ) ds k0=k o(k0) Rk0(2s) Rk0(4s) then the generalized Ohno conjecture (1.4) is true. Later in this chapter, we shall explore the truth of the converse of this theorem. We next turn to a more detailed discussion of Scholz' re ection. Let k0=k be an extension of degree 3 with resolvent eld equal to F = k( p ) (which has degree 1 or 2 over k). The main idea of Scholz' re ection is that cubic extensions k0=k with resolvent eld F roughly correspond to cubic extensions ^k0=k with resolvent eld ^ F. This comes about as follows. The compositum L = k0F is a cyclic cubic extension of F. Let B be the eld B = F( p 3) = F ^ F. Then the degree [B : k] is a divisor of 4, and the compositum N = k0B is a cyclic cubic extension of B. Since B contains the cube roots of unity, by Kummer theory the cubic extension N=B has the form N = B( 1=3) for some nonzero 2 B. 5.2 Conductors and discriminants of cubic extensions In this section, we establish the basic notation of conductors, di erents, and dis criminants of cubic extensions. This material is derived from Hasse [9] and Martinet Payans [11]. Before we continue, we need to recall a few results from class eld theory that would allow us to analyze our Dirichlet series identities. Theorem 5.2 (Isomorphism Theorem) There is a onetoone correspondence be tween the nite abelian extensions L of F and the open subgroups U = UL of the idele class group JF = A F =F such that the Galois group Gal(L=F) is isomorphic to JF=U. Moreover, if L=F is a nite abelian extension and K is an intermediate eld L K F, then the corresponding subgroups satisfy F UL UK UF A F . For each character of the group JF trivial on U, let f denote its conductor. It is an integral ideal in F. 59 Theorem 5.3 (ConductorDiscriminant Formula) Let L=F be a nite abelian exten sion of number elds corresponding to the open subgroup U of the idele class group JF . Then the relative discriminant DL=F of L=F is given by DL=F = Y f ; where ranges over all the characters of JF trivial on U. Let k0=k be an extension of degree 3 with Galois closure L and resolvent eld F. The relative discriminants of k0=k, L=k, and F=k respectively, considered as ideals in ok, are denoted by Dk0=k, DL=k, and DF=k, respectively. The di erents of these extensions, as ideals in the rings of the integers of the corresponding over eld, are denoted by dk0=k, dL=k, and dF=k, respectively. The relative discriminants and di erents are related by means of the relative norms Dk0=k = Nk0=k(dk0=k); DL=k = NL=k(dL=k); DF=k = NF=k(dF=k): According to the notation introduced in Chapter 1, we can write dk0=k = N(Dk0=k); dL=k = N(DL=k); dF=k = N(DF=k): If k0=k is noncyclic, then F=k turns out to be quadratic and L=F cyclic cubic. By the isomorphism theorem, this extension corresponds to an open subgroup U of index 3 in JF . There are two nontrivial cubic characters and 2 with kernel equal to U. By the conductordiscriminant formula, the discriminant of L=F is DL=F = f2 , since and 2 have the same conductor. Therefore by the tower law for discriminants (see Prop. 13 of Chap. VII4 in [18]) we have DL=k = D3 F=k NF=k(f )2: Next, we shall discuss the concepts of conductors and discriminants over the idele class group. For a place v of F, let iv be the natural injection of F v into the idele 60 class group JF . Thus, iv(x) is the coset of the idele with v component equal to x and all other components equal to 1. Let be a character of JF which is trivial on U. We de ne the v component to be v(x) = (iv(x)) for x 2 F v . For a nite place v, the kernel of v contains either the full unit group o v of F (in which case we set fv = 0) or some subgroup 1+$fv v ov for a smallest positive integer fv, where $v is a uniformizer in Fv. Then the conductor of v is ' v = $fv v . For an in nite place v, the kernel of a nite order character v is either all of F v in which case we set ' v = 1, or possibly just the positive real numbers R+ in the event v is real. In the latter case, we set ' v = 1. The idelic conductor ' is de ned to be the idele with v component equal to ' v for all places v. The conductor is wellde ned as an element of A F =A0 F;1, where A0 F;1 = Y vj1 F0 v Y v1 o v where F0 v represents the connected component of 1 in F v . Hence, F0 v = C if v is complex and R+ if v is real. For a place v of F and a place w of L lying above v, let f 1; : : : ; mg be a basis of Lw over Fv. Let (i) j range over the m conjugates of j . For in nite places v, we de ne the relative discriminant Lw=Fv of the extension Lw=Fv to be the square of the determinant of the matrix ( (i) j ), which is an element of F v . This relative discriminant is wellde ned only modulo multiplication by elements of F2 v , the group of squares of elements of F v . By convention, we stipulate C=C = R=R = 1 and C=R = 1. For nite places, the maximal compact subring of Lw is a free ovmodule, and thus we may select a basis f jg. Then Lw=Fv is de ned using this basis. With this special choice of the j , the relative discriminant is wellde ned module o2v , the group of squares of elements of o v . The vadic part v;L=F of the relative discriminant of L=F is de ned by v;L=F = Y wjv Lw=Fv 2 F v : 61 The idelic relative discriminant L=F is taken to be the idele whose vadic com ponent is v;L=F . That this is an idele is a consequence of the fact that v;L=F 2 o v for almost all v. This discriminant is wellde ned modulo multiplication by elements of A2 F;0 = Y v1 o2v : This idelic de nition of discriminant was rst advanced in [8], where many basic properties are established. There is simple relationship between the conductors and discriminants as ideals with their counterparts as ideles. Let IF denote the group of fractional ideals of F. There is a natural homomorphism id : A F ! IF described in Chap. V3 of [18]. Then we have f = id(' ) DL=F = id( L=F ): Moreover, we have an idelic analogous of the conductordiscriminant formula: Theorem 5.4 (Idelic ConductorDiscriminant Formula) Let L=F be a nite abelian extension of number elds corresponding to the open subgroup U of the idele class group JF . Then the idelic relative discriminant L=F of L=F is given by L=F = Y ' ; where ranges over all the characters of JF trivial on U. 5.3 The resolvent eld identity Nakagawa's proof establishes by means of Scholz' re ection that the terms correspond ing to extensions k0=k in C ( ) on one side of Ohno's series identity correspond to the terms for extensions k0=k in C (3 ) on the other side. To simplify our work with these identities, we introduce some notation for the Euler products in the identities. 62 Following equation (2.4) of Chapter 2, we de ne the following Euler factors: Ek0;v(s) = 8>>>>>>>>>>>>>>< >>>>>>>>>>>>>>: (1 + q2s v )2 if (k0=k; v) = (1); 1 + q4s v if (k0=k; v) = (2u); 1 + q2s v if (k0=k; v) = (2r); 1 q2s v + q4s v if (k0=k; v) = (3u); 1 if (k0=k; v) = (3r); (5.1) where (k0=k; v) denotes the splitting type of the place v of k in the extension k0=k. We are omitting the common factors that will cancel out in the generalized Ohno Nakagawa identity. From Theorem 3.1, the dual Euler factors are ^E k0;v(s) = Ek0;v(s) for v  3, and for v j 3 we have ^E k0;v(s) = 8>>>>>>>>>>>>>>< >>>>>>>>>>>>>>: q4s v (1 + q12s v + 2q14s v ) if (k0=k; v) = (1); q4s v (1 + q12s v ) if (k0=k; v) = (2u); q2s v (1 + q14s v ) if (k0=k; v) = (2r); q4s v (1 + q12s v q14s v ) if (k0=k; v) = (3u); 1 if (k0=k; v) = (3r); (5.2) so long as 3 is unrami ed in k. Again, we have omitted the factors that cancel out in the conjectured OhnoNakagawa identity. Then the cancelled OhnoNakagawa identity has the form X k02K ds k0=k o(k0) Y v1 ^E k0;v(s) = 3r2+m3ns X k02K ds k0=k o(k0) Y v1 Ek0;v(s): (5.3) By the same reasoning as behind Theorem 5.1, this conjecture is true if and only if X k02C (3 ) ds k0=k o(k0) Y v1 ^E k0;v(s) = 3r2+m3ns X k02C ( ) ds k0=k o(k0) Y v1 Ek0;v(s) (5.4) holds for all 2 k =k2. 63 It is important to note that all the exponents of qs v in all the Euler factors are even, and yet the exponent of 3s in the OhnoNakagawa identity is odd in a sense we will presently make clear, and that this implies there is a natural splitting of the OhnoNakagawa identity. First, we need to express the Euler products as ordinary Dirichlet series. Each nonarchimedean place v of k corresponds to a prime ideal pv in the ring o of integers of k, satisfying qv = N(pv) = (o : pv), the absolute norm of pv. The absolute norm of the ideal 3o generated by 3 in o is just 3n where n = [k : Q]. Then the two sides of our conjecture expand into series of the following form: 3r2+m3ns X k02K ds k0=k o(k0) Y v1 Ek0;v(s) = X k02K X a Ck0;a N(33Dk0=ka2)s X k02K ds k0=k o(k0) Y v1 ^E k0;v(s) = X k02K X a ^ Ck0;a N(Dk0=ka2)s ; where the coe cients Ck0;a, ^ Ck0;a are ordinary rational numbers with denominator a divisor of 6. Here the sum over a ranges over all integral ideals of o; however, due to the nature of the Euler products we may assume that the prime power factor of a corresponding to any prime ideal pv is pj v for 0 j 2, except for the prime ideals pv lying over 3 in the dual series, which may have exponents 0 j 4. In order for this conjectured identity to hold, the sum of the coe cients Ck01 ;a1 for given M = N(33Dk01 =ka21 ) for varying k01 2 K and a1 in o must equal the sum of the coe cients ^ Ck02 ;a2 for M = N(Dk02 =ka22 ) and varying k02 2 K and a2 in o. The terms cancelling in this subidentity would satisfy N(Dk02 =k) = N(3Dk01 =kc2) for some fractional ideal c in k. In the case k = Q, this equality of norms together with the fact that k1 and k2 have opposite splitting types at 1 implies that Dk2 = 3Dk1 modulo multiplication by squares, where Dk1 and Dk2 are the discriminants, as signed integers, of k1 and k2 64 respectively. The resolvent eld of k1 is then F = Q( p Dk1 ), while the resolvent eld of k2 is the dual ^ F = Q( p 3Dk1 ). This proves that the OhnoNakagawa identity for Q holds only if all the resolvent eld identities (5.4) are true. This completes the proof of Theorem 1.4, which we restate as follows: Theorem 5.5 For a squarefree integer d, let C (d) denote the collection of all ex tensions k=Q of degree at most 3 with resolvent eld Q( p d). Then for d > 0, we have X k2C (3d) ds k o(k) Y p ^E k;p(s) = 313s X k2C (d) ds k o(k) Y p Ek;p(s); X k2C (3d) ds k o(k) Y p ^E k;p(s) = 33s X k2C (d) ds k o(k) Y p Ek;p(s): Everything in the above identities is the same as in the whole OhnoNakagawa identity, just split according to the resolvent elds. For general ground elds k, the norm equality N(Dk02 =k) = N(3Dk01 =kc2) does not strictly imply that Dk02 =k = 3Dk01 =kc2 as ideals (since there are possibly di erent prime ideals of the same norm). However, based on the role of Scholz' re ection in Nakagawa's proof, it is still natural to suppose that the identity splits according to the resolvent elds. After this, we shall work on simplifying the resolvent OhnoNakagawa identity (5.4) by means of class eld theory. 65 CHAPTER 6 Examples of the resolvent OhnoNakagawa identity In this chapter, we shall use known tabulations of extensions of degree at most 3 to verify nite analogues of the resolvent OhnoNakagawa identity. The examples in this chapter provide precise numerical evidence for Conjectures 1.1 and 1.2; the approach is di erent from the equalities of class numbers established in Ohno's original paper [14]. Here, instead of calculating class numbers of integral binary cubic forms, we use existing tables of number elds and calculations of their splitting types at di erent places to check the conjectures recast as an equality of nite sums of nite Euler products. These equalities come from eld extensions with bounded rami cation, while the discriminants of the binary cubic forms involved may be enormous and far beyond the tables calculated by Ohno. 6.1 The nite OhnoNakagawa identity Just as in Chapter 5, choose 2 k =k2, and let F = k( p ) and ^ F = k( p 3 ). Let S be a nite set of places of k containing all in nite places and all places v dividing 3dF=kd ^ F=k. Let CS( ) = CS(F) be the set of all extensions k0=k of degree at most 3 with resolvent eld F and which are unrami ed for all places v =2 S. Class eld theory implies the set CS( ) is a nite set of extensions k0=k. Similarly, CS(3 ) = CS( ^ F) is a nite set of extensions. Thus, for all extensions k0=k in CS( ) and in CS(3 ), the relative discriminant dk0=k is divisible only by qv for places v 2 S. The terms in the Dirichlet series a Ms for which M is divisible only by qv for v 2 S must cancel out on both sides of the conjectured resolvent eld identity (5.4). This proves the following 66 theorem: Theorem 6.1 The generalized resolvent eld conjecture (1.6) is true if and only if for all 2 k =k2 and all nite sets of places S containing all places v j 1 and v j 3dF=kd ^ F=k, where F = k( p ), ^ F = k( p 3 ), we have X k02CS( ^ F) ds k0=k o(k0) Y v2S ^E k0;v(s) = 3r2+m3ns X k02CS(F) ds k0=k o(k0) Y v2S Ek0;v(s): (6.1) Recall that n = [k : Q], r2 is the number of complex places of k, o(k0) is the automor phism order of k0=k (so 1,2,3, or 6) as de ned on p. 21, Ek0;v and ^E k0;v are the Euler factors de ned in (5.1) and (5.2), and m is the number of real embeddings of k for which is positive. In particular, by Nakagawa's theorem all these nite identities are true when k = Q. The crucial aspect of this theorem is that there are only nitely many terms on both sides of the identity. We shall call this the nite OhnoNakagawa identity for S and . In this chapter, we con rm this identity for a fair number of cases based on eld data from various sources. Section 6.1 presents numerous con rmations of Nakagawa's theorem based on the data of elds of degree at most 3 over Q, while Section 6.2 presents con rmations of our new conjecture over k = Q(i). 6.2 Resolvent identities over Q To verify the nite OhnoNakagawa identity for a given nite set of places S of k and element 2 k , we need a list of the extensions k0=k contained in CS( ), their relative discriminants, and their splitting types at the places v 2 S. We have obtained this data from several independent sources, which we shall identify below. Again, as we have established, the identities are known consequences of Nakagawa's theorem, but this veri cation is quite di erent from Ohno's original data, and we feel these examples of identities are worth describing in detail. 67 Cohen et al. give a survey of counting number elds in [3]. The Bordeaux com putational number theory group has made available tables of number elds of degree at most 7 and discriminants below speci c bounds at http://pari.math.ubordeaux1.fr/pub/pari/packages/nftables/ We originally consulted the les in the Bordeaux archive called T20.gp, T22.gp, T31.gp, T33.gp, where the two numbers refer to degree n and number of real places r1 of the number elds. A typical line in one of these les would be of the form: [321,[1,1,4,1],1,[]] which lists the discriminant 321 of the number eld, the vector of coe cients of a generating polynomial x3x24x+1, the class number and the structure of the class group of the number eld. We are only interested in the discriminant and generating polynomial. These provided veri cation of the identities over Q at least for small sets of places S. In particular, for S = f2; 3g, we may extract from the les those elds with discriminant of the form 2a3b. In addition to Q, there are 7 quadratic elds and 9 cubic elds up to conjugacy. We list these elds in Table 6.1 sorted by the squarefree part of their discriminant Dk. As a reminder, these lists include only one of each conjugate triple of noncyclic cubic extensions of Q. Thus, in our identity, we must use o(k0=k) = 3 for cyclic cubic extensions and o(k0=k) = 1 for noncyclic cubic extensions. We can identify the cyclic cubic extensions k0=k over k = Q as those cubic extensions with discriminant equal to a perfect square. Table 6.1 includes only one cyclic cubic eld of discriminant 81. The next task in verifying the identities is to determine the Euler factors Ek0;p(s) and ^E k0;p(s) for each prime p = 2; 3. This requires determining the splitting type of k0 over kv = Qp. We use the generating polynomial supplied by the Bordeaux tables to carry this out, and we use the padic package in Maple to factor this polynomial padically. Here is the Maple procedure that accomplishes this 68 # Take a number field of degree <=3 and find its splitting type at p splittype:=proc(field,p) local x,pol,n,m: pol:=poly(field,x): n:=degree(pol): m:=nops([rootp(pol,p)]): if n=m then RETURN(1) elif ( field[1] mod p ) = 0 then # ramified if n=2 or m=1 then RETURN(3) else RETURN(5) fi: else # unramified if n=2 or m=1 then RETURN(2) else RETURN(4) fi: fi: end: This procedure assumes that field is a vector describing the number eld as con tained in the Bordeaux tables. Thus, according to the Bordeaux format, the entry field[1] is the discriminant of the eld. First, this procedures uses another pro cedure poly(field,x) that extracts the generating polynomial of the number eld with indeterminate x. Then it determines the degree n of this polynomial (1, 2, or 3), and the number m of roots of the polynomial in Qp through the rootp command in Maple's padic package. If n = m, then the polynomial splits completely over Qp and the type is (1). If not, it then tests to see if p is rami ed in k0 over k = Q by simply checking whether or not p divides the discriminant field[1]. Then in either case, the type is quadratic if n = 2 or n = 3 and m = 1, which means that the generating polynomial has an irreducible quadratic factor over Qp. In the procedure, types (1), (2u), (2r), (3u) and (3r) are numbered 1, 2, 3, 4, and 5. The results of 69 these calculations are also shown in Table 6.1. Given the determination of types, our Maple programs determine the Euler factors by substituting q = qv and x = q2s v into the corresponding entry of the arrays of formats for the Euler factors listed below eulerfac:=[ (1+x)^2, (1+x^2), (1+x), (1x+x^2), 1]: eulerfacdual:= [ x^2*(1+q*x+2*q*x^2), x^2*(1+q*x), x*(1+q*x^2), x^2*(1+q*xq*x^2), 1]: using the dual factor only for p = 3, when qv = 3fv under the assumption that 3 is unrami ed in kv with residue degree fv. The original Euler factors were calculated in [5] and presented here in equation (2.4), while the dual Euler factors for v j 3 were calculated in Theorem 3.1. These procedures allow us to evaluate the two sides of the nite OhnoNakagawa identity in Theorem 6.1. We present them in the form stated in Theorem 6.1 as sums of partial Euler products. The simplest identity covered by our conjecture is the case where S = f3g and = 1 or 3, since 3 is the only prime dividing 3dF=kd ^ F=k. In that case, the only elds entering the identities are Q, q1, k1 and k3, since these are the only elds for which the absolute value of the discriminant is a power of 3. Then the identities below have the Euler products (for only the prime p = 3) for the elds of discriminant 3, 243 on one side and the elds of discriminant 1, 81 on the other side. The Euler factor for 3 is determined by our recipes with the splitting type read from the last column of Table 6.1. = 1 1 2 3s 32 s 1 + 3 34 s + 243s = 313s 1 6 1 + 32 s 2 + 1 3 81s = 3 1 6 34 s 1 + 3 32 s + 6 34 s + 1 3 81s = 33s 1 2 3s 1 + 32 s + 243s 70 Both of these identities may be easily checked by hand to be true, as Nakagawa's theorem implies. Next we turn to the full list of elds which are unrami ed outside S = f2; 3g. For = 1; 2; 3; 6, we have m = 0, while for = 1;2;3;6 we have m = 1. For the purpose of comparison, we shall group the identities in pairs and 3 (mod squares) since these pairs have the same elds on both sides. = 1 1 2 3s 1 + 24 s 32 s 1 + 3 34 s + 108s + 243s 1 + 24 s + 2 972s = 313s 1 6 1 + 22 s 2 1 + 32 s 2 + 1 3 81s 1 22 s + 24 s = 3 1 6 1 + 22 s 2 34 s 1 + 3 32 s + 6 34 s + 1 3 81s 1 22 s + 24 s = 33s 1 2 3s 1 + 24 s 1 + 32 s + 108s + 243s 1 + 24 s + 2 972s As a small explanation, both of the above identities concern the two elds in Table 6.1 with = 1 and the ve elds with = 3. The Euler factors are determined according to the recipes given above, with the types read o the last two columns of Table 6.1. The remaining pairs of identities follow below = 2 1 2 24s 1 + 22 s 32 s 1 + 3 34 s + 216s 1 + 22 s = 313s 1 2 8s 1 + 22 s 1 + 34 s = 6 1 2 8s 1 + 22 s 34 s 1 + 3 32 s = 33s 1 2 24s 1 + 22 s 1 + 32 s + 216s 1 + 22 s = 3 1 2 4s 1 + 22 s 34 s 1 + 3 32 s + 324s 1 + 22 s = 313s 1 2 12s 1 + 22 s 1 + 32 s 71 = 1 1 2 12s 1 + 22 s 32 s 1 + 3 34 s = 33s 1 2 4s 1 + 22 s 1 + 34 s + 324s 1 + 22 s = 6 1 2 8s 1 + 22 s 34 s 1 + 3 32 s + 6 34 s + 648s 1 + 22 s = 313s 1 2 24s 1 + 22 s 1 + 32 s + 1944s 1 + 22 s = 2 1 2 24s 1 + 22 s 32 s 1 + 3 34 s + 1944s 1 + 22 s = 33s 1 2 8s 1 + 22 s 1 + 32 s 2 + 648s 1 + 22 s These identities may be veri ed by elementary algebra to be correct; however, we also used Maple's algebraic simpli cation tools to verify them by computer. We next proceeded to check cases where the set S of places contains all primes up to and including a given prime p. We denote these sets of extensions by Sp. It turn out that for even relatively small primes p such as p = 11, the extensions may have discriminant as large as 22355272112 = 144074700, which is beyond the published Bordeaux tables. To go further, we used the program cubic written by Karim Belabas. The algorithm is established in [1], and the source code is available at http://www.math.ubordeaux.fr/~belabas/research/software/cubic1.2.tgz We made some minor modi cations in cubic to allow it to restrict output to only elds which are unrami ed for all p > 11. Table 6.2 gives the number of elds in CSp of both positive and negative discriminant, with the largest discriminant in each set also displayed. Fields are counted only up to conjugacy. With this data and our Maple procedures, we veri ed the nite OhnoNakagawa identity (6.1) for all cases comprised by CS11 . 72 As one simple further example, we shall take S = f2; 3; 7g (thus omitting 5) and = 7 and = 21. The negative discriminants counted are 7 (quadratic) and 567 = 347 (cubic), and the positive discriminants are 21 (quadratic) and 756 = 22337 (cubic). The identities (6.1) turn out to be = 21 1 2 7s 1 + 22 s 2 34 s 1 + 3 32 s 1 + 72 s + 567s 1 22 s + 24 s 1 + 72 s = 313s 1 2 21s 1 + 24 s 1 + 32 s 1 + 72 s + 756s 1 + 72 s = 7 1 2 21s 1 + 24 s 32 s 1 + 3 34 s 1 + 72 s + 756s 1 + 72 s = 33s 1 2 7s 1 + 22 s 2 1 + 34 s 1 + 72 s + 567s 1 22 s + 24 s 1 + 72 s Again, both identities may be veri ed by elementary algebra, although they must be true due to Nakagawa's theorem. After using the Bordeaux tables and Belabas' program to complete the above tests of the nite OhnoNakagawa identities, we learned of the program of John Jones and David Roberts which enumerates low degree elds with prescribed rami cation, which is exactly what we need to test these identities. The JonesRoberts algorithms are described in [10], and made available at the website http://hobbes.la.asu.edu/NFDB/ We used this program to con rm the list of elds provided by Belabas' program, as well as the OhnoNakagawa identities. It can be used to enumerate elds which are unrami ed except for primes at most 17, and thus provide more con rmation of Nakagawa's theorem. 73 6.3 Resolvent identities over Q(i) Here we take k = Q(i) and consider the extensions k0=k of degree at most 3. Such elds k0 have degree 2, 4 or 6, and all in nite places are complex. The Bordeaux tables include les T40.gp and T60.gp which list all quartic and sextic totally com plex elds up to conjugacy and with maximal discriminant 999988 and 199664, respectively. As it turns out, this is not large enough to verify the identity even for S = f1 + i; 3g. For example, by our earlier list for Q, the sextic eld k( 3 p 3) has discriminant (4)3(243)2 = 3779136. The papers [15] and [2] provide information about enumerating sextic elds. Fortunately, the JonesRoberts program allows us to enumerate all elds of degree at most 6 with prescribed rami cation at a small set of primes, and as we shall see this allows us to verify the nite OhnoNakagawa identity for k = Q(i) and S = f1 + i; 3g. At the website http://hobbes.la.asu.edu/NFDB/, we rst conducted a search for elds of degree 4, r1 = 0, r2 = 2 with arbitrary size discriminant, but rami cation possible only at p1 = 2 and p2 = 3. This would include all quadratic extensions of Q(i) which are unrami ed outside S = f1 + i; 3g. This produced a list of 29 degree 4 polynomials corresponding to each possible eld up to conjugacy. We next used Jones' program to determine the list of sextic elds with r1 = 0 and rami cation only at p1 = 2 and p2 = 3. This produced a list of 140 polynomials, which would include all cubic extensions of Q(i) unrami ed outside S = f1 + i; 3g. The lists contain one polynomial for each isomorphism class of eld matching the conditions imposed. The next task is to extract from these lists precisely those polynomials generating extensions of Q(i). For that purpose, we use the following basic fact from eld theory: Lemma 6.1 Let L=Q be a nite extension of degree n, and let K=Q be an extension of degree m j n. Let be an element of L such that L = Q( ). Then K is a sub eld of L if and only if the minimal polynomial of over K has degree n m . This will be a 74 factor of the minimal polynomial of over Q. Proof. Let p(x) 2 Q[x] be the monic minimal polynomial of 2 L; then the degree of p(x) is [L : Q] = n. Let q(x) be the monic minimal polynomial of over K which is assumed to have degree n=m where m = [K : Q]. Since the compositum LK is the same as the eld K( ), then [LK : K] = [K( ) : K] = n=m. Then by the tower law [LK : Q] = [LK : K][K : Q] = (n=m)m = n = [L : Q]. This proves LK = L and hence that K L. By minimality, q(x) is a factor of p(x). The converse, where we assume K L, immediately follows from the tower law [L : K] = [L : Q]=[K : Q]. Thus, to extract the extensions of Q(i), we simply have to check if the polynomials in the lists provided by Jones' program factor over Q(i). For example, the rst quartic eld in Jones' list is p(x) = x4 x2 + 1: In PARI, the discriminant of the number eld generated by a root is calculated by the command nfinit(p).disc, where p is the polynomial expression. This example has discriminant Dk0 = 144. We can calculate the factorization of p(x) over Q(i) by means of the command factornf( x^4 x^2 +1, y^2+1) with the result being p(x) = x4 x2 + 1 = (x2 ix 1)(x2 + ix 1) Thus, this number eld is a quadratic extension of Q(i). When we apply this test to the lists of polynomials produced by Jones' program, we nd that 5 of the quar tic polynomials and 13 of the sextic polynomials generate extensions of Q(i). These are presented in Tables 6.3 and 6.4. These tables also contain the absolute discrim inants Dk0 and the absolute norms of the relative discriminants dk0=k = N(Dk0=k) 75 calculated from the tower law given that DQ(i) = 4. From basic facts about dis criminants of towers of number elds (see [18], Proposition VIII.4.13), since Dk = 4, we have Dk0 = D2 k N(Dk0=k) = 16 N(Dk0=k) for [k0 : k] = 2, and Dk0 = D3 k N(Dk0=k) = 64 N(Dk0=k) for [k0 : k] = 3. The cubic factors of the sextic polynomials in Table 6.4 are given in Table 6.5. The quadratic and cubic extensions of k = Q(i) are given as k( ) where is a root of the quartic or sextic polynomial in our lists. The two factors of the polynomials over k may give nonconjugate extensions of k. We can also use PARI's command factornf to test each of the quadratic and cubic extensions of Q(i) to see if they contain the quadratic and cubic number elds listed in Table 6.1. The results are presented in the rst column of Table 6.3, where three elds k0 are identi ed as the biquadratic elds q1(i), q2(i), and q3(i), and in the fourth column of Table 6.4, where we see 9 of the sextic polynomials factor over the cubic elds kj , 1 j 9. Since k = Q(i) has class number 1, the relative discriminant of k0=k is of the form Dk0=k = Dk0=kZ[i], where Dk0=k is determined as an element of (Z[i] r f0g)=f 1g. Also, if k0 = k( ) for some 2 Z[i] and the monic minimal polynomial of over Z[i] is q(x), then the discriminant (q) of q(x) as a polynomial is equal to a square of a nonzero element in Z[i] times the generator of the relative discriminant of k0=k. Thus, (q) = u2Dk0=k for some u 2 Z[i] r f0g. Then N( (q)) = N(u)2 dk0=k. The fourth column of Table 6.5 shows the polynomial discriminant (q) of q(x), and the fth column shows N(u)2. The solutions to N(u) = 1, 2, and 4, resp., in Z[i] are u 2 f 1; ig, f 1 ig, and f 2; 2ig, resp. Then u2 2 f 1g, f 2ig, and 4, resp. Since 1 = i2 in Z[i], these calculations allow us to determine Dk0=k modulo squares from the calculations of (q). This is shown in the sixth column of Table 6.5. This is an easier calculation for the quadratic extensions of Q(i) in that the discriminant of each quadratic polynomial in Table 6.3 is also a relative discriminant of the extension. When the relative discriminant of q(x) is an integer multiple of 1 i, 76 then the conjugate factor q(x) has conjugate relative discriminant. Since 1 + i does not equal 1 i modulo squares in Z[i], these two factors q(x), q(x) give rise to non conjugate extensions over k. That explains why the factors q and q for dk0=k = 512, 4608, 23328 and 209952 in Tables 6.3 and 6.5 gives rise to nonconjugate extensions over k. The resolvent eld of each of the listed extensions is k( p Dk0=k) = k( p ) where is the squarefree part of Dk0=k. Since we are considering elds unrami ed outside S = f1+i; 3g, the integers of Z[i] which are divisible only by primes over S are equal modulo squares to precisely one of = 1; i; 1 i; 3; 3i; 3(1 i): In our tables, we have factored out squares and identi ed in the last column. The next issue is to completely determine for the factorization p(x) = q(x)q(x) whether the two factor polynomials q(x) and q(x) generate conjugate or nonconjugate extensions k0=k. They are conjugate over Q, but not necessarily over k. In the quartic case, the two extensions are k( p ) and k( p ) where is the generator of the relative discriminant modulo squares. By Kummer theory, these are the same extension if and only if 2 k2. Since k2 \ Q = Q2, this means N( ) must be a positive square in Q. Thus, for the nonsquare discriminants 32 and 288, the factors q(x) and q(x) generate two di erent quadratic extensions k0=k, which are not Galois over Q. This explains our notation for the seven quadratic extensions k0 of Q(i) in Table 6.3. There are relations among the splitting elds of the quartic polynomials. The quartics corresponding to Q3 and Q5 both have Galois group D4, while the others have Galois group C2 C2. The splitting elds of both the Q3 and Q5 quartics contains Q2. In general, suppose q(x) is a monic irreducible polynomial over k = Q(i), and that q(x) is its conjugate polynomial. If is a root of q(x), then is a root of q(x). Let K = k( ) and K = k( ). If K=k is a cyclic cubic extension, then K = K if and only if K is a Galois extension of degree 6 of Q, since then conjugation is 77 an automorphism of order 2 of K=Q. The Galois group is then cyclic C6 or the symmetric group S3. In the former case, K contains a cyclic cubic extension of Q, which is unrami ed outside S. The only possibility is the cyclic eld k1 of discriminant 81. From Table 6.4, the second sextic factors over k1, and thus the splitting eld is the compositum K2 = k1 k = k1(i), which is cyclic of degree 6 over Q. If the Galois group of K=Q is S3, then K contains a conjugate triple of nonconjugate cubic elds. Thus, this can be determined again by factoring over the cubic elds in Table 6.1. The cyclic extensions K=k have relative discriminant generator equal to a square in k, and from our list we see there are just three possible sextics all of which have DK=k = 81. The rst sextic factors over the cyclic cubic eld k1, while the second one in our list factors over k7. Thus, both those sextics have the same cubic extensions of k arising from the two cubic factors over k. The other two sextics each give rise to two distinct but conjugate cyclic cubic extensions of k. All the other cases listed in Table 6.5 correspond to noncyclic cubic extensions of k = Q(i), since Dk0=k is not a square in k. Suppose the three roots of q(x) generate the three conjugate cubic extensions K1, K2 and K3 over k. Suppose that the compositum of these extensions is the S3extension L of k. Then the roots of q(x) generate the extensions K1, K2, and K3, and their compositum is L. The triple fKjg is the same as fKjg if and only if L = L, which again means that L=Q is a Galois extension of degree 12. Assuming L = L, the Galois group G of L=Q contains S3 as a normal subgroup corresponding to the sub eld k = Q(i), and contains complex conjugation as an order 2 automorphism. Write S3 = h ; j 3 = 2 = 1; = 1i, with all these automorphisms xing k. Let be complex conjugation on k extended to L. Since S3 is a normal subgroup of G, the three order 2 elements , and 2 are permuted by conjugation by . Thus, at least one is xed by conjugation by . We may relabel to be one of the xed ones so that = . This implies that h ; i is an order 78 4 subgroup of G, and thus corresponds to a real cubic extension of Q. That means that the original sextic polynomial factors over this cubic eld, which would have to be in the list of elds unrami ed outside S on page 85. We may check which of the sextics in Table 6.4 factors over the cubics listed on page 85, and if so then we conclude L = L. This accounts for our notation for the 17 cubic extensions of Q(i) (up to conjugacy) listed in Table 6.5. Also, Table 6.4 shows the cubic elds kj named in Table 6.1 over which these sextic polynomials factor. The splitting eld of the K4 sextic contain the splitting elds
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Title  Generalizing the Theorem of Nakagawa on Binary Cubic Forms to Number Fields 
Date  20120701 
Author  Dioses, Jorge 
Keywords  binary cubic forms, class field theory, cubic extensions, Dirichlet series 
Department  Mathematics 
Document Type  
Full Text Type  Open Access 
Abstract  The goal of this thesis is to study possible generalizations of a theorem of Nakagawa, first stated as a conjecture by Ohno, that gives a relationship between cubic fields of positive and negative discriminant. This theorem is described as an equation of Dirichlet series whose coefficients are class numbers of binary cubic forms. Its proof makes an extensive use of class field theory. Our approach for generalizing this result to cubic extensions of an arbitrary number field is to write the series in terms of ideles following the works of Datskovsky and Wright. By comparing the residues at their poles, we are able to deduce a conjecture that is a direct generalization of the original theorem. In the process of refining this generalization, we obtain some results concerning local integrals and series over idele group character. Moreover, we use tables of number fields which are currently available and computer algebra systems to provide strong evidence for the validity of the proposed conjecture. 
Note  Dissertation 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  GENERALIZING THE THEOREM OF NAKAGAWA ON BINARY CUBIC FORMS TO NUMBER FIELDS By JORGE DIOSES Bachelor of Science in Mathematics Ponti cal Catholic University of Peru Lima, Lima, Peru 1997 Licentiate in Mathematics Ponti cal Catholic University of Peru Lima, Lima, Peru 2000 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial ful llment of the requirements for the Degree of DOCTOR OF PHILOSOPHY July, 2012 COPYRIGHT c By JORGE DIOSES July, 2012 GENERALIZING THE THEOREM OF NAKAGAWA ON BINARY CUBIC FORMS TO NUMBER FIELDS Dissertation Approved: Dr. David Wright Dissertation Advisor Dr. Heidi Ho er Dr. Anthony Kable Dr. Chris Francisco Dr. Sheryl Tucker Dean of the Graduate College iii TABLE OF CONTENTS Chapter Page 1 Introduction 1 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Basic notation and review of zeta functions of binary cubic forms 13 2.1 Notation for number elds and local elds . . . . . . . . . . . . . . . 13 2.2 Binary cubic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Zeta functions of binary cubic forms and Dirichlet series . . . . . . . 19 2.4 Fourier transforms and the dual Dirichlet series . . . . . . . . . . . . 23 3 Local integrals of a Fourier transform 29 3.1 Statement of the integral to be calculated . . . . . . . . . . . . . . . 29 3.2 Reductions of the local integral . . . . . . . . . . . . . . . . . . . . . 30 3.3 Evaluation of the local integral . . . . . . . . . . . . . . . . . . . . . 33 3.3.1 Type (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.2 Type (2u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.3 Type (2r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3.4 Type (3u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3.5 Type (3r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4 Veri cation of a simple identity . . . . . . . . . . . . . . . . . . . . . 41 4 Residues of the Dirichlet series and generalizing OhnoNakagawa 43 4.1 Filtrations of the Dirichlet series . . . . . . . . . . . . . . . . . . . . . 43 iv 4.2 Poles and residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.3 Residue of the dual Dirichlet series at s = 1 . . . . . . . . . . . . . . 48 4.4 Residue of the dual Dirichlet series at s = 5=6 . . . . . . . . . . . . . 50 4.5 Generalizing Ohno's conjecture . . . . . . . . . . . . . . . . . . . . . 52 5 Decomposing the Dirichlet series according to the resolvent eld 57 5.1 The resolvent eld of an extension k0=k of degree at most 3 . . . . . . 57 5.2 Conductors and discriminants of cubic extensions . . . . . . . . . . . 59 5.3 The resolvent eld identity . . . . . . . . . . . . . . . . . . . . . . . . 62 6 Examples of the resolvent OhnoNakagawa identity 66 6.1 The nite OhnoNakagawa identity . . . . . . . . . . . . . . . . . . . 66 6.2 Resolvent identities over Q . . . . . . . . . . . . . . . . . . . . . . . . 67 6.3 Resolvent identities over Q(i) . . . . . . . . . . . . . . . . . . . . . . 74 7 Expressing the resolvent Dirichlet series as sums of idele class group characters 91 7.1 Simpli cation of the generalized OhnoNakagawa conjecture . . . . . 91 7.2 Shintani's Dirichlet series as sums of idele class group characters . . . 93 BIBLIOGRAPHY 99 v LIST OF TABLES Table Page 6.1 Fields of degree 3 unrami ed outside 2,3. . . . . . . . . . . . . . . 85 6.2 Number of elds (up to conjugacy) of degree 3 unrami ed for primes > p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.3 Polynomials generating quadratic extensions of Q(i) unrami ed outside 1 + i, 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.4 Sextic polynomials generating cubic extensions of Q(i) unrami ed out side 1 + i, 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.5 Cubic polynomials generating cubic extensions of Q(i) unrami ed out side 1 + i, 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.6 Splitting type of 2 in k0=Q, and corresponding type of (1+i)Z[i] in k0=k. 89 6.7 Splitting type of 3 in k0=Q, and corresponding type of 3Z[i] in k0=k. . 90 vi CHAPTER 1 Introduction 1.1 Overview A fundamental problem in number theory is to describe in as precise a manner as possible the collection of algebraic number elds, meaning the extensions of nite degree of the rational number eld Q. For example, all the quadratic extensions may be precisely described as Q( p d) where d ranges over all squarefree integers not equal to 1. One corollary of this is that there is a simple correspondence between elds of positive discriminant Q( p d) and elds of negative discriminant Q( p d). Generalizing these simple statements to even just cubic elds is highly nontrivial. For instance, the smallest negative discriminant of an extension of degree 3 of Q is 23, corresponding to the polynomial x3 x + 1, while the smallest positive discriminant is 49 corresponding to x3 x2 2x + 1. Tables of discriminants of cubic elds, both positive and negative, were calculated in the late nineteenth and early twentieth century, and there was no apparent correlation between the two lists of positive and negative discriminants. In [14] Ohno found a correspondence between these elds, which was most easily explained in terms of class numbers of integral binary cubic forms. This correspondence was stated as a conjecture which was later proved in [13] by Nakagawa using class eld theory. The goal of this thesis is to investigate possible generalizations of this result to the case of quadratic and cubic extensions of an arbitrary number eld. It relies on the work of Datskovsky and Wright in [5] where the original de nitions are extended to global elds of characteristic not equal to 2 or 3. 1 In the second section of Chapter 1, we revise the original statements of the theorem of Nakagawa when the base eld is Q. We also state in simple terms a conjecture that would generalize this result to any number eld. The conjecture is shown as an equation between Dirichlet series, considered as sums over number elds of degree at most 3. We present in Chapter 2 the basic de nitions and notations for number elds, binary cubic forms, zeta functions, and Dirichlet series. We also de ne the appropriate Haar measures that are needed throughout this thesis. In Chapter 3, we evaluate certain local integrals that play an important role in the generalization process. These local integrals de ne the Dirichlet series of one of the sides of our conjecture. We work under the assumption that 3 is unrami ed in the given number eld. The calculation of the residues at its poles of the Dirichlet series mentioned above is given in Chapter 4. We use a technique given by Datskovsky andWright to accomplish this. Based on this, we are able to give a justi cation for our conjecture. In Chapter 5, we introduce additional de nitions that allow us to decompose our conjectured identity into a family of identities between sums over elds. Each side of these identities corresponds to a special eld. We also introduce some terminology and review some basic facts of class eld theory. They will be invoked in the last chapter of this thesis. We test numerically the validity of the conjecture in Chapter 6. In order to do so, we make use of available tables of cubic and quadratic elds. We do this for the quadratic imaginary eld Q(i). This provides strong evidence for the proposed identity. We also verify Nakagawa's theorem. Finally, in Chapter 7, we reduce both sides of our conjectured identity to sums over characters of an idele class group. We proved some results mentioned in the rst chapter. Using the results from Chapter 5, we express these Dirichlet series as sums 2 containing only ideles. 1.2 Statement of results In 1972, Shintani created the theory of zeta functions associated to the space of binary cubic forms in the paper [17], and used that theory to establish the basic analytic properties of two Dirichlet series 1(s) = X1 m=1 h1(m) + 1 3h2(m) ms 2(s) = X1 m=1 h(m) ms (1.1) where h(m) denotes the number of SL2(Z)equivalence classes of integral binary cubic forms of discriminant m, and, for m > 0, h1(m) and h2(m) denote the numbers of classes of discriminant m with isotropy group of order 1 and 3, respectively. Shintani showed that these Dirichlet series converge absolutely for Re(s) > 1, that they have meromorphic continuations to the entire splane, and that they satisfy a functional equation. In addition, he proved that they were holomorphic except for simple poles at s = 1 and s = 5=6, and he gave formulas for the residues of the Dirichlet series at these poles. With this information, Shintani was able to improve a theorem of Davenport [7] on the meanvalue of classnumbers of integral binary cubic forms, by giving a more precise formula for the error term. The functional equation that Shintani discovered actually expresses 1(1s) and 2(1 s) as linear combinations of the dual Dirichlet series ^ 1(s) = X1 m=1 ^h1(m) + 1 3 ^h 2(m) ms ^ 2(s) = X1 m=1 ^h (m) ms where the dual classnumbers are the numbers of SL2(Z)equivalence classes of integral binary cubic forms Fx(u; v) = x1u2 + x2u2v + x3uv2 + x4v3 where the middle coe cients x2, x3 are both divisible by 3. This turns out to be the natural dual lattice to the lattice of integral binary cubic forms. 3 Datskovsky andWright introduced adelic terminology and notation into Shintani's work in the papers [19, 5, 6, 4], and generalized Shintani's results to the space of binary cubic forms over an arbitrary global eld of characteristic di erent from 2 and 3. In [5], it was observed in Proposition 4.1 that Shintani's functional equation has a natural diagonalization. To state this diagonalization, we write (s) = 1(s) 1 p 3 2(s) ^ (s) = ^ 1(s) 1 p 3 ^ 2(s) We de ne the associated gamma factors to be r (s) = 2s 33s 2s (s) s 2 + 1 4 1 6 s 2 + 1 4 1 3 : The diagonalized functional equations are then r (1 s) (1 s) = 3 r (s)^ (s) for either choice of sign . In [14], Ohno observed that these diagonalized functional equations would be especially symmetric if there were an identity between the original Dirichlet series and the dual Dirichlet series. This suggested an identity between classnumbers for the lattice of integral forms and classnumbers for its dual lattice. By extensive computation of classnumbers, Ohno veri ed numerically the conjecture that for the original Shintani series (1.1) ^ 1(s) = 33s 2(s) ^ 2(s) = 313s 1(s) These identities imply that ^ (s) = 3 1 2 3s (s): With that identity, the diagonalized functional equations become " (1 s) (1 s) = " (s) (s) 4 where " (s) = 2s 3 3 2 s 2s (s) s 2 + 1 4 1 6 s 2 + 1 4 1 3 : Ohno also proved by means of Shintani's functional equation that his two conjectured identities above are logically equivalent. Once Ohno's conjecture was stated, work on the truth of it was swift, and Nakagawa proved the conjecture in 1998 in [13]. Nakagawa's proof is based partly on an idea of Scholz [16] relating the 3class group of quadratic elds Q( p d) to the 3class group of Q( p 3d). The research presented here is concerned with the generalization of Ohno's con jecture and Nakagawa's theorem to Shintani Dirichlet series for the space of binary cubic forms over a number eld k. Since the ring of integers o of the number eld k need not have class number 1, the direct generalization of Shintani's Dirichlet series to number elds is more easily expressed in terms of eld extensions k0=k of degree at most 3 than it is in terms of classnumbers of binary cubic forms. To explain this, we rst present an identity proved in [5] for Shintani's original series. Let (s) denote Riemann's zeta function, and k(s) the Dedekind zeta function of the number eld k. Let dk denote the absolute value of the discriminant of k=Q. For a eld k of degree at most 3 over Q, we de ne o(k) = 6 if k = Q and o(k) = [k : Q] otherwise. We also de ne Rk(s) = 8>>>>>>< >>>>>>: (s)3 if k = Q; (s) k(s) if [k : Q] = 2; k(s) if [k : Q] = 3: Then Shintani's Dirichlet series are proved in [5] to be equal to 1(s) = 2 (4s) (6s 1) X k tot. real ds k o(k) Rk(2s) Rk(4s) 2(s) = 2 (4s) (6s 1) X k complex ds k o(k) Rk(2s) Rk(4s) where the rst series ranges over totally real k=Q of degree at most 3, and the sec ond series ranges over k=Q with one complex in nite place and degree at most 3. 5 Nakagawa also provided a proof of this identity in [12]. We can directly use this terminology to state the generalization of Shintani's series to number elds k. First, the series range over the extensions k0=k of degree at most 3. Again we de ne o(k0) = 6 if k0 = k and o(k0) = [k0 : k] otherwise. De ne dk0=k to be the absolute norm of the relative discriminant of the extension k0=k, and put Rk0(s) = 8>>>>>>< >>>>>>: k(s)3 if k0 = k; k(s) k0(s) if [k0 : k] = 2; k0(s) if [k0 : k] = 3: To state the generalized Shintani series over a number eld k, it remains to de ne the notion of signature of an extension k0=k of degree less or equal to 3. For each real place v (or real embedding) of k, either the places of k0 lying over v are either all real, in which case we say v(k0=k) = +, or there is a unique complex place w lying over v, and then we say v(k0=k) = . This situation is unique to extensions of degree at most 3. We call the vector (k0=k) = ( v(k0=k))v real the signature of k0=k. If r1 is the number of real places of k, then there are 2r1 possible signatures; we denote the set of such signatures by A. For each possible signature of k, we denote the set of extensions k0=k of degree at most 3 which have signature by K . At last, we may state the Shintani series for each signature 2 A as (s) = k(4s) k(6s 1) X k02K ds k0=k o(k0) Rk0(2s) Rk0(4s) (1.2) where the sum ranges over all extensions k0 in K . (Note that the factor of 2 has disappeared; in [19, 5] forms are considered equivalent relative to the group GL2 rather than SL2, and that accounts for the factor of 2.) Datskovsky and Wright prove in [5] these series extend to meromorphic functions of s which are holomorphic everywhere except for simple poles at s = 1 and s = 5=6. That paper also provides precise expressions for the residues of these series at 1 and 5=6. Finally, that paper 6 also proves a functional equation expressing (1s) as linear combinations of dual Dirichlet series ^ (s). By utilizing the adelic approach of [5], an expression nearly identical to equation (1.2) may be derived for the dual Dirichlet series ^ (s). (Note: To be consistent with Nakagawa's theorem and Shintani's original notation, we modify the de nition of ^ (s) in DatskovkyWright by a constant factor, to be explained in Chapter 2.) The only di erences are due to new local factors at primes v lying over 3. These new local factors were calculated for Q by Nakagawa in Lemma 3.6, p. 121, of [13]. In order to give the formula for ^ (s), we rst need to de ne the splitting type of the prime v of k in an extension k0 of degree at most 3: Type (1): k0 k kv = k3 v, or k2 v, or kv; Type (2u): k0 k kv = kv F, or F, where F=kv is quadratic unrami ed; Type (2r): k0 k kv = kv F, or F, where F=kv is quadratic rami ed; Type (3u): k0 k kv = F where F=kv is cubic unrami ed; Type (3r): k0 k kv = F where F=kv is cubic rami ed: Then we use the technique of DatskovskyWright to prove the following: Theorem 1.1 Let k be a number eld of degree n in which 3 is unrami ed. Then the dual Dirichlet series for each signature over k may be expressed as ^ (s) = k(4s) k(6s 1) X k02K ds k0=k o(k0) Rk0(2s) Rk0(4s) Y vj3 Tk0;v(s) (1.3) 7 where for each prime v j 3 we have qv = j3j1 v , and we de ne the rational functions Tk0;v(s) = 8>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>: q4s v 1 + q12s v + 2q14s v (1 + q2s v )2 if v is of type (1) in k0; q4s v 1 + q12s v 1 + q4s v if v is of type (2u) in k0; q2s v 1 + q14s v 1 + q2s v if v is of type (2r) in k0; q4s v 1 + q12s v q14s v 1 q2s v + q4s v if v is of type (3u) in k0; 1 if v is of type (3r) in k0: The proof of this theorem appears in Chapter 3. We expect that a similar theorem holds when 3 is rami ed in k, but we have not yet completed the necessary local calculations. The goal of this thesis is to relate the collection of dual Dirichlet series ^ (s) to the collection of original series (s). Just as in Nakagawa's theorem, it will emerge that the proper generalization relates ^ (s) to (s), where the signature is the negative of in the sense that for each real place v of k we have v = + if and only if ( )v = . At the end of this introduction, we shall explain why this involution should occur. Datskovsky and Wright proved that the series (s) and ^ (s) have meromorphic continuations to the entire splane which are holomorphic except for simple poles at s = 1 and s = 5=6, and they gave explicit formulas for the residues of (s) at 1 and 5=6. In order to test the relationship between ^ (s) and (s), in Chapter 4 we use the results of DatskovskyWright to calculate similar formulas for the residues of ^ (s) at 1 and 5=6. Based on those formulas, we prove the following: Theorem 1.2 Let k be a number eld of degree n with r1 real places and r2 complex places. For every signature over k for which we have m real places v with v = +, the identity ^ (s) = 3r2+m3ns (s) 8 is true at s = 1 and 5=6. Moreover, this is the only expression of the form 3A+Bs for which this theorem is true. Notice that this theorem makes no condition on whether 3 is rami ed or not in k. That led us to conjecture this generalization of the OhnoNakagawa identity: Conjecture 1.1 (Generalized Ohno Conjecture) Let k be a number eld of degree n with r1 real places and r2 complex places. For every signature over k for which we have m real places v with v = +, we have: ^ (s) = 3r2+m3ns (s) (1.4) To see how this is consistent with Nakagawa's theorem, in that case we have n = 1, r1 = 1 and r2 = 0. Then in this conjecture + corresponds to 1 in Shintani's notation, and corresponds to 2. For signature = +, we have m = 1 and thus ^ 2(s) = 313s 1(s), while for = we have m = 0 and ^ 1(s) = 33s 2(s). This is precisely Nakagawa's theorem. The remainder of this thesis is dedicated to the reduction of this conjecture to manageable pieces which we hope may be proved by class eld theory and the ideas of Nakagawa. First, by direct substitution of equations (1.2) and (1.3) into conjecture (1.4), we see that a number of factors directly cancel out, leaving only X k02K ds k0=k o(k0) Rk0(2s) Rk0(4s) Y vj3 Tk0;v(s) = 3r2+m3ns X k02K ds k0=k o(k0) Rk0(2s) Rk0(4s) (1.5) In examining this proposed identity, we see that the Euler products multiplying ds k0=k are all of the form X a ca N(a)2s where a ranges over the integral ideals of k, N(a) denotes the absolute norm of a, and the coe cients ca are ordinary rational numbers. The important point here is the factor 2 in the exponent. Since 33ns = N(3o)3s for the ideal generated by 3 in k, the di erence must be made up in the norms of the relative discriminants dk0=k. To be more precise, consider an extension k01 =k counted on the left side of the identity with a corresponding term equal to a constant multiple of 9 dk01 =k N(a1)2 s . This may be `counteracted' by a term dk02 =k N(3o)3 N(a2)2 s on the right side if and only if the ratio of norms of the relative discriminants dk02 =k=dk01 =k is divisible by an odd power of N(3o). This is one suggestion that the conjectured identity (1.5) may be split into a sequence of identities by restricting the elds k0 included in the summations on each side. That idea is also promoted by the one of the key ideas of Nakagawa's proof, which in turn was based on a theorem of Scholz [16]. For an extension k0=k, let L=k be its Galois closure. If k0=k has degree at most 3, then L=k contains a unique subextension F = k( p ) of degree at most 2, generated by the square root of the discriminant of any generating element of k0 over k. We call F the resolvent eld of k0=k. Note that F = k if k0=k is trivial or a cyclic cubic extension. We de ne the dual resolvent eld to F to be ^ F = k( p 3 ). Thus, F and ^ F are sub elds of the extension F( p 3) obtained by adjoining the cube roots of unity to F. The relative discriminants of F=k and ^ F=k are, up to multiplication by the square of an ideal of k, equal to and 3 , respectively. It will turn out that the odd power of 3 in the identity (1.5) will be accounted by restricting the terms on the left to those extensions k0=k with resolvent eld equal to ^ F and the terms on the right to those k0=k with resolvent eld F. This motivates the following re ned conjecture: Conjecture 1.2 (Resolvent Field Identity) Let k be a number eld of degree n with r1 real places and r2 complex places. Let be a nonzero element of k, and de ne C ( ) to be the set of extensions k0=k of degree at most 3 whose resolvent eld is k( p ). Let m be the number of real embeddings of k for which the image of is a positive number. Then, using the notation for Shintani's series which we have previously developed, we have the identity X k02C (3 ) ds k0=k o(k0) Rk0(2s) Rk0(4s) Y vj3 Tk0;v(s) = 3r2+m3ns X k02C ( ) ds k0=k o(k0) Rk0(2s) Rk0(4s) (1.6) Since the generalized Ohno conjecture is just the sum of all these resolvent eld 10 identities for ranging over the nonzero elements of k modulo multiplication by squares (we will denote this set by k =k2), we may at least state: Theorem 1.3 If the resolvent eld identity is true for all nonzero in k, then the generalized Ohno conjecture (1.4) is true for k. For the one known case k = Q, we shall explain in detail why the converse is true, thus proving: Theorem 1.4 For k = Q, the resolvent eld identity is true for all nonzero in Q. The last part of this thesis is devoted to the analysis necessary to eventually complete the proof of the resolvent eld identity in terms of class eld theory. For nonzero 2 k, the elds k0 2 C ( ) have Galois closure L containing F = k( p ). If [k0 : k] 2, then F = k0. If [k0 : k] = 3, then L is a cyclic cubic extension of F. If k0=k is cyclic cubic, then L = k0 and F = k. If k0=k is noncyclic cubic, then L is an S3extension of k containing F, and L contains three cubic extensions of k all conjugate to k0. Class eld theory implies the abelian Galois extensions of a number eld F correspond to the open subgroups (and hence characters) of the idele class group JF of F. Basic notation and statements of class eld theory will be provided in a later section. In this case, the cyclic cubic extensions L=F correspond bijectively to the open subgroups of index 3 in the idele class group JF . These open subgroups in turn correspond in a onetotwo way to nontrivial complex characters of JF satisfying 3 = 1. These observations will allow us to rewrite the resolvent eld series in Theorem 1.3 as a sum over idele class characters of order dividing 3. To state this precisely, for any resolvent eld F let X(F) denote the group of characters of the idele class group JF of F such that 3 = 1 and also NF=k = 1 if [F : k] = 2, where NF=k denotes the relative norm. If F = k, the latter condition is omitted. The conductor of such a character is an integral ideal f in F. Let N(f ) denote the absolute norm of the conductor of . Then for the extension k0=k 11 corresponding to the kernel of , we have dk0=k = dF=k N(f ). The idele class character induces a character on all prime ideals P of F where P  f . By abuse of notation, we will write this character's value as (P). For prime ideal factors of the conductor, we will extend this de nition by setting (P) = 0. Then we will prove: Theorem 1.5 Let k be a number eld, be a nonzero element of k, and F = k( p ). Then, using the notation we have previously developed for the collection C ( ) of ex tensions k0=k and for the character group X(F), we have the identity X k02C ( ) ds k0=k o(k0) Rk0(2s) Rk0(4s) = ds F=k o(F) X 2X(F) N(f )s Y P 1 + (P) N(P)2s (1.7) where o(F) = 3 if F = k and 1 otherwise, and the product is taken over all prime ideals of F. We also will produce a similar expression for the dual discriminant eld series as a sum over cubic characters of the idele class group J^ F , although this is complicated by the presence of the Euler factors Tk0;v for places vj3. The cubic characters of JF and J^ F are related to the cubic characters of the compositum F ^ F. Using the fact that the compositum F ^ F contains the cube roots of unity, Scholz was able to use this relationship to deduce a relationship between the threeclassnumbers of F and ^ F in [16]. The completion of our project and the proof of the generalized Ohno conjecture at least for elds k where 3 is unrami ed would follow from re ning Scholz' ideas to establish the resolvent eld series identity for all nonzero in k. Our future research will be directed toward providing this proof. 12 CHAPTER 2 Basic notation and review of zeta functions of binary cubic forms 2.1 Notation for number elds and local elds Our notation for number elds, local elds, adele rings and idele rings, etc., will be fairly consistent with that presented in [19, 5, 6] and somewhat with Weil [18]. Let k be a number eld of degree n over the rational number eld Q. For any place v of k, we use kv to denote the completion of k at the place v. Let r1 and r2 be the numbers of real and complex, respectively, places of k, i.e. places such that kv = R and C, respectively. Then r1 + 2r2 = n. These together form the set of in nite places of k, and we shall write v j 1 for these places. For all local elds kv, we normalize the absolute value j jv to be the modulus of any additive Haar measure on kv. This means that (aU) = jajv (U) for any open subset U of kv, any Haar measure of kv, and any nonzero element a of kv. For real places v, this means jxjv is the customary absolute value on R, while for complex places v this means jxjv is the square of the usual absolute value on C. For real places v, we choose the Haar measure dvx on kv such that R 1 0 dvx = 1, i.e. the usual Lebesgue measure on R. For complex places v, we choose dvx so that the measure of the unit circle fx 2 C : jxjv 1g is 2 . That means dvx is twice the usual Lebesgue measure on C; the reason for this choice is that it is more convenient for dvx to represent the di erential form jdx ^ dxj in integration formulas. For any nite place v of k, we write v  1, and we denote the maximal compact subring of kv by ov. The unique prime ideal of ov is denoted pv, and we choose a generator of the principal ideal pv and name it v, called a uniformizer of kv. The 13 modulus qv of kv is the order of the nite eld ov=pv. The additive Haar measure dvx on kv is normalized so that R ov dvx = 1. The absolute value jajv on kv is normalized so that dv(ax) = jajvdvx, which implies that j vjv = q1 v . If we work with a nonarchimedean local eld K without reference to a global eld k, we shall denote the maximal compact subring by O, the unique prime ideal by P, a uniformizer by , the modulus by q, the normalized Haar measure by dx, and the normalized absolute value by j j. For any ring R (always commutative with identity), we denote the subgroup of invertible elements by R . Hence, for a eld K, K denotes the nonzero elements of K. Thus, the unit subgroup of ov is denoted by o v , and this is the same as the subgroup of elements x satisfying jxjv = 1. For a number eld k, we denote the ring of adeles by A = Ak = Q0 v kv and the group of ideles by A k = Q0 v k v , where these are restricted direct products in the usual sense. The ideles correspond to units in the ring of adeles, but the restricted product topology is not the same as the subspace topology. The idele norm jajA of an idele a 2 A k is the modulus of multiplication by a relative to any Haar measure on the adeles Ak. This means jajA = Q v javjv, where av denotes the component of a at the place v. For all x 2 k embedded along the diagonal in A k , we have the product formula jxjA = 1. We choose for the Haar measure on the additive group of adeles Ak the restricted tensor product measure dAx = N v dvxv, where x = (xv)v. The number eld k embeds along the diagonal as a discrete subgroup of A such that A=k is compact. With this choice of measure, the induced measure of the quotient is R A=k dAx = d1=2 k , in terms of the absolute value of the discriminant of k (see Prop. V.4.7 in [18]). On the multiplicative group k v of nonzero elements, we choose Haar measures d vx = dvx jxjv if v is an in nite place, and for nite places we choose d v x so that the unit group o v has measure 1. On the ideles A k , we choose the Haar measure 14 d A x = N v d v xv. By Prop. V.4.9 of [18], the measure of the set C(m), the image in A k =k of all ideles x satisfying 1 jxjA m, is k logm with k = 2r1(2 )r2hkRk ek ; where hk is the class number of k, Rk is the regulator of k, and ek is the number of roots of unity in k. Let A1 denote the subgroup of ideles x with idele norm jxjA = 1. Then k is a subgroup of A1, and the quotient group A1=k is compact. (See Theorem IV.4.6 in [18].) There is an embedding of the group of positive real numbers R+ into the ideles A , which we will denote z(t), satisfying jz(t)jA = t for all t 2 R+. One such embedding is de ned as the idele z(t) = (zv(t))v where zv(t) = 1 for all nite places v and zv(t) = t1=n for all in nite places v, with n = [k : Q]. Then we may decompose the Haar measure for A in the following way Z A (x) d A x = Z 1 0 Z A1 (z(t)x) d1 Ax dt t ; for any integrable function . Then the measure of A1=k induced by d1 Ax is k. Finally, we introduce our notation for the Dedekind zeta function k(s). Let o = ok denote the ring of integers of k, and for each integral ideal a of o let N(a) denote the absolute norm of a, i.e. the cardinality of the quotient ring o=a. The Dedekind zeta function is k(s) = X a N(a)s = Y p 1 N(p)s 1 where the sum extends over all integral ideals a and the product extends over all prime ideals p. Equivalently, this may be written as a product over all nite places of v as k(s) = Y v1 (1 qs v )1: This zeta function converges locally uniformly for Re(s) > 1 and has a meromorphic continuation to the entire splane which is holomorphic except for a simple pole at 15 s = 1 with residue Res s=1 k(s) = k d1=2 k : These facts and more may be found in VII.6 of [18]. 2.2 Binary cubic forms This notation is based on [17, 19, 5]. A binary cubic form is an expression Fx(u; v) = x1u3+x2u2v+x3uv2+x4v3 where xj , 1 j 4 are the coe cients. We will generally think of a fourdimensional vector x = (x1; x2; x3; x4) as a binary cubic form. The module of binary cubic forms with coe cients in a ring R is denoted by VR. There is a natural representation of the group G = GL2 on V given by Fg x(u; v) = 1 det g Fx 0 B@ (u; v) 0 B@ a b c d 1 CA 1 CA for x 2 V and g = 0 B@ a b c d 1 CA 2 G. We say two forms x; y are Gequivalent if y = g x for some g 2 G. The discriminant of the form x is a homogeneous polynomial P(x) of degree 4 (see p. 35 in [5]). This polynomial satis es P(g x) = (det g)2P(x) for g 2 G and x 2 V . A binary cubic form x is de ned to be singular if and only if P(x) = 0. We denote the hypersurface of singular forms in V by S, and the subset of nonsingular forms by V 0. Both subsets are Ginvariant. Just as in Shintani [17], we de ne the bilinear form [x; y] = x1y4 1 3 x2y3 + 1 3 x3y2 x4y1 on V . For the involution g = (det g)1g on G, this form satis es [g x; g y] = [x; y] for all x; y 2 V and g 2 G. We next turn to binary cubic forms over a eld K. The splitting eld K(x) of a binary cubic form x 2 VK is the smallest extension of K (in a given algebraic closure 16 of K) for which the form factors into linear factors de ned over K(x). The splitting eld is either K, a quadratic or cubic cyclic extension of K, or an S3extension of K. The key connection between the space of binary cubic forms and eld extensions of K is the following fact: Proposition 2.1 Two nonsingular binary cubic forms x; y 2 VK are GKequivalent if and only if their splitting elds are the same K(x) = K(y). When we wish to refer to the points of G and V de ned over a ring R, we will write GR and VR, for example, Gk, Vk, Gkv , Vkv , GA, VA, etc. We shall choose the Haar measure dvx on Vkv to be simply the product of the four coordinate measures dvxj and similarly for the measure dAx on VA. We should point out that this choice is di erent from that of DatskovskyWright [5], where the selfdual measure was chosen relative to a standard biinvariant for on VA. We are changing this choice so that the de nition of the dual Dirichlet series will be in agreement with the papers of Shintani, Ohno, and Nakagawa in the case of k = Q. For a local eld K, we choose the invariant measure dg on g 2 GK in the same manner as in [5]. To review, we de ne UK to be the maximal compact subgroups the orthogonal group O(2) if K =R, the unitary group U(2) if K = C, and the group GO if K is nonarchimedean with maximal compact subring O. We de ne BK to be the Borel subgroup of lower triangular matrices in GK. Then by the Iwasawa decomposition GK = UKBK. We choose the invariant measure du on u 2 UK so that UK has measure 1. We de ne a rightinvariant measure db on b 2 Bk satisfying Z BK (b) db = Z K d t1 Z K d t2 Z K dc (n(c)a(t1; t2)) where the Haar measures on K and K are as previously de ned and we use the notation n(c) = 0 B@ 1 0 c 1 1 CA and a(t1; t2) = 0 B@ t1 0 0 t2 1 C A. Here is any integrable function 17 on GK. Then the measure dg on GK is de ned by Z GK (g) dg = Z UK du Z BK db (ub): In the case of a nonarchimedean eld K with maximal compact subring O, if is the characteristic function of the maximal compact subgroup GO, then Z GK (g) dg = Z UK du Z BO db = Z BO db = Z O d t1 Z O d t2 Z O dc = 1; by our choice of Haar measure on K and K . This con rms this measure satis es the condition that GO has measure 1. On the adelizations VA and GA, we take the invariant measures to be the restricted product measures dAx = N v dvxv and dAg = N v dvgv. Then Vk is a discrete cocom pact subgroup of VA such that R VA=Vk dAx = d2 k. On the quotient group GA=Gk, the set F(m) of all points g with 1 j det gjA m has measure k logm, where k = 2 r1 hkRk ek d3=2 k k(2): (This constant comes from the volume calculation after Prop. 6.3, p. 528, in [19], and the volume normalization on p. 66 in [5].) Let G1 A denote the subgroup of all g 2 GA with j det gjA = 1. Then Gk is a discrete subgroup of GA with quotient G1 A=Gk of nite invariant volume. De ne an embedding w(t) of t 2 R+ into GA by w(t) = a(z( p t); z( p t)), using the embedding z : R+ ! A de ned in the previous section. Then we have j detw(t)jA = t, and we may decompose the invariant volume on GA as follow Z GA=Gk (g) dAg = Z 1 0 Z G1 A=Gk (w(t)g) d1 Ag ! dt t ; for integrable functions (g) on GA=Gk. With this de nition, we see that the measure of G1 A=Gk with respect to d1 Ag is k. 18 Finally, we shall describe the orbits of nonsingular binary cubic forms over a local eld kv. By Proposition 2.1, these correspond to the possible splitting elds kv(x) over kv. For a complex place v, there is only one such splitting eld, namely, C, and thus there is only one nonsingular orbit. For a real place v, the splitting eld may be R or C corresponding to whether the discriminant P(x) is positive or negative in kv. For a nite place v, there are nitely many possible splitting elds kv(x) which we group into ve basic types: Type (1): kv(x) = kv, Type (2u): kv(x)=kv is quadratic unrami ed, Type (2r): kv(x)=kv is quadratic rami ed, Type (3u): kv(x)=kv is cubic unrami ed, Type (3r): kv(x)=kv is the Galois closure of a rami ed cubic extension Kv=kv. In the rst four types, we abbreviate Kv = kv(x), while in the last type Kv is a possibly nonGalois cubic extension whose Galois closure is kv(x). There is only one orbit of each type (1), (2u) and (3u), while there may be more than one orbits of types (2r) and (3r). We shall adopt the description of these given on pp. 3536 in [5]. In particular, for each orbit we shall choose orbital representatives x 2 Vkv as described in that paper. Those are arranged so that P(x ) is a relative discriminant (Kv=kv) of Kv=kv. Occasionally, we shall use Av to denote the set of orbits GkvnV 0 kv . 2.3 Zeta functions of binary cubic forms and Dirichlet series The adelic version of Shintani's zeta function associated to the space (G; V ) of binary cubic forms is Z(!; ) = Z GA=Gk !(det g) X x2V 0 k (g x) dAg; where ! is a quasicharacter on A k which is trivial on k and (x) is a SchwartzBruhat function on the adelization VA. Here Gk is a discrete subgroup embedded along the 19 diagonal in GA, and for every nonsingular form x 2 V 0 k the stabilizer subgroup Gk;x is a nite subgroup of order 1, 2, 3 or 6, depending on the splitting eld k(x) as described in [5]. Here we shall simplify the notation because we will not be referring to nonprincipal quasicharacters. We shall put !(x) = jxj2s A , and de ne Z(s; ) = Z GA=Gk j det gj2s A X x2V 0 k (g x) dAg; which is proved in [19] to be absolutely and locally uniformly convergent for Re(s) > 1, and to have a meromorphic continuation to the entire splane which is holomorphic except for simple poles at s = 1 and s = 5=6. By rewriting the inner summation over Gkequivalence classes and then exchanging summation and integration, we obtain Z(s; ) = X x2GknV 0 k 1 jGk;xj Z GA j det gj2s A (g x) dAg; where x ranges over representatives of the Gkequivalence classes of nonsingular forms in V 0 k. Assuming the SchwartzBruhat function = v v is of product form, the integral above has an Euler product Z GA j det gj2s A (g x) dAg = Y v Z Gkv j det gvj2s v v(gv x) dvgv: For each place v, the form x belongs to one v of nitely many orbits in V 0 kv over kv. Thus, there is an element gx;v 2 Gkv such that x v = gx;v x, using the standard orbital representatives mentioned in Section 2.2 (see page 67 of [5]). Note that P(x v ) = P(gx;v x) = (det gx;v)2 P(x). Then the Euler factor may be rewritten by substitution as Z Gkv j det gvj2s v v(gv x) dvgv = Z Gkv j det gvj2s v v(gvg1 x;v x v ) dvgv = j det gx;vj2s v Z Gkv j det gvj2s v v(gv x v ) dvgv = jP(x v )js v jP(x)js v Z Gkv j det gvj2s v v(gv x v ) dvgv 20 Note that for x 2 V 0 k we have jP(x)jA = Q v jP(x)jv = 1 by the product formula. Also, if k0=k is an extension of degree at most 3 whose Galois closure is the splitting eld k(x) of x, then P(x v ) is the vadic component of the adelic relative discriminant (k0=k) as de ned in [8], and so Y v jP(x v )jv = d1 k0=k in terms of the absolute norm of the relative discriminant of k0=k. We introduce the following notation for the local zeta functions of the space of binary cubic forms Z v (s; v) = Z Gkv j det gvj2s v v(gv x v ) dvgv: Then returning to the Euler product, we have Z GA j det gj2s A (g x) dAg = Y v Z Gkv j det gvj2s v v(gv x) dvgv = ds k0=k Y v Z v (s; v): In this notation, v denotes the local orbit corresponding to the form x or its corre sponding extension k0=k of degree at most 3. At this point, we are ready to convert the zeta function from a sum over orbits x 2 GknV 0 k to a sum over extensions k0=k of degree at most 3. We shall put o(k0=k) = jGk;xj in all cases where k(x)=k is a Galois extension of degree at most 3. When the splitting eld k(x)=k is an S3extension, it is the Galois closure of any of three conjugate cubic subextensions k0=k. We shall allow our series to include the same term for each of those cubic subextensions, and to compensate we set o(k0) = o(k0=k) = 3 instead of jGk;xj = 1. With these conventions, we now have the following series expansion of the adelic zeta function Z(s; ) = X k0=k ds k0=k o(k0) Y v Z v (s; v): (2.1) Again the local orbits v depend on the local type of k0=k over v. 21 The Euler products in (2.1) include factors for the in nite places v. Our next step toward producing Shintani's Dirichlet series is to factor out the local zeta functions for the in nite places. This is where the signatures come into play. If v is a complex place, there is only one nonsingular orbit. If v is a real place, there are two nonsingular orbits: one for forms of positive discriminant and one for forms of negative discriminant. We shall denote these choices by v = + and just as in the de nition of signature in Chapter 1.2. We write the signature as a vector = ( v)vj1 of these choices for all in nite places, and we denote the set of signatures by A (or A1 if we wish to emphasize that this is a choice of orbits at the in nite places). For 2 A1, we de ne Z (s; 1) = Y vj1 Z v (s; v): where 1 denotes the part of the SchwartzBruhat function corresponding to the in nite places of k. Thus, 1 is a rapidly decreasing C1function on the real vector space Q vj1 kv of dimension n. As in Chapter 1.2, for each signature , we de ne K to be the collection of all extensions k0=k of degree at most 3 which have signature . Then at last we have the decomposition of the adelic zeta function as follows Z(s; ) = X 2A1 Z (s; 1) (s; 0) (2.2) with (s; 0) = X k02K ds k0=k o(k0) Y v1 Z v (s; v) (2.3) where 0 = v1 v. Again, take note that the orbit v for each nite place is determined by the extension k0=k. For each nite place v, the SchwartzBruhat function v is a locally constant function of compact support on kv. Thus, (s; 0) does turn out to be essentially a Dirichlet series. To obtain Shintani's series in particular, we make a standard choice of these SchwartzBruhat functions at nite places, namely, v = 0;v, the characteristic 22 function of the submodule Vov of all binary cubic forms with coe cients in ov, the maximal compact subring of kv. With this choice, in the case of k = Q (or any eld k of class number 1) the sum over V 0Q in the de nition of the zeta function Z(s; ) reduces to a sum over V 0Z , the set of nonsingular integral binary cubic forms, and we have the identity Z(s; ) = Z1(s; 1) = Z GR=GZ j det gjs X x2V 0 Z 1(g x) dRg; exactly the zeta function de ned by Shintani in [17]. Thus, the natural generalization of Shintani's Dirichlet series is (s) = X k02K ds k0=k o(k0) Y v1 Z v (s; 0;v) DatskovskyWright calculated these local integrals to be Z v (s; 0;v) = (1 q16s v )1(1 q2s v )1 (2.4) 8>>>>>>>>>>>>>>< >>>>>>>>>>>>>>: (1 + q2s v )2; if v is of type (1); 1 + q4s v ; if v is of type (2u); 1 + q2s v ; if v is of type (2r); 1 q2s v + q4s v ; if v is of type (3u); 1; if v is of type (3r); where qv denotes the module of the nite place v. 2.4 Fourier transforms and the dual Dirichlet series The analytic properties of Shintani's Dirichlet series are derived by means of the Poisson Summation Formula and the use of Fourier transforms. We will establish our conventions for Fourier transform in this section and then review the functional equation for the adelic zeta function de ned in the previous section. Then we will ex 23 tract the dual Dirichlet series, which are the other components to the OhnoNakagawa identity. We have to de ne standard nontrivial additive characters h iv on the local elds kv for all places v of k and the adelic additive character hxi = Q vhxviv on adeles x = (xv)v 2 Ak. First, for Q, we choose standard additive characters on Q1 = R and Qp for any prime p as follows hxiR = exp(2 i x) hxip = exp(2 i fxgp) where fxgp is de ned as P1 j=m aj pj 2 Q in terms of the standard padic expansion x = P1 j=m aj pj with coe cients 0 aj p 1. Note that all these characters are trivial on Z. Also, if x 2 Q, then by the theorem of partial fractions we have x P p fxgp 2 Z, where the sum extends over all prime numbers p. That proves that the adelic character hxi = hxiA = Q vhxviv is trivial on Q embedded along the diagonal in AQ. To de ne the additive characters on extensions kv, we use the trace from kv to R or Qp. Thus, for in nite places v, hxiv = hTrkv=R(x)iR, and for nite places v j p we have hxiv = hTrkv=Qp(x)ip. For x 2 k, we have Trk=Q(x) = X vj1 Trkv=R(x) and Trk=Q(x) = X vjp Trkv=Qp(x) for all nite primes p. This implies that the adelic additive character hxi = Q vhxviv on Ak is trivial on k embedded along the diagonal in Ak. As inWeil [18], Defn. VII.2.4, we choose a di erental idele = ( v)v such that, for all v j 1 we have v = 1 and, for 24 all v  1 the largest ideal contained in the kernel of h iv is 1 v ov. Then vov is the di erent of kv=Qp. By Prop. VII.2.6 in Weil [18], we have j jA = d1 k . Using the measure dvx de ned in Section 2.1, we de ne the Fourier transform on SchwartzBruhat functions on kv by ^ (x) = Z kv (y)hyxiv dvy: Then ^ is also a SchwartzBruhat function on kv, and we have the inversion formula (x) = j vj1 v Z kv ^ (y)hyxiv dvy: Similarly, for an adelic SchwartzBruhat function , we de ne ^ (x) = Z Ak (y)hyxiA dAy; with inversion formula (x) = dk Z Ak ^ (y)hyxiA dAy: The selfdual measure on Ak would then be d1=2 k dAx. To extend these concepts to the vector space V of binary cubic forms, we compose the abovede ned additive characters with the bilinear alternating form [x; y] on V de ned in Section 2.2. We use the notation hx; yi = h[x; y]i with subscripts v and A as needed. With Fourier transform for functions on Vkv de ned by ^ (x) = Z Vkv (y) hx; yiv dvy we have the inversion formula (x) = j3j1 v j vj2v Z Vkv ^ (y) hx; yiv dvy; 25 due to the coe cients 1 3 in the bilinear form. A negative sign is not needed due to the bilinear form being alternating. For the adelic Fourier transform ^ , we have (x) = d2 k Z VA ^ (y) hx; yiA dAy; since the product formula implies j3jA = 1. It follows that the selfdual measure on VA is d2 k dAx, where dAx is the measure chosen in Section 2.2. In Theorem 6.1 of [19], Wright generalized Shintani's proof for Q to establish the functional equation Z(s; ^ ) = Z(1 s; ) for the adelic zeta function de ned in Section 2.3. In this functional equation ^ is de ned relative to the selfdual measure on VA. To comply with our choice of measure, we must replace ^ by d2 k ^ , which gives the functional equation Z(s; ^ ) = d2 kZ(1 s; ): We next carry out the same unfolding process for Z(s; ^ ) into Dirichlet series that we did for Z(s; ) in Section 2.3. If = v v has product form, then ^ = v^ v also has product form, and in the end we nd that Z(s; ^ ) = X 2A1 Z (s; ^ 1) (s; ^ 0) with ^ (s; ^ 0) = X k02K ds k0=k o(k0) Y v1 Z v (s; ^ v) with all the same notational conventions as in Section 2.3. We now again make the choice that for all nite places v we have v = 0;v is the characteristic function of Vov . Then the Fourier transform ^ 0;v may be computed to be the characteristic function of ( 1 v ov) (3 1 v ov) (3 1 v ov) ( 1 v ov) Vkv , or more brie y ^ 0;v(x) = 0;v( v(x1; 1 3x2; 1 3x3; x4)), as mentioned on p. 69 in [5] (with a slight typographical 26 error entering 3 instead of 1 3 as it should be). We would like to factor out the di erental element as much as possible. First we de ne 0;v(x) = 0;v(x1; 1 3x2; 1 3x3; x4), so that ^ 0;v(x) = 0;v( vx). Note that 0 B@ v 0 0 v 1 CA x = vx by the de nition of our representation of GL2 on V . Then by changing variables in the integral de ning the local zeta function, we have Z v (s; ^ 0;v) = Z Gkv j det gvj2s v ^ 0;v(gv x v ) dvgv = Z Gkv j det gvj2s v 0;v( 0 B@ v 0 0 v 1 CA gv x v ) dvgv = j vj4s v Z Gkv j det gvj2s v 0;v(gv x ) dvgv = j vj4s v Z v (s; 0;v): Using the fact that Q v j vjv = d1 k , we have, for this choice of v = 0;v for all nite places v, Z(s; ^ ) = d4s k X 2A1 Z (s; ^ 1) ^ (s) with ^ (s) = X k02K ds k0=k o(k0) Y v1 Z v (s; 0;v): This completes our de nition of the dual Dirichlet series ^ (s). It is our major goal now to work out the relationship between ^ (s) and (s) conjectured in Chapter 1.2. As a consequence of the functional equation for the adelic zeta function, we have the relation X 2A1 Z (s; ^ 1) ^ (s) = d24s k X 2A1 Z (1 s; 1) (1 s): (2.5) Later, as needed, we shall review the known facts about the residues of these Dirich let series, and the functional equation satis ed by the local zeta functions at in nite 27 places. In the next chapter, we calculate explicit expressions for Z v (s; 0;v), compa rable to those produced for Z v (s; 0;v) in [5]. Note that if v is a nite place that does not lie over the prime 3, then 3 is a unit in kv and it follows that 0;v = 0;v. Thus, in that case, the evaluation of the local zeta function is given by equation (2.4). In the next chapter, we shall consider exclusively places v j 3. 28 CHAPTER 3 Local integrals of a Fourier transform 3.1 Statement of the integral to be calculated In this chapter, we address the problem of calculating for a nite place v the local zeta function Z v (s; 0;v) as described at the end of Section 2.4. Since our work in this chapter will be exclusively over a local eld, we shall simplify our notation by letting K be a nonarchimedean local eld and using the notation of Sections 2.1 and 2.2. To review, we denote by O, P, , q, respectively, the maximal compact subring of O, the unique prime ideal P of O, a uniformizer or any generator of P so that P = O, and the order of the nite eld O=P, respectively. We normalize the absolute value jaj for a 2 K to be the modulus of multiplication by a with respect to any additive Haar measure on K. Thus, j j = q1. We denote by dy the additive Haar measure on K for which the measure of O is 1, and by dg the invariant measure on GK so that GO has measure 1. Our goal is to evaluate the local integral Z (s; 0) = Z GK j det gj2s 0(g x ) dg where s is a complex number with Re(s) > 1, is an orbit of nonsingular binary cubic forms in VK, x is the standard representative of , and 0(x) = 0(x1; 1 3x2; 1 3x3; x4) where 0 is the characteristic function of the compact subset VO. Hence, 0 is the characteristic function of the subset O 3O 3O O VO. If 3 is a unit in K, then 0 = 0, and this evaluation was completed in [5]. Thus, in this chapter we assume that K is a 3 eld of characteristic not equal to 3. For the de nition of Fourier 29 transform given in Section 2.4 and a di erental element 2 K, we saw in that section that Z (s; ^ 0) = j j4s Z (s; 0) which explains how this integral arises as a Fourier transform. The result we will prove in this chapter is the following: Theorem 3.1 Assuming that 3 is a uniformizer of K, we have Z (s; 0) = (1 q16s)1(1 q2s)1 8>>>>>>>>>>>>>>< >>>>>>>>>>>>>>: q4s(1 + q12s + 2q14s) if is of type (1); q4s(1 + q12s) if is of type (2u); q2s(1 + q14s) if is of type (2r); q4s(1 + q12s q14s) if is of type (3u); 1 if is of type (3r): 3.2 Reductions of the local integral We begin with some simple reductions in this calculation. First, we write GK = UKBK using the Iwasawa decomposition de ned in Section 2.2 of Chapter 2. For any u in UK = GO, we have that u x is in VO if and only if x is in VO. Hence 0(u x) = 0(x) for all x 2 VK. The same holds for 0. Then, considering the measure on GK de ned above, we have Z (s; 0) = Z UK du Z BK db j det(ub)j2s 0(ub x ) = Z UK du j det uj2s Z BK db j det bj2s 0(u (b x )) = Z UK du Z BK db j det bj2s 0(b x ) = Z BK db j det bj2s 0(b x ) : 30 The last integral gives us a motivation to de ne, for any SchwartzBruhat function , the integral transform I (s; ) = Z BK db j det bj2s (b x ) : Since any b in BK can be written as b = n(y)a(t; u) for some y 2 K and t, u in K , we can express this last integral as I (s; ) = Z K d t Z K d u Z K dy jtuj2s (n(y)a(t; u) x ) : Considering the operator D on the space of SchwartzBruhat functions de ned by D = a( 2; ) , or more explicitly, (D )(x) = (x) (a( 2; 1) x), we conclude that I (s;D ) = I (s; ) Z K d t Z K d u Z K dy jtuj2s (a( 2; 1) (n(y)a(t; u) x )) = I (s; ) Z K d t Z K d u Z K dy jtuj2s (n( y)a( 2t; 1u) x ) = I (s; ) j j1+6sI (s; ) = (1 q16s) I (s; ) ; where we have made the substitutions y 7! 1y, t 7! 2t, and u 7! u. In particular, for = 0 and putting 1 = D 0 we obtain Z (s; 0) = (1 q16s)1 I (s; 1) : So in order to calculate the value of this zeta function explicitly, we simply need to evaluate I (s; 1). By de nition, 1(x) = 0(x) 0( 3x1; 2x2; 1x3; x4) = 0(x1; 1 3x2; 1 3x3; x4) 0( 3x1; 1 3 2x2; 1 3 1x3; x4) : In other words, 1 is the characteristic function of (O 3O 3O O) n (P3 3P2 3P O). 31 In cases (1) and (2) we take x = (0; 1; ; ). Case (1) corresponds to = 0 and = 1 and so x = (0; 1; 1; 0). The integral I (s; 1) becomes now I (s; 1) = Z K d t Z K d u Z K dy jtj2s1 juj2s 1 0; t; 2y + u; y(y + u) t : Case (2) corresponds to = and = where O0 = O[ ] is the maximal compact subring of a quadratic extension K0 = K( ) of K and so x = (0; 1; + ; ). If the extension is unrami ed, case (2u), we take to be a unit not congruent to any unit in O module . For the rami ed extension, case (2r), we take to be the uniformizer . Given this choice of x we can write the integral I (s; 1) for case (2), after making an appropriate substitution, as I (s; 1) = Z K d t Z K d u Z K dy jtj2s1 juj2s 1 0; t;Tr(y + u ); N(y + u ) t ; where Tr and N are the relative trace and norm, respectively, for the extension K0 over K. In case (3) we take x = (1; + 0 + 00; 0 + 00 + 0 00; 0 00) where O0 = O[ ] is the maximal compact subring of a cubic extension K0 = K( ) of K. For case (3u) we take to be a unit not congruent to any unit in O module . For the case (3r) we take to be the uniformizer . For this choice of x we can write I (s; 1), following a change of variables, as I (s; 1) = Z K d t Z K d u Z K dy jtj2s1juj6s 1 t;Tr(y + u ); S(y + u ) t ; N(y + u ) t2 ; where Tr, S, and N are the relative trace, second symmetric function, and norm, respectively, for this cubic extension. In the next section we will calculate the value of the integral I (s; 1). By the reductions done in this section we know that this will su ce to prove Theorem 3.1. The calculation will be done assuming that 3 is a uniformizer in K, i.e. 3O = P. 32 When that is the case, 1 becomes the characteristic function of (O P P O) n (P3 P3 P2 O). 3.3 Evaluation of the local integral Let us complete the task of evaluating I (s; 0) for each of the possible types of . 3.3.1 Type (1) In the corresponding integral, since 2 is a unit, we can make the following change of variables (t; u; y) 7! (t=4; u; (y u)=2) to obtain I (s; 1) = Z K d t Z K d u Z K dy jtj2s1 juj2s 1 0; t; y; y2 u2 t : The integral is nonzero only if t; y; y2 u2 t belongs to (P P O)n(P3 P2 O) and in this case its value is I (s; 1) = Z K jtj2s1 d t Z K juj2s d u Z K dy : The integral is therefore nonzero in the following cases: (a) t 2 P , y 2 P, y2 u2 2 tO. These conditions imply that u2 2 P and so u 2 P = O. Thus, the integral in this case becomes I(a) = Z O jtj2s1 d t Z P juj2s d u Z P dy =q12s X1 l=1 Z lO juj2s d u q1 =q2s X1 l=1 q2sl = q4s 1 q2s : 33 (b) t 2 2O , y 2 P, y2 u2 2 tO. Again as before, u2 2 P and so u 2 P = O. Hence, the integral is now I(b) = Z 2O jtj2s1 d t Z P juj2s d u Z P dy =q24s X1 l=1 Z lO juj2s d u q1 =q14s X1 l=1 q2sl = q16s 1 q2s : (c) t 2 lO , y 2 O , y2 u2 2 tO, l 3. From these conditions it follows that (y u)(y + u) 2 Pl P. But P is a prime ideal so either y u 2 P or y + u 2 P. In the former case, since y + u = 2y (y u) 2 O , we conclude that y u 2 Pl1 and so u 2 y + Pl1 = y(1 + Pl2). In the latter case, since y u = 2y (y + u) 2 O , we obtain that y + u 2 Pl1 and so u 2 y+Pl1 = y(1+Pl2). This means that u 2 y(1+Pl2)ty(1+Pl2) for a xed y. Before we continue we observe that according to the de nition of the measures given in Chapter 2, we have d u = 1 1 q1 du juj : Moreover, since O =(1 + P) = (O=P) and (1 + Pm1)=(1 + Pm) = O=P for m 2, we have that the measure d u satis es Z 1+Pm d u = 1 (q 1)qm1 m 1 : 34 We can nally calculate the integral for this case. I(c) = X1 l=3 Z lO jtj2s1 d t Z O dy Z y(1+Pl2) juj2s d u + Z y(1+Pl2) juj2s d u = X1 l=3 Z lO jtj2s1 d t Z O dy 2q2s Z 1+Pl2 juj2s d u = X1 l=3 q(12s)l (1 q1)q1 2q2s (q 1)ql3 =2q12s X1 l=3 q2sl = 2q18s 1 q2s : Now, combining these results we obtain the desired result for type (1) I (s; 1) = I(a) + I(b) + I(c) = q4s(1 + q12s + 2q14s) 1 q2s : 3.3.2 Type (2u) When K0 = K( ) is a unrami ed quadratic extension of K with maximal compact subring O0 = O[ ], we have that is a unit and is also a uniformizer of K0. Hence, y +u is in (P0)m if and only if both y and u are in P. The corresponding integral is nonzero only if t;Tr(y + u ); N(y + u ) t is in (P P O) n (P3 P2 O) and its value is given by I (s; 1) = Z K jtj2s1 d t Z K juj2s d u Z K dy : The integral is nonzero for the following cases: (a) t 2 O , Tr(y +u ) 2 P, N(y +u ) 2 tO. Under these conditions, y +u 2 P0 and so y 2 P and u 2 P. This integral was calculated above, type (1) case (a), and its value is I(a) = Z O jtj2s1 d t Z P juj2s d u Z P dy = q4s 1 q2s : 35 (b) t 2 2O , Tr(y + u ) 2 P, N(y + u ) 2 tO. With these conditions, as before, y 2 P and u 2 P. We have already calculated this integral, type (1) case (b), and we found that I(b) = Z 2O jtj2s1 d t Z P juj2s d u Z P dy = q16s 1 q2s : Hence, we have found the value of the integral for type (2u) I (s; 1) = I(a) + I(b) = q4s(1 + q12s) 1 q2s : 3.3.3 Type (2r) When K0 = K( ) is a rami ed quadratic extension of K with maximal compact subring O0 = O[ ], we have that is a uniformizer of K0 and N( ) is a uniformizer of K. Then N(y + u ) is in Pm if and only jyj qm=2 and juj q(m1)=2. The associated integral is nonzero only if t;Tr(y + u ); N(y + u ) t is in (P P O)n(P3 P2 O) and this case it reduces to I (s; 1) = Z K jtj2s1 d t Z K juj2s d u Z K dy : The integral is nonzero for the following cases: (a) t 2 O , Tr(y + u ) 2 P, N(y + u ) 2 tO. These conditions imply that jyj q1=2 and juj 1 and so y 2 P and u 2 O. Then the integral is I(a) = Z O jtj2s1 d t Z O juj2s d u Z P dy =q12s X1 l=0 Z lO juj2s d u q1 =q2s X1 l=0 q2sl = q2s 1 q2s : 36 (b) t 2 2O , Tr(y + u ) 2 P, N(y + u ) 2 tO. These conditions imply that jyj q1 and juj q1=2 and so y 2 P and u 2 P. Then the value of the integral is already known, see type (1) case (b). I(b) = Z 2O jtj2s1 d t Z P juj2s d u Z P dy = q16s 1 q2s : Therefore, we have calculated the value of the integral for type (2r) I (s; 1) = I(a) + I(b) = q2s(1 + q4s) 1 q2s : 3.3.4 Type (3u) If K0 = K( ) is a unrami ed cubic extension of K with maximal compact subring O0 = O[ ], then is a unit and is a uniformizer of K0. Thus, y + u is in (P0)m if and only if both y and u are in Pm. Additionally, we can assume that Tr( ) = 0 and S( ) and N( ) are units of K. And for any y and u in K, we have Tr(y + u ) = 3y S(y + u ) = 3y2 + u2 S( ) N(y + u ) = y3 + yu2 S( ) + u3 N( ) : The corresponding integral is nonzero only if t;Tr(y + u ); S(y + u ) t ; N(y + u ) t2 is in (O P P O) n (P3 P3 P2 O) and its value is I (s; 1) = Z K jtj2s1 d t Z K juj6s d u Z K dy : Therefore, the integral is nonzero in the following cases: (a) t 2 O , Tr(y + u ) 2 P, S(y + u ) 2 tP , N(y + u ) 2 t2O. The rst and last conditions implies that y +u 2 O0 and so y 2 O and u 2 O. But to satisfy the 37 third condition, we actually require u 2 P. So the integral becomes I(a) = Z O jtj2s1 d t Z P juj6s d u Z O dy =1 X1 l=1 Z lO juj6s d u 1 = X1 l=1 q6sl = q6s 1 q6s : (b) t 2 O , Tr(y + u ) 2 P, S(y + u ) 2 tP , N(y + u ) 2 t2O. These conditions imply that y + u 2 P0 and so y 2 P and u 2 P. With this the integral is now I(b) = Z O jtj2s1 d t Z P juj6s d u Z P dy =q2s+1 X1 l=1 Z lO juj6s d u q1 =q2s X1 l=1 q6sl = q4s 1 q6s : (c) t 2 2O , Tr(y + u ) 2 P, S(y + u ) 2 tP , N(y + u ) 2 t2O. With these conditions we have y +u 2 P2 and so y 2 P2 and u 2 P2. And the integral in this case is given by I(c) = Z 2O jtj2s1 d t Z P2 juj6s d u Z P2 dy =q4s+2 X1 l=2 Z lO juj6s d u q2 =q4s X1 l=2 q6sl = q8s 1 q6s : (d) t 2 3O , (Tr(y + u ); S(y + u )) 2 (P tP ) n (P3 tP 2), N(y + u ) 2 t2O. The rst and last conditions imply that y + u 2 (P0)2 and so y 2 P2 and 38 u 2 P2. However, the remaining condition will be valid only if u 2 2O . So the integral becomes I(d) = Z 3O jtj2s1 d t Z 2O juj6s d u Z P2 dy =q6s+3 q12s q2 =q16s : Putting together these partial results gives us the value of the integral for type (3u) I (s; 1) = I(a) + I(b) + I(c) + I(d) = q4s(1 + q12s q14s) 1 q2s : 3.3.5 Type (3r) If K0 = K( ) is a rami ed cubic extension of K with maximal compact subring O0 = O[ ], then is a uniformizer of K0 and N( ) is a uniformizer of K. Hence, N(y + u ) is in Pm if and only if jyj qm=3 and juj q(m1)=3. Moreover, we assume that Tr( ) 2 P, S( ) 2 P, and N( ) 2 O . For any y and u in K we have Tr(y + u ) = 3y + uTr( ) S(y + u ) = 3y2 + 2yuTr( ) + t2 S( ) N(y + u ) = y3 + y2uTr( ) + yu2 S( ) + u3 N( ) : As before, the integral is nonzero only if t;Tr(y + u ); S(y + u ) t ; N(y + u ) t2 is in (O P P O) n (P3 P3 P2 O) and its value is I (s; 1) = Z K jtj2s1 d t Z K juj6s d u Z K dy : Therefore, to have a nonzero integral we have to consider following cases: (a) t 2 O , Tr(y + u ) 2 P, S(y + u ) 2 tP , N(y + u ) 2 t2O. From these assumptions, we conclude that jyj 1 and juj q1=3, i.e. y 2 O and u 2 O. 39 The integral reduces in this case to I(a) = Z O jtj2s1 d t Z O juj6s d u Z O dy =1 X1 l=0 Z lO juj6s d u 1 = X1 l=0 q6sl = 1 1 q6s : (b) t 2 O , Tr(y + u ) 2 P, S(y + u ) 2 tP , N(y + u ) 2 t2O. Using these assumptions, we obtain jyj q2=3 and juj q1=3, i.e. y 2 P and u 2 P. The integral, which was calculated for type (3u) case (b), is I(b) = Z O jtj2s1 d t Z P juj6s d u Z P dy = q4s 1 q6s : (c) t 2 2O , Tr(y+u ) 2 P, S(y+u ) 2 tP , N(y+u ) 2 t2O. These assumptions imply that jyj q4=3 and juj q1, i.e. y 2 P2 and u 2 P. So our integral is I(c) = Z 2O jtj2s1 d t Z P juj6s d u Z P2 dy =q4s+2 X l=1 Z lO juj6s d u q2 =q4s X1 l=1 q6sl = q2s 1 q6s : These results can be combined to get the integral for type (3r) I (s; 1) = I(a) + I(b) + I(c) = 1 1 q2s : This concludes the evaluation of I (s; 0) in all cases, and therefore completes the proof of Theorem 3.1. 40 3.4 Veri cation of a simple identity In this section we use an identity which was shown to be true by Datskovsky and Wright in order to verify the validity of Theorem 3.1. Consider any locally integrable function on VK. By formula (2.4) on page 38 of [5] we have the following identity Z VK (x) dx = X bKc Z GK j det gj2 (g x ) dg ; where the sum is taken over all the GK{orbits in VK, bK = q12e(1q1)(1q2), e is the order of in K, and c = jP(x )j o( ) where P(x ) is the discriminant of x and o( ) is the order of the stabilizer in GK of any x 2 . In particular, we can take to be ^ 0. The left hand side is simply Z VK ^ 0(x) dx = j3j j j2 0(0) = j3j j j2 by the Fourier inversion formula and our choice of the measure dx on VK. The right hand side becomes X bKc Z (1; ^ 0) = bKj j4 X c Z (1; 0) by the de nition of the orbital zeta function. These formulas are valid for any p{ eld whose characteristic is not 2 or 3. We want to very the above identity for the kind of elds we have been considering in this chapter, that is for 3{ elds for which 3 is a uniformizer. Therefore, under these assumptions using Theorem 3.1 and after canceling out common terms, the identity we are trying to show reduces to X c Z (1; 0) = b1 K j3j j j2 = q2(1 q1)1(1 q2)1 : For convenience, we write (j) to indicate that is an orbit of type (j), where j take the values 1, 2u, 3u, 2r, 3r. We recall that c (1) = 1=6, c (2u) = 1=2, and c (3u) = 1=3. Moreover, X (2r) c (2r) = 1 q and X (3r) c (3r) = 1 q2 ; 41 where the sums are taken over all the orbits of type (2r) and (3r), respectively. Since j j = q1, the orbital zeta function of 0 at 1 can be written by Theorem 3.1 and the de nition of , as Z (1; 0) = (1 q5)1(1 q2)1 8>>>>>>>>>>>>>>< >>>>>>>>>>>>>>: q4(1 + q1 + 2q3) if is of type (1); q4(1 + q1) if is of type (2u); q2(1 + q3) if is of type (2r); q4(1 + q1 q3) if is of type (3u); 1 if is of type (3r): We can now use all this information to calculate X c Z (1; 0) = 1 6 Z (1)(1; 0) + 1 2 Z (2u)(1; 0) + 1 3 Z (3u)(1; 0) + 1 q Z (2r)(1; 0) + 1 q2 Z (3r)(1; 0) =(1 q5)1(1 q2)1 1 6 q4(1 + q1 + 2q3) + 1 2 q4(1 + q1) + 1 3 q4(1 + q1 q3) + 1 q q2(1 + q3) + 1 q2 1 =(1 q5)1(1 q2)1(q2 + q3 + q4 + q5 + q6) =q2(1 q5)1(1 q2)1(1 + q1 + q2 + q3 + q4) =q2(1 q1)1(1 q2)1 which concludes the veri cation of the identity. 42 CHAPTER 4 Residues of the Dirichlet series and generalizing OhnoNakagawa In [5], formulas were calculated for the residues of the Dirichlet series (s) at its poles s = 1 and s = 5=6, but the calculation of the residue formulas for the dual Dirichlet series ^ (s) was left incomplete. The method of that paper required the calculation of local integrals of a SchwartzBruhat function, which was completed for nonarchimedean local elds for the characteristic functions 0, but not for its Fourier transform ^ 0. As we saw in the previous chapter, these new local integrals are more di cult to calculate. We shall use instead the ltration method of [6] to calculate the residues of the dual Dirichlet series, and thus our results in this chapter will be valid for all number elds k, and not simply those where 3 is unrami ed. At the end of this chapter, we shall use the complete set of residues of both (s) and ^ (s) to deduce what the correct generalization of Ohno's conjecture should be. Our method will verify this conjecture at least at s = 1 and s = 5=6. 4.1 Filtrations of the Dirichlet series It will be necessary to generalize the decomposition of the adelic zeta function given in equation (2.2) on page 22. Instead of singling out just the in nite places, we shall now allow an arbitrary nite set S of places of k to be distinguished. We shall assume that S contains all in nite places as well as possibly some nite places. For a place v of k, let Av denote the nite set of Gkv orbits of nonsingular forms in Vkv . Let AS = Y v2S Av. Then = ( v) will denote a choice from AS, meaning a choice of an orbit at each place in S. We will call these choices orbit vectors over S. 43 Then just as in Section 2.3 of Chapter 2, assuming the SchwartzBruhat function = v v has product type and that v = 0;v for all nite places v =2 S, the adelic zeta function may be decomposed as Z(s; ) = X 2AS Z ;S(s; ) ;S(s) where now Z ;S(s; ) = Y v2S Z v (s; v); (4.1) Z v (s; v) = Z Gkv j det gvj2s v v(gv x v ) dvgv ;S(s) = X k02K ds k0=k o(k0) Y v =2S Z v (s; 0;v): Here x v is the standard choice of representative for the chosen orbit v of binary cubic forms over kv (described in Prop. 2.1, p. 35 of [5]). Also, K denotes the set of extensions k0=k of degree at most 3 so that k0 k kv corresponds to the choice of orbit v for each place v 2 S. The choice of S is built into the choice of the orbit vector 2 AS, but we will indicate S explicitly in the notation ;S because the technique we wish to use in this chapter is to extend the distinguished set S of places until relations between the various Dirichlet series are easier to detect. In particular, suppose T is a nite set of places of k that contains S. We denote the set of places in T which do not belong to S as T n S. Then there is a natural restriction mapping from orbit vectors in AT to orbit vectors in AS. We will generally use to denote a choice of orbits in AS and to denote a choice of orbits in AT . If v = v for all v 2 S, we say jS = , meaning that restricted to S agrees with . We can decompose the Dirichlet series ;S(s) in terms of the series ;T (s) with 44 jS = as follows ;S(s) = X k02K ds k0=k o(k0) Y v =2S Z v (s; 0;v) = X jS= X k02K ds k0=k o(k0) Y v2TnS Z v (s; 0;v) Y v =2T Z v (s; 0;v) where the sum ranges over those 2 AT whose restriction to S is , and we have explicitly written the Euler product in terms of the local zeta integral factors. Recall that for v =2 T, v is determined as the orbit corresponding to k0=k at v. Thus, it should be kept in mind that for v =2 T the orbit v is a function of the extension k0=k. Rearranging the above sum and product gives ;S(s) = X jS= 2 4 Y v2TnS Z v (s; 0;v) 3 5 ;T (s) (4.2) This is the main ltration formula for the original Dirichlet series. We can extend these ideas to the dual Dirichlet series as well and obtain the formulas ^ ;S(s) = X k02K ds k0=k o(k0) Y v =2S Z v (s; 0;v) (4.3) = X jS= 2 4 Y v2TnS Z v (s; 0;v) 3 5 ^ ;T (s) (4.4) The main idea exploited in this chapter is that if the set of places T contains all places lying over 3, then 3 is a unit for all v =2 T, and therefore 0;v = 0;v for all v =2 T. Consequently ^ ;T (s) = ;T (s) for all 2 AT . Thus, the component series in the two decompositions of ;S and ^ ;S(s) are the same, and the sole di erence lies in the nitely many local zeta function factors Z v (s; 0;v) for v 2 T n S. 45 4.2 Poles and residues First, let us review the slightly more general framework of the earlier papers of DatskovskyWright. In [19, 4], it is proved that the adelic zeta function Z(!; ) = Z GA=Gk !(det g) X x2V 0 k (g x) dAg; has a meromorphic continuation to the entire complex manifold k of quasicharacters ! on A =k . This continuation is holomorphic except for simple poles at ! = !0, !2, !1=3 and !5=3 where is any character satisfying 3 = 1. In this thesis, we are restricting the quasicharacters to principal ones ! = !2s = j j2s A for complex s. Thus, Z(s; ) = Z GA=Gk j det gj2s A X x2V 0 k (g x) dAg; is holomorphic in the entire splane with the exception of simple poles at s = 0, 1 6 , 5 6 , and 1. The decomposition (2.2) of the zeta function Z(s; ) in terms of the Dirichlet series (s) allows us to prove that the Dirichlet series have meromorphic continuations to the entire splane which are holomorphic except for simple poles at s = 1 and s = 5=6. Very general residue formulas are stated in Theorem 6.2, p. 71, in [5]. The measure on GA used in that paper is not the tensor product measure de ned in Section 2 of Chapter 2. Thus, after tracing through the notation presented in [19, 5], we nd the following residue formulas Res s=1 ;S(s) = k 2 p dk S k;S(2) c [1 + b ] (4.5) Res s=5=6 ;S(s) = k 6dk S k;S 1 3 c a where k and dk are de ned in Chapter 2, S = Y v2S v1 (1 q1 v ); k;S(s) = Y v =2S (1 qs v )1; (partial Dedekind zeta function, analytically continued) 46 and the a = Q v2S a v , b = Q v2S b v , and c = Q v2S c v are constants describing the structure of the orbit v de ned on pages 58, 61 and 38 of [5]. To describe a v , b v and c v , suppose that the orbit v corresponds to the local extension k0w =kv of degree at most 3 (up to conjugacy) or equivalently the simple algebra k0 k kv which has dimension 3 over kv. In [5], these extensions are classi ed into ve types: (1), (2u), (2r), (3u) and (3r), where the number is the degree of the extension, and the letter indicates whether that extension is unrami ed or ram i ed. Let v = (k0w =kv) be the relative discriminant, which is an element of k v determined uniquely modulo multiplication by squares of units, and let o( v) be the number of automorphisms of k0 kv over kv. For types (1), (2), and (3), respectively, we have o( v) = 6, 2, and 3 or 1, respectively, the last depending on whether the local extension is Galois or not. Then from [5], page 38 and top of page 36, we have c v = j v jv o( v) : (4.6) On page 61 of [5], b v is simply de ned as 3, 1, 0, resp. for types (1), (2), (3), resp. The de nition of a v is the most involved and is described in a chart on page 58 of [5]. To obtain the residue formulas for the dual Dirichlet series ^ ;S(s), we choose a set T of places that contains S and all nite places v j 3. Then, as we mentioned before, for all orbit vectors 2 AT , we have ^ ;T (s) = ;T (s). Our ltration formulas now give the following relations among the residues of all these series, for r = 1 and 5=6: Res s=r ;S(s) = X jS= 2 4 Y v2TnS Z v (r; 0;v) 3 5Res s=r ;T (s) (4.7) Res s=r ^ ;S(s) = X jS= 2 4 Y v2TnS Z v (r; 0;v) 3 5Res s=r ;T (s) In the next sections, we shall use properties of the local zeta functions together with these ltrations to calculate formulas for the residues of the dual Dirichlet series. 47 4.3 Residue of the dual Dirichlet series at s = 1 In this section, we abbreviate k = k 2 p dk , all the factors in the residue at s = 1 that depend on k but on nothing else. The ltration formula (4.7) together with the residue formulas (4.5) yield Res s=1 ^ ;S(s) = X jS= 2 4 Y v2TnS Z v(1; 0;v) 3 5Res s=1 ;T (s) = X jS= 2 4 Y v2TnS Z v(1; 0;v) 3 5 k T k;T (2) c [1 + b ] = k T k;T (2) X jS= 2 4 Y v2TnS Z v(1; 0;v) 3 5 c [1 + b ]: Note that the local zeta function was shown in [5] to be holomorphic for Re(s) > 1=6, and thus we can simply substitute s = 1 in the residue calculation. By the organizing principle that a sum of products may be rearranged as a product of sums, we will now manipulate the above residue formulas. Both c and b factor as products over the places v 2 T; however, before factoring out the terms corresponding to places in S, we must split the residue formula and then factor, using the facts that k;T (s) = k;S(s) Y v2TnS (1 qs v ) T = S Y v2TnS (1 q1 v ) c = c Y v2TnS c v b = b Y v2TnS b v 48 This leads to Res s=1 ^ ;S(s) = k T k;T (2) X jS= 2 4 Y v2TnS Z v(1; 0;v) 3 5 c + X jS= 2 4 Y v2TnS Z v(1; 0;v) 3 5 c b = k S k;S(2) c X jS= 2 4 Y v2TnS (1 q1 v )(1 q2 v )c v Z v(1; 0;v) 3 5+ c b X jS= 2 4 Y v2TnS (1 q1 v )(1 q2 v )c vb v Z v(1; 0;v) 3 5 = k S k;S(2) c Y v2TnS " (1 q1 v )(1 q2 v ) X v2Av c v Z v(1; 0;v) # + c b Y v2TnS " (1 q1 v )(1 q2 v ) X v2Av c vb v Z v(1; 0;v) # The sums that appear can be simpli ed by means of Fourier transform formulas proved in [5]. Keeping in mind that we are using the measure such that Vov has measure 1, the formula (2.4) on page 38 in [5], restated as Proposition 5.1 on page 52, implies that, for nite places v, (1 q1 v )(1 q2 v ) X v2Av c vZ v(1; 0;v) = Z Vkv 0;v(x) dvx : Since 0;v is the characteristic function of ov 3ov 3ov ov Vkv , the integral is j3j2v . Thus, we have the rst sum in the residue formula equal to (1 q1 v )(1 q2 v ) X v2Av c vZ v(1; 0;v) = j3j2v : The other part of the residue at 1 involves Theorem 5.2, Proposition 5.2 and the de nition of the singular invariant distributions 3 from [5]. Assume throughout that v is a nite place and kv is a p eld. Theorem 5.2 on page 61 says that (1 q1 v ) X v2Av c vb v Z v(1; 0;v) = 3( 0;v) 49 and the distribution 3 is de ned on page 54 as 3( v) = 3(2; v) = Z k v d v t Z kv dvx Z kv dvy jtj2v v(0; t; x; y); for Govsymmetric functions v, where ov is the maximal compact subring in kv. Then it is straightforward to evaluate 3( 0;v) = Z k v d v t Z kv dvx Z kv dvy jtj2v 0;v(0; t; x; y) = Z 3ov jtj2v d v t j3jv = j3j3v (1 q2 v )1 Combining this with our earlier equation for 3( 0;v) produces (1 q1 v ) X v2Av c v b v Z v(1; 0;v) = j3j3v (1 q2 v )1: This gives the second sum in our residue formula at 1 as (1 q1 v )(1 q2 v ) X v2Av c v b v Z v(1; 0;v) = j3j3v : This leads to the full residue formula at s = 1: Res s=1 ^ ;S(s) = k S k;S(2) c Y v2TnS j3j2v 2 41 + b Y v2TnS j3jv 3 5 : Our last task is to account for our assumption that T contains all the places v j 3. De ne jxjS = Q v2S jxjv for any x 2 k. By our assumptions on T, we have j3jT = j3jA = 1, by the idele product formula. Then Q v2TnS j3jv = j3jT =j3jS = j3j1 S . Then our nal residue formula at 1 is Res s=1 ^ ;S(s) = k 2 p dk S k;S(2) c j3j2 S 1 + b j3j1 S : (4.8) 4.4 Residue of the dual Dirichlet series at s = 5=6 Next, we turn to the residue formula at 5=6. Here we abbreviate k = k 6dk , all the factors in the residue at s = 5=6 that depend on k but on nothing else. We use the 50 same notation and arguments at the beginning of Section 4.3 along with the formula a = a Y v2TnS a v , but start with the residue formula for ;S at 5=6. This leads to Res s=5=6 ^ ;S(s) = X jS= 2 4 Y v2TnS Z v(5=6; 0;v) 3 5 Res s=5=6 ;T (s) = X jS= 2 4 Y v2TnS Z v(5=6; 0;v) 3 5 k T k;T 1 3 c a = k T k;T 1 3 X jS= 2 4 Y v2TnS Z v(5=6; 0;v) 3 5 c a = k S k;S 1 3 c a X jS= 2 4 Y v2TnS (1 q1 v )(1 q1=3 v ) c v a v Z v(5=6; 0;v) 3 5 = k S k;S 1 3 c a Y v2TnS " (1 q1 v )(1 q1=3 v ) X v2Av c v a v Z v(5=6; 0;v) # after exchanging the sum and product in exactly the same way as in the preceding section. The sum in the above formula corresponds to a second Fourier inversion formula. For nite places v, Theorem 5.1 in [5] states that (1 q1 v ) X v2Av c v a v Z v(5=6; 0;v) = 4( 0;v); for the distribution 4 de ned on pp. 33 and 34 of [5] by the integral 4( v) = 4(1=3; v) = Z k v d v t Z kv dvx Z kv dvy Z kv dvz jtj1=3 v v(t; x; y; z) if v is any Govsymmetric function. Replacing v by 0;v, this is easy to calculate as 4( 0;v) = Z k v d v t Z kv dvx Z kv dvy Z kv dvz jtj1=3 v 0;v(t; x; y; z) = Z ov jtj1=3 v d v t j3j2v = j3j2v (1 q1=3 v )1 51 Therefore, the sum in the residue formula becomes (1 q1)(1 q1=3) X v2Av c v a v Z v(5=6; 0;v) = j3j2v : Putting this altogether gives Res s=5=6 ^ ;S(s) = k S k;S 1 3 c a Y v2TnS j3j2v : Assuming again that T contains all places lying over 3, we can nally give the formula for the residue at 5=6: Res s=5=6 ^ ;S(s) = k 6dk S k;S 1 3 c a j3j2 S : (4.9) 4.5 Generalizing Ohno's conjecture Ohno's conjecture (see [14, 13]) takes the form ^ 1(s) = 33s 2(s) ^ 2(s) = 313s 1(s) where 1(s), 2(s) are Shintani's Dirichlet series (see [17]) corresponding to integral binary cubic forms of positive and negative discriminant, respectively, and ^ 1(s), ^ 2(s) are the analogous series for the dual lattice. This is the case k = Q with the set of places S limited to just the one in nite place v = 1. The completion of Q at the place 1 is simply the real numbers R, and there are two GRorbits of nonsingular real binary cubic forms, namely, the totally real forms of positive discriminant, and the complex forms of negative discriminant. We will attempt to generalize this pattern to any number eld k by taking S to be the set of in nite places v j 1 of k. For any choice of orbits 2 A1, we will de ne as another choice in the following way. First, we decompose A1 as a direct product A1 = Q vj1 Av and = ( v)v. If kv = C, there is only one GCorbit, and we de ne v = v, the lone orbit. If kv = R, there are two GRorbits, and we simply 52 de ne v to be the other orbit besides v. Then for 2 A1, we de ne in the natural componentwise fashion. Our goal is to establish a formula of the form ^ (s) = 3A+Bs (s) for any 2 A1, for some constants A and B dependent on k and . The reason for the comparison of series for the orbit types and lies in the theorem of Scholz in [16] about the relationship between the 3class numbers of quadratic eld of positive and negative discriminant. There the key tool is to adjoin the cube roots of unity to the Galois closure of a noncyclic cubic eld, with quadratic resolvent eld of discriminant D. The extended eld now contains another family of conjugate cubic elds with quadratic resolvent eld of discriminant 3D. That correspondence changes the sign of the discriminants of the cubic elds. Hopefully, this mechanism will be made more precise in the course of our research. First, we collect the residue calculations of this chapter as well as the original calculations of DatskovskyWright into a convenient reference theorem: Theorem 4.1 For a nite set S of places of the number eld k containing all in nite places, the residues of the Shintani Dirichlet series ;S(s) and the dual series ^ ;S(s), as de ned in Sections 2.3 and 2.4, are given by the following formulas: Res s=1 ;S(s) = k 2 p dk S k;S(2) c [1 + b ] Res s=1 ^ ;S(s) = k 2 p dk S k;S(2) c j3j2 S 1 + b j3j1 S Res s=5=6 ;S(s) = k 6dk S k;S 1 3 c a Res s=5=6 ^ ;S(s) = k 6dk S k;S 1 3 c a j3j2 S When S is the set of in nite places, we have j3jS = j3j1 = 3n, where n = [k : Q]. 53 Then, using the preceding theorem, we have ^ (1) (1) = 32n c c 1 + b 3n 1 + b ^ (5=6) (5=6) = 32n c c a a : The next step in simplifying these formulas is to manipulate the formulas for a , b and c when corresponds to an orbit type over R or C. Since we are only considering places v j 1, there are only three local orbit types v to consider, which we will denote as v = 0 if kv = C, v = + if kv = R and v corresponds to the binary cubic forms of positive discriminant, and v = if kv = R and v corresponds to the binary cubic forms of negative discriminant. Recall that in Section 4.2, equation (4.6), we reviewed the de nitions of b and c established in [5], and we give these again strictly for the archimedean places c0 = 1 6 ; c+ = 1 6 ; c = 1 2 ; b0 = 3; b+ = 3; b = 1: Considering ratios, we have c0 c0 = 1; c+ c = 1 3 ; c c+ = 3: For = ( v)v 2 A1, suppose that for m of the real places v we have v = + and then for the other r1 m places we have v = . Then by taking products of the formulas from the previous paragraph we get c c = 3m 1 3 r1m = 32mr1 ; b = 3r2+m; b = 3r2+r1m: Then 1 + b 3n 1 + b = 1 + 3r1+r2mn 1 + 3r2+m = 1 + 3r2m 1 + 3r2+m = 3r2m; since n = r1+2r2. It is noteworthy that under our choices this ratio simpli es to just a power of 3. Then combining these results we obtain ^ (1) (1) = 32n 32mr1 3r2m = 3m2nr1r2 (4.10) 54 For the value of the ratio at 5=6, we need the formulas for a given on page 58 in [5]. a0 = 3 p 3 4 2 1 3 6 ; a+ = 3 p 3 2 1 3 3 ; a = 3 2 1 3 3 : Then the ratios are a0 a0 = 1; a+ a = p 3; a a+ = 1 p 3 : For 2 A1 with m real places v such that v = +, as before, we have a a = 1 p 3 m ( p 3)r1m = 3 1 2 r1m: Then ^ (5=6) (5=6) = 32n 32mr1 3 1 2 r1m = 3m2n1 2 r1 : (4.11) Both values are consistent with ^ (s)= (s) being powers of 3. We may now solve for the constants A;B in this conjectural form: ^ (1) (1) = 3A+B = 3m2nr1r2 ^ (5=6) (5=6) = 3A+5 6B = 3m2n1 2 r1 : This leads to the linear equations: A + B = m 2n r1 r2 A + 5 6 B = m 2n 1 2 r1; which have the unique solution A = m + r2 B = 3n: Then the proposed generalization of Ohno's Conjecture (and Nakagawa's theorem) is ^ (s) (s) = 3m+r23ns : (4.12) The calculations presented here establish Theorem 1.2 and motivate Conjecture 1.1. 55 Finally, let us compare this conjecture to Nakagawa's theorem for k = Q. In that case we have n = [Q : Q] = 1, r1 = 1, and r2 = 0. Then our conjecture would say ^ 2(s) 1(s) = 31+03s = 313s; ^ 1(s) 2(s) = 30+03s = 33s: This is exactly the theorem of Nakagawa mentioned at the beginning of this section. 56 CHAPTER 5 Decomposing the Dirichlet series according to the resolvent eld Datskovsky and Wright established the expression of Shintani Dirichlet (s) series as a sum over extensions k0=k of degree at most 3, as mentioned in the introduction, and in Chapter 3 of this thesis we established the analogous formula for the dual Dirichlet series ^ (s). Then after cancelling common factors, as described in Chapter 1.2 at equation (1.5), our generalization of Ohno's conjecture becomes X k02K ds k0=k o(k0) Rk0(2s) Rk0(4s) Y vj3 Tk0;v(s) = 3r2+m3ns X k02K ds k0=k o(k0) Rk0(2s) Rk0(4s) In this chapter, we shall decompose this identity according to the resolvent elds of the extensions k0=k, and give the proofs of Theorems 1.3 and 1.4 in Chapter 1.2. 5.1 The resolvent eld of an extension k0=k of degree at most 3 If k0=k has degree strictly less than 3, we simply de ne the resolvent eld to be F = k0. If k0=k is a cubic extension, it is either cyclic, in which case we de ne the resolvent eld to be F = k, or it is noncyclic and its Galois closure over k contains a unique quadratic eld F, which is called the resolvent eld in that case. Each resolvent eld F has degree at most 2 over k, and thus can be expressed in the form F = k( p ) for some nonzero element of k. By Kummer theory, k( p 1) = k( p 2) if and only if 1= 2 2 k2, the subgroup of squares in k . Thus, the possible resolvent elds F of k0=k bijectively correspond to the cosets in k =k2. For each 2 k , de ne C ( ) to be the set of all extensions k0=k of degree at most 3 which have resolvent eld equal to F = k( p ). 57 For each real embedding : k ! R of k, and for any 2 k , either > 0 or < 0. Thus, we can de ne a signature of by setting v = + if > 0 and v = otherwise. This signature is the same as the signature of the extension k( p )=k as de ned in Chapter 1.2. If v = +, then k( p ) k kv = R R, and if v = , then k( p ) k kv = C. For any extension k0=k 2 C ( ), since the Galois closure of k0 over k contains k( p ), this shows that k0 k kv must be a direct sum of three copies of R. Similar reasoning in case v = proves that the signature of any k0 2 C ( ) is the same as the signature of . Thus, for any signature , the set of extensions K is the disjoint union of C ( ) over representatives of all cosets 2 k =k2 with signature . Finally, for any resolvent eld F = k( p ), we de ne the dual resolvent eld to be ^ F = k( p 3 ). The reason for this choice of dual is that the compositum F ^ F must contain the eld F0 = k( p 3) generated by the cube roots of unity over k. Kummer theory says that any cyclic cubic extension F0=F for which F contains the cube roots of unity must be of the form F0 = F( 3 p ) for some 2 F , and that fact plays a special role in Scholz' re ection theorem in [16] and Nakagawa's proof. Note that duality is symmetric in that the dual eld of ^ F is just F. If the signature of is , then clearly the signature of 3 is . We summarize these observations about the resolvent elds in the following propo sition: Proposition 5.1 For any signature of the eld k and its negative , the sets of extensions K and K , respectively, are the disjoint unions of the subsets C ( ) and C (3 ) as ranges over representatives of each coset in k =k2 which has signature . By summing over the coset representatives 2 k =k2 with signature , this proposition directly proves what we stated as Theorem 1.3 in Chapter 1.2. 58 Theorem 5.1 If for every 2 k , we have X k02C (3 ) ds k0=k o(k0) Rk0(2s) Rk0(4s) Y vj3 Tk0;v(s) = 3r2+m3ns X k02C ( ) ds k0=k o(k0) Rk0(2s) Rk0(4s) then the generalized Ohno conjecture (1.4) is true. Later in this chapter, we shall explore the truth of the converse of this theorem. We next turn to a more detailed discussion of Scholz' re ection. Let k0=k be an extension of degree 3 with resolvent eld equal to F = k( p ) (which has degree 1 or 2 over k). The main idea of Scholz' re ection is that cubic extensions k0=k with resolvent eld F roughly correspond to cubic extensions ^k0=k with resolvent eld ^ F. This comes about as follows. The compositum L = k0F is a cyclic cubic extension of F. Let B be the eld B = F( p 3) = F ^ F. Then the degree [B : k] is a divisor of 4, and the compositum N = k0B is a cyclic cubic extension of B. Since B contains the cube roots of unity, by Kummer theory the cubic extension N=B has the form N = B( 1=3) for some nonzero 2 B. 5.2 Conductors and discriminants of cubic extensions In this section, we establish the basic notation of conductors, di erents, and dis criminants of cubic extensions. This material is derived from Hasse [9] and Martinet Payans [11]. Before we continue, we need to recall a few results from class eld theory that would allow us to analyze our Dirichlet series identities. Theorem 5.2 (Isomorphism Theorem) There is a onetoone correspondence be tween the nite abelian extensions L of F and the open subgroups U = UL of the idele class group JF = A F =F such that the Galois group Gal(L=F) is isomorphic to JF=U. Moreover, if L=F is a nite abelian extension and K is an intermediate eld L K F, then the corresponding subgroups satisfy F UL UK UF A F . For each character of the group JF trivial on U, let f denote its conductor. It is an integral ideal in F. 59 Theorem 5.3 (ConductorDiscriminant Formula) Let L=F be a nite abelian exten sion of number elds corresponding to the open subgroup U of the idele class group JF . Then the relative discriminant DL=F of L=F is given by DL=F = Y f ; where ranges over all the characters of JF trivial on U. Let k0=k be an extension of degree 3 with Galois closure L and resolvent eld F. The relative discriminants of k0=k, L=k, and F=k respectively, considered as ideals in ok, are denoted by Dk0=k, DL=k, and DF=k, respectively. The di erents of these extensions, as ideals in the rings of the integers of the corresponding over eld, are denoted by dk0=k, dL=k, and dF=k, respectively. The relative discriminants and di erents are related by means of the relative norms Dk0=k = Nk0=k(dk0=k); DL=k = NL=k(dL=k); DF=k = NF=k(dF=k): According to the notation introduced in Chapter 1, we can write dk0=k = N(Dk0=k); dL=k = N(DL=k); dF=k = N(DF=k): If k0=k is noncyclic, then F=k turns out to be quadratic and L=F cyclic cubic. By the isomorphism theorem, this extension corresponds to an open subgroup U of index 3 in JF . There are two nontrivial cubic characters and 2 with kernel equal to U. By the conductordiscriminant formula, the discriminant of L=F is DL=F = f2 , since and 2 have the same conductor. Therefore by the tower law for discriminants (see Prop. 13 of Chap. VII4 in [18]) we have DL=k = D3 F=k NF=k(f )2: Next, we shall discuss the concepts of conductors and discriminants over the idele class group. For a place v of F, let iv be the natural injection of F v into the idele 60 class group JF . Thus, iv(x) is the coset of the idele with v component equal to x and all other components equal to 1. Let be a character of JF which is trivial on U. We de ne the v component to be v(x) = (iv(x)) for x 2 F v . For a nite place v, the kernel of v contains either the full unit group o v of F (in which case we set fv = 0) or some subgroup 1+$fv v ov for a smallest positive integer fv, where $v is a uniformizer in Fv. Then the conductor of v is ' v = $fv v . For an in nite place v, the kernel of a nite order character v is either all of F v in which case we set ' v = 1, or possibly just the positive real numbers R+ in the event v is real. In the latter case, we set ' v = 1. The idelic conductor ' is de ned to be the idele with v component equal to ' v for all places v. The conductor is wellde ned as an element of A F =A0 F;1, where A0 F;1 = Y vj1 F0 v Y v1 o v where F0 v represents the connected component of 1 in F v . Hence, F0 v = C if v is complex and R+ if v is real. For a place v of F and a place w of L lying above v, let f 1; : : : ; mg be a basis of Lw over Fv. Let (i) j range over the m conjugates of j . For in nite places v, we de ne the relative discriminant Lw=Fv of the extension Lw=Fv to be the square of the determinant of the matrix ( (i) j ), which is an element of F v . This relative discriminant is wellde ned only modulo multiplication by elements of F2 v , the group of squares of elements of F v . By convention, we stipulate C=C = R=R = 1 and C=R = 1. For nite places, the maximal compact subring of Lw is a free ovmodule, and thus we may select a basis f jg. Then Lw=Fv is de ned using this basis. With this special choice of the j , the relative discriminant is wellde ned module o2v , the group of squares of elements of o v . The vadic part v;L=F of the relative discriminant of L=F is de ned by v;L=F = Y wjv Lw=Fv 2 F v : 61 The idelic relative discriminant L=F is taken to be the idele whose vadic com ponent is v;L=F . That this is an idele is a consequence of the fact that v;L=F 2 o v for almost all v. This discriminant is wellde ned modulo multiplication by elements of A2 F;0 = Y v1 o2v : This idelic de nition of discriminant was rst advanced in [8], where many basic properties are established. There is simple relationship between the conductors and discriminants as ideals with their counterparts as ideles. Let IF denote the group of fractional ideals of F. There is a natural homomorphism id : A F ! IF described in Chap. V3 of [18]. Then we have f = id(' ) DL=F = id( L=F ): Moreover, we have an idelic analogous of the conductordiscriminant formula: Theorem 5.4 (Idelic ConductorDiscriminant Formula) Let L=F be a nite abelian extension of number elds corresponding to the open subgroup U of the idele class group JF . Then the idelic relative discriminant L=F of L=F is given by L=F = Y ' ; where ranges over all the characters of JF trivial on U. 5.3 The resolvent eld identity Nakagawa's proof establishes by means of Scholz' re ection that the terms correspond ing to extensions k0=k in C ( ) on one side of Ohno's series identity correspond to the terms for extensions k0=k in C (3 ) on the other side. To simplify our work with these identities, we introduce some notation for the Euler products in the identities. 62 Following equation (2.4) of Chapter 2, we de ne the following Euler factors: Ek0;v(s) = 8>>>>>>>>>>>>>>< >>>>>>>>>>>>>>: (1 + q2s v )2 if (k0=k; v) = (1); 1 + q4s v if (k0=k; v) = (2u); 1 + q2s v if (k0=k; v) = (2r); 1 q2s v + q4s v if (k0=k; v) = (3u); 1 if (k0=k; v) = (3r); (5.1) where (k0=k; v) denotes the splitting type of the place v of k in the extension k0=k. We are omitting the common factors that will cancel out in the generalized Ohno Nakagawa identity. From Theorem 3.1, the dual Euler factors are ^E k0;v(s) = Ek0;v(s) for v  3, and for v j 3 we have ^E k0;v(s) = 8>>>>>>>>>>>>>>< >>>>>>>>>>>>>>: q4s v (1 + q12s v + 2q14s v ) if (k0=k; v) = (1); q4s v (1 + q12s v ) if (k0=k; v) = (2u); q2s v (1 + q14s v ) if (k0=k; v) = (2r); q4s v (1 + q12s v q14s v ) if (k0=k; v) = (3u); 1 if (k0=k; v) = (3r); (5.2) so long as 3 is unrami ed in k. Again, we have omitted the factors that cancel out in the conjectured OhnoNakagawa identity. Then the cancelled OhnoNakagawa identity has the form X k02K ds k0=k o(k0) Y v1 ^E k0;v(s) = 3r2+m3ns X k02K ds k0=k o(k0) Y v1 Ek0;v(s): (5.3) By the same reasoning as behind Theorem 5.1, this conjecture is true if and only if X k02C (3 ) ds k0=k o(k0) Y v1 ^E k0;v(s) = 3r2+m3ns X k02C ( ) ds k0=k o(k0) Y v1 Ek0;v(s) (5.4) holds for all 2 k =k2. 63 It is important to note that all the exponents of qs v in all the Euler factors are even, and yet the exponent of 3s in the OhnoNakagawa identity is odd in a sense we will presently make clear, and that this implies there is a natural splitting of the OhnoNakagawa identity. First, we need to express the Euler products as ordinary Dirichlet series. Each nonarchimedean place v of k corresponds to a prime ideal pv in the ring o of integers of k, satisfying qv = N(pv) = (o : pv), the absolute norm of pv. The absolute norm of the ideal 3o generated by 3 in o is just 3n where n = [k : Q]. Then the two sides of our conjecture expand into series of the following form: 3r2+m3ns X k02K ds k0=k o(k0) Y v1 Ek0;v(s) = X k02K X a Ck0;a N(33Dk0=ka2)s X k02K ds k0=k o(k0) Y v1 ^E k0;v(s) = X k02K X a ^ Ck0;a N(Dk0=ka2)s ; where the coe cients Ck0;a, ^ Ck0;a are ordinary rational numbers with denominator a divisor of 6. Here the sum over a ranges over all integral ideals of o; however, due to the nature of the Euler products we may assume that the prime power factor of a corresponding to any prime ideal pv is pj v for 0 j 2, except for the prime ideals pv lying over 3 in the dual series, which may have exponents 0 j 4. In order for this conjectured identity to hold, the sum of the coe cients Ck01 ;a1 for given M = N(33Dk01 =ka21 ) for varying k01 2 K and a1 in o must equal the sum of the coe cients ^ Ck02 ;a2 for M = N(Dk02 =ka22 ) and varying k02 2 K and a2 in o. The terms cancelling in this subidentity would satisfy N(Dk02 =k) = N(3Dk01 =kc2) for some fractional ideal c in k. In the case k = Q, this equality of norms together with the fact that k1 and k2 have opposite splitting types at 1 implies that Dk2 = 3Dk1 modulo multiplication by squares, where Dk1 and Dk2 are the discriminants, as signed integers, of k1 and k2 64 respectively. The resolvent eld of k1 is then F = Q( p Dk1 ), while the resolvent eld of k2 is the dual ^ F = Q( p 3Dk1 ). This proves that the OhnoNakagawa identity for Q holds only if all the resolvent eld identities (5.4) are true. This completes the proof of Theorem 1.4, which we restate as follows: Theorem 5.5 For a squarefree integer d, let C (d) denote the collection of all ex tensions k=Q of degree at most 3 with resolvent eld Q( p d). Then for d > 0, we have X k2C (3d) ds k o(k) Y p ^E k;p(s) = 313s X k2C (d) ds k o(k) Y p Ek;p(s); X k2C (3d) ds k o(k) Y p ^E k;p(s) = 33s X k2C (d) ds k o(k) Y p Ek;p(s): Everything in the above identities is the same as in the whole OhnoNakagawa identity, just split according to the resolvent elds. For general ground elds k, the norm equality N(Dk02 =k) = N(3Dk01 =kc2) does not strictly imply that Dk02 =k = 3Dk01 =kc2 as ideals (since there are possibly di erent prime ideals of the same norm). However, based on the role of Scholz' re ection in Nakagawa's proof, it is still natural to suppose that the identity splits according to the resolvent elds. After this, we shall work on simplifying the resolvent OhnoNakagawa identity (5.4) by means of class eld theory. 65 CHAPTER 6 Examples of the resolvent OhnoNakagawa identity In this chapter, we shall use known tabulations of extensions of degree at most 3 to verify nite analogues of the resolvent OhnoNakagawa identity. The examples in this chapter provide precise numerical evidence for Conjectures 1.1 and 1.2; the approach is di erent from the equalities of class numbers established in Ohno's original paper [14]. Here, instead of calculating class numbers of integral binary cubic forms, we use existing tables of number elds and calculations of their splitting types at di erent places to check the conjectures recast as an equality of nite sums of nite Euler products. These equalities come from eld extensions with bounded rami cation, while the discriminants of the binary cubic forms involved may be enormous and far beyond the tables calculated by Ohno. 6.1 The nite OhnoNakagawa identity Just as in Chapter 5, choose 2 k =k2, and let F = k( p ) and ^ F = k( p 3 ). Let S be a nite set of places of k containing all in nite places and all places v dividing 3dF=kd ^ F=k. Let CS( ) = CS(F) be the set of all extensions k0=k of degree at most 3 with resolvent eld F and which are unrami ed for all places v =2 S. Class eld theory implies the set CS( ) is a nite set of extensions k0=k. Similarly, CS(3 ) = CS( ^ F) is a nite set of extensions. Thus, for all extensions k0=k in CS( ) and in CS(3 ), the relative discriminant dk0=k is divisible only by qv for places v 2 S. The terms in the Dirichlet series a Ms for which M is divisible only by qv for v 2 S must cancel out on both sides of the conjectured resolvent eld identity (5.4). This proves the following 66 theorem: Theorem 6.1 The generalized resolvent eld conjecture (1.6) is true if and only if for all 2 k =k2 and all nite sets of places S containing all places v j 1 and v j 3dF=kd ^ F=k, where F = k( p ), ^ F = k( p 3 ), we have X k02CS( ^ F) ds k0=k o(k0) Y v2S ^E k0;v(s) = 3r2+m3ns X k02CS(F) ds k0=k o(k0) Y v2S Ek0;v(s): (6.1) Recall that n = [k : Q], r2 is the number of complex places of k, o(k0) is the automor phism order of k0=k (so 1,2,3, or 6) as de ned on p. 21, Ek0;v and ^E k0;v are the Euler factors de ned in (5.1) and (5.2), and m is the number of real embeddings of k for which is positive. In particular, by Nakagawa's theorem all these nite identities are true when k = Q. The crucial aspect of this theorem is that there are only nitely many terms on both sides of the identity. We shall call this the nite OhnoNakagawa identity for S and . In this chapter, we con rm this identity for a fair number of cases based on eld data from various sources. Section 6.1 presents numerous con rmations of Nakagawa's theorem based on the data of elds of degree at most 3 over Q, while Section 6.2 presents con rmations of our new conjecture over k = Q(i). 6.2 Resolvent identities over Q To verify the nite OhnoNakagawa identity for a given nite set of places S of k and element 2 k , we need a list of the extensions k0=k contained in CS( ), their relative discriminants, and their splitting types at the places v 2 S. We have obtained this data from several independent sources, which we shall identify below. Again, as we have established, the identities are known consequences of Nakagawa's theorem, but this veri cation is quite di erent from Ohno's original data, and we feel these examples of identities are worth describing in detail. 67 Cohen et al. give a survey of counting number elds in [3]. The Bordeaux com putational number theory group has made available tables of number elds of degree at most 7 and discriminants below speci c bounds at http://pari.math.ubordeaux1.fr/pub/pari/packages/nftables/ We originally consulted the les in the Bordeaux archive called T20.gp, T22.gp, T31.gp, T33.gp, where the two numbers refer to degree n and number of real places r1 of the number elds. A typical line in one of these les would be of the form: [321,[1,1,4,1],1,[]] which lists the discriminant 321 of the number eld, the vector of coe cients of a generating polynomial x3x24x+1, the class number and the structure of the class group of the number eld. We are only interested in the discriminant and generating polynomial. These provided veri cation of the identities over Q at least for small sets of places S. In particular, for S = f2; 3g, we may extract from the les those elds with discriminant of the form 2a3b. In addition to Q, there are 7 quadratic elds and 9 cubic elds up to conjugacy. We list these elds in Table 6.1 sorted by the squarefree part of their discriminant Dk. As a reminder, these lists include only one of each conjugate triple of noncyclic cubic extensions of Q. Thus, in our identity, we must use o(k0=k) = 3 for cyclic cubic extensions and o(k0=k) = 1 for noncyclic cubic extensions. We can identify the cyclic cubic extensions k0=k over k = Q as those cubic extensions with discriminant equal to a perfect square. Table 6.1 includes only one cyclic cubic eld of discriminant 81. The next task in verifying the identities is to determine the Euler factors Ek0;p(s) and ^E k0;p(s) for each prime p = 2; 3. This requires determining the splitting type of k0 over kv = Qp. We use the generating polynomial supplied by the Bordeaux tables to carry this out, and we use the padic package in Maple to factor this polynomial padically. Here is the Maple procedure that accomplishes this 68 # Take a number field of degree <=3 and find its splitting type at p splittype:=proc(field,p) local x,pol,n,m: pol:=poly(field,x): n:=degree(pol): m:=nops([rootp(pol,p)]): if n=m then RETURN(1) elif ( field[1] mod p ) = 0 then # ramified if n=2 or m=1 then RETURN(3) else RETURN(5) fi: else # unramified if n=2 or m=1 then RETURN(2) else RETURN(4) fi: fi: end: This procedure assumes that field is a vector describing the number eld as con tained in the Bordeaux tables. Thus, according to the Bordeaux format, the entry field[1] is the discriminant of the eld. First, this procedures uses another pro cedure poly(field,x) that extracts the generating polynomial of the number eld with indeterminate x. Then it determines the degree n of this polynomial (1, 2, or 3), and the number m of roots of the polynomial in Qp through the rootp command in Maple's padic package. If n = m, then the polynomial splits completely over Qp and the type is (1). If not, it then tests to see if p is rami ed in k0 over k = Q by simply checking whether or not p divides the discriminant field[1]. Then in either case, the type is quadratic if n = 2 or n = 3 and m = 1, which means that the generating polynomial has an irreducible quadratic factor over Qp. In the procedure, types (1), (2u), (2r), (3u) and (3r) are numbered 1, 2, 3, 4, and 5. The results of 69 these calculations are also shown in Table 6.1. Given the determination of types, our Maple programs determine the Euler factors by substituting q = qv and x = q2s v into the corresponding entry of the arrays of formats for the Euler factors listed below eulerfac:=[ (1+x)^2, (1+x^2), (1+x), (1x+x^2), 1]: eulerfacdual:= [ x^2*(1+q*x+2*q*x^2), x^2*(1+q*x), x*(1+q*x^2), x^2*(1+q*xq*x^2), 1]: using the dual factor only for p = 3, when qv = 3fv under the assumption that 3 is unrami ed in kv with residue degree fv. The original Euler factors were calculated in [5] and presented here in equation (2.4), while the dual Euler factors for v j 3 were calculated in Theorem 3.1. These procedures allow us to evaluate the two sides of the nite OhnoNakagawa identity in Theorem 6.1. We present them in the form stated in Theorem 6.1 as sums of partial Euler products. The simplest identity covered by our conjecture is the case where S = f3g and = 1 or 3, since 3 is the only prime dividing 3dF=kd ^ F=k. In that case, the only elds entering the identities are Q, q1, k1 and k3, since these are the only elds for which the absolute value of the discriminant is a power of 3. Then the identities below have the Euler products (for only the prime p = 3) for the elds of discriminant 3, 243 on one side and the elds of discriminant 1, 81 on the other side. The Euler factor for 3 is determined by our recipes with the splitting type read from the last column of Table 6.1. = 1 1 2 3s 32 s 1 + 3 34 s + 243s = 313s 1 6 1 + 32 s 2 + 1 3 81s = 3 1 6 34 s 1 + 3 32 s + 6 34 s + 1 3 81s = 33s 1 2 3s 1 + 32 s + 243s 70 Both of these identities may be easily checked by hand to be true, as Nakagawa's theorem implies. Next we turn to the full list of elds which are unrami ed outside S = f2; 3g. For = 1; 2; 3; 6, we have m = 0, while for = 1;2;3;6 we have m = 1. For the purpose of comparison, we shall group the identities in pairs and 3 (mod squares) since these pairs have the same elds on both sides. = 1 1 2 3s 1 + 24 s 32 s 1 + 3 34 s + 108s + 243s 1 + 24 s + 2 972s = 313s 1 6 1 + 22 s 2 1 + 32 s 2 + 1 3 81s 1 22 s + 24 s = 3 1 6 1 + 22 s 2 34 s 1 + 3 32 s + 6 34 s + 1 3 81s 1 22 s + 24 s = 33s 1 2 3s 1 + 24 s 1 + 32 s + 108s + 243s 1 + 24 s + 2 972s As a small explanation, both of the above identities concern the two elds in Table 6.1 with = 1 and the ve elds with = 3. The Euler factors are determined according to the recipes given above, with the types read o the last two columns of Table 6.1. The remaining pairs of identities follow below = 2 1 2 24s 1 + 22 s 32 s 1 + 3 34 s + 216s 1 + 22 s = 313s 1 2 8s 1 + 22 s 1 + 34 s = 6 1 2 8s 1 + 22 s 34 s 1 + 3 32 s = 33s 1 2 24s 1 + 22 s 1 + 32 s + 216s 1 + 22 s = 3 1 2 4s 1 + 22 s 34 s 1 + 3 32 s + 324s 1 + 22 s = 313s 1 2 12s 1 + 22 s 1 + 32 s 71 = 1 1 2 12s 1 + 22 s 32 s 1 + 3 34 s = 33s 1 2 4s 1 + 22 s 1 + 34 s + 324s 1 + 22 s = 6 1 2 8s 1 + 22 s 34 s 1 + 3 32 s + 6 34 s + 648s 1 + 22 s = 313s 1 2 24s 1 + 22 s 1 + 32 s + 1944s 1 + 22 s = 2 1 2 24s 1 + 22 s 32 s 1 + 3 34 s + 1944s 1 + 22 s = 33s 1 2 8s 1 + 22 s 1 + 32 s 2 + 648s 1 + 22 s These identities may be veri ed by elementary algebra to be correct; however, we also used Maple's algebraic simpli cation tools to verify them by computer. We next proceeded to check cases where the set S of places contains all primes up to and including a given prime p. We denote these sets of extensions by Sp. It turn out that for even relatively small primes p such as p = 11, the extensions may have discriminant as large as 22355272112 = 144074700, which is beyond the published Bordeaux tables. To go further, we used the program cubic written by Karim Belabas. The algorithm is established in [1], and the source code is available at http://www.math.ubordeaux.fr/~belabas/research/software/cubic1.2.tgz We made some minor modi cations in cubic to allow it to restrict output to only elds which are unrami ed for all p > 11. Table 6.2 gives the number of elds in CSp of both positive and negative discriminant, with the largest discriminant in each set also displayed. Fields are counted only up to conjugacy. With this data and our Maple procedures, we veri ed the nite OhnoNakagawa identity (6.1) for all cases comprised by CS11 . 72 As one simple further example, we shall take S = f2; 3; 7g (thus omitting 5) and = 7 and = 21. The negative discriminants counted are 7 (quadratic) and 567 = 347 (cubic), and the positive discriminants are 21 (quadratic) and 756 = 22337 (cubic). The identities (6.1) turn out to be = 21 1 2 7s 1 + 22 s 2 34 s 1 + 3 32 s 1 + 72 s + 567s 1 22 s + 24 s 1 + 72 s = 313s 1 2 21s 1 + 24 s 1 + 32 s 1 + 72 s + 756s 1 + 72 s = 7 1 2 21s 1 + 24 s 32 s 1 + 3 34 s 1 + 72 s + 756s 1 + 72 s = 33s 1 2 7s 1 + 22 s 2 1 + 34 s 1 + 72 s + 567s 1 22 s + 24 s 1 + 72 s Again, both identities may be veri ed by elementary algebra, although they must be true due to Nakagawa's theorem. After using the Bordeaux tables and Belabas' program to complete the above tests of the nite OhnoNakagawa identities, we learned of the program of John Jones and David Roberts which enumerates low degree elds with prescribed rami cation, which is exactly what we need to test these identities. The JonesRoberts algorithms are described in [10], and made available at the website http://hobbes.la.asu.edu/NFDB/ We used this program to con rm the list of elds provided by Belabas' program, as well as the OhnoNakagawa identities. It can be used to enumerate elds which are unrami ed except for primes at most 17, and thus provide more con rmation of Nakagawa's theorem. 73 6.3 Resolvent identities over Q(i) Here we take k = Q(i) and consider the extensions k0=k of degree at most 3. Such elds k0 have degree 2, 4 or 6, and all in nite places are complex. The Bordeaux tables include les T40.gp and T60.gp which list all quartic and sextic totally com plex elds up to conjugacy and with maximal discriminant 999988 and 199664, respectively. As it turns out, this is not large enough to verify the identity even for S = f1 + i; 3g. For example, by our earlier list for Q, the sextic eld k( 3 p 3) has discriminant (4)3(243)2 = 3779136. The papers [15] and [2] provide information about enumerating sextic elds. Fortunately, the JonesRoberts program allows us to enumerate all elds of degree at most 6 with prescribed rami cation at a small set of primes, and as we shall see this allows us to verify the nite OhnoNakagawa identity for k = Q(i) and S = f1 + i; 3g. At the website http://hobbes.la.asu.edu/NFDB/, we rst conducted a search for elds of degree 4, r1 = 0, r2 = 2 with arbitrary size discriminant, but rami cation possible only at p1 = 2 and p2 = 3. This would include all quadratic extensions of Q(i) which are unrami ed outside S = f1 + i; 3g. This produced a list of 29 degree 4 polynomials corresponding to each possible eld up to conjugacy. We next used Jones' program to determine the list of sextic elds with r1 = 0 and rami cation only at p1 = 2 and p2 = 3. This produced a list of 140 polynomials, which would include all cubic extensions of Q(i) unrami ed outside S = f1 + i; 3g. The lists contain one polynomial for each isomorphism class of eld matching the conditions imposed. The next task is to extract from these lists precisely those polynomials generating extensions of Q(i). For that purpose, we use the following basic fact from eld theory: Lemma 6.1 Let L=Q be a nite extension of degree n, and let K=Q be an extension of degree m j n. Let be an element of L such that L = Q( ). Then K is a sub eld of L if and only if the minimal polynomial of over K has degree n m . This will be a 74 factor of the minimal polynomial of over Q. Proof. Let p(x) 2 Q[x] be the monic minimal polynomial of 2 L; then the degree of p(x) is [L : Q] = n. Let q(x) be the monic minimal polynomial of over K which is assumed to have degree n=m where m = [K : Q]. Since the compositum LK is the same as the eld K( ), then [LK : K] = [K( ) : K] = n=m. Then by the tower law [LK : Q] = [LK : K][K : Q] = (n=m)m = n = [L : Q]. This proves LK = L and hence that K L. By minimality, q(x) is a factor of p(x). The converse, where we assume K L, immediately follows from the tower law [L : K] = [L : Q]=[K : Q]. Thus, to extract the extensions of Q(i), we simply have to check if the polynomials in the lists provided by Jones' program factor over Q(i). For example, the rst quartic eld in Jones' list is p(x) = x4 x2 + 1: In PARI, the discriminant of the number eld generated by a root is calculated by the command nfinit(p).disc, where p is the polynomial expression. This example has discriminant Dk0 = 144. We can calculate the factorization of p(x) over Q(i) by means of the command factornf( x^4 x^2 +1, y^2+1) with the result being p(x) = x4 x2 + 1 = (x2 ix 1)(x2 + ix 1) Thus, this number eld is a quadratic extension of Q(i). When we apply this test to the lists of polynomials produced by Jones' program, we nd that 5 of the quar tic polynomials and 13 of the sextic polynomials generate extensions of Q(i). These are presented in Tables 6.3 and 6.4. These tables also contain the absolute discrim inants Dk0 and the absolute norms of the relative discriminants dk0=k = N(Dk0=k) 75 calculated from the tower law given that DQ(i) = 4. From basic facts about dis criminants of towers of number elds (see [18], Proposition VIII.4.13), since Dk = 4, we have Dk0 = D2 k N(Dk0=k) = 16 N(Dk0=k) for [k0 : k] = 2, and Dk0 = D3 k N(Dk0=k) = 64 N(Dk0=k) for [k0 : k] = 3. The cubic factors of the sextic polynomials in Table 6.4 are given in Table 6.5. The quadratic and cubic extensions of k = Q(i) are given as k( ) where is a root of the quartic or sextic polynomial in our lists. The two factors of the polynomials over k may give nonconjugate extensions of k. We can also use PARI's command factornf to test each of the quadratic and cubic extensions of Q(i) to see if they contain the quadratic and cubic number elds listed in Table 6.1. The results are presented in the rst column of Table 6.3, where three elds k0 are identi ed as the biquadratic elds q1(i), q2(i), and q3(i), and in the fourth column of Table 6.4, where we see 9 of the sextic polynomials factor over the cubic elds kj , 1 j 9. Since k = Q(i) has class number 1, the relative discriminant of k0=k is of the form Dk0=k = Dk0=kZ[i], where Dk0=k is determined as an element of (Z[i] r f0g)=f 1g. Also, if k0 = k( ) for some 2 Z[i] and the monic minimal polynomial of over Z[i] is q(x), then the discriminant (q) of q(x) as a polynomial is equal to a square of a nonzero element in Z[i] times the generator of the relative discriminant of k0=k. Thus, (q) = u2Dk0=k for some u 2 Z[i] r f0g. Then N( (q)) = N(u)2 dk0=k. The fourth column of Table 6.5 shows the polynomial discriminant (q) of q(x), and the fth column shows N(u)2. The solutions to N(u) = 1, 2, and 4, resp., in Z[i] are u 2 f 1; ig, f 1 ig, and f 2; 2ig, resp. Then u2 2 f 1g, f 2ig, and 4, resp. Since 1 = i2 in Z[i], these calculations allow us to determine Dk0=k modulo squares from the calculations of (q). This is shown in the sixth column of Table 6.5. This is an easier calculation for the quadratic extensions of Q(i) in that the discriminant of each quadratic polynomial in Table 6.3 is also a relative discriminant of the extension. When the relative discriminant of q(x) is an integer multiple of 1 i, 76 then the conjugate factor q(x) has conjugate relative discriminant. Since 1 + i does not equal 1 i modulo squares in Z[i], these two factors q(x), q(x) give rise to non conjugate extensions over k. That explains why the factors q and q for dk0=k = 512, 4608, 23328 and 209952 in Tables 6.3 and 6.5 gives rise to nonconjugate extensions over k. The resolvent eld of each of the listed extensions is k( p Dk0=k) = k( p ) where is the squarefree part of Dk0=k. Since we are considering elds unrami ed outside S = f1+i; 3g, the integers of Z[i] which are divisible only by primes over S are equal modulo squares to precisely one of = 1; i; 1 i; 3; 3i; 3(1 i): In our tables, we have factored out squares and identi ed in the last column. The next issue is to completely determine for the factorization p(x) = q(x)q(x) whether the two factor polynomials q(x) and q(x) generate conjugate or nonconjugate extensions k0=k. They are conjugate over Q, but not necessarily over k. In the quartic case, the two extensions are k( p ) and k( p ) where is the generator of the relative discriminant modulo squares. By Kummer theory, these are the same extension if and only if 2 k2. Since k2 \ Q = Q2, this means N( ) must be a positive square in Q. Thus, for the nonsquare discriminants 32 and 288, the factors q(x) and q(x) generate two di erent quadratic extensions k0=k, which are not Galois over Q. This explains our notation for the seven quadratic extensions k0 of Q(i) in Table 6.3. There are relations among the splitting elds of the quartic polynomials. The quartics corresponding to Q3 and Q5 both have Galois group D4, while the others have Galois group C2 C2. The splitting elds of both the Q3 and Q5 quartics contains Q2. In general, suppose q(x) is a monic irreducible polynomial over k = Q(i), and that q(x) is its conjugate polynomial. If is a root of q(x), then is a root of q(x). Let K = k( ) and K = k( ). If K=k is a cyclic cubic extension, then K = K if and only if K is a Galois extension of degree 6 of Q, since then conjugation is 77 an automorphism of order 2 of K=Q. The Galois group is then cyclic C6 or the symmetric group S3. In the former case, K contains a cyclic cubic extension of Q, which is unrami ed outside S. The only possibility is the cyclic eld k1 of discriminant 81. From Table 6.4, the second sextic factors over k1, and thus the splitting eld is the compositum K2 = k1 k = k1(i), which is cyclic of degree 6 over Q. If the Galois group of K=Q is S3, then K contains a conjugate triple of nonconjugate cubic elds. Thus, this can be determined again by factoring over the cubic elds in Table 6.1. The cyclic extensions K=k have relative discriminant generator equal to a square in k, and from our list we see there are just three possible sextics all of which have DK=k = 81. The rst sextic factors over the cyclic cubic eld k1, while the second one in our list factors over k7. Thus, both those sextics have the same cubic extensions of k arising from the two cubic factors over k. The other two sextics each give rise to two distinct but conjugate cyclic cubic extensions of k. All the other cases listed in Table 6.5 correspond to noncyclic cubic extensions of k = Q(i), since Dk0=k is not a square in k. Suppose the three roots of q(x) generate the three conjugate cubic extensions K1, K2 and K3 over k. Suppose that the compositum of these extensions is the S3extension L of k. Then the roots of q(x) generate the extensions K1, K2, and K3, and their compositum is L. The triple fKjg is the same as fKjg if and only if L = L, which again means that L=Q is a Galois extension of degree 12. Assuming L = L, the Galois group G of L=Q contains S3 as a normal subgroup corresponding to the sub eld k = Q(i), and contains complex conjugation as an order 2 automorphism. Write S3 = h ; j 3 = 2 = 1; = 1i, with all these automorphisms xing k. Let be complex conjugation on k extended to L. Since S3 is a normal subgroup of G, the three order 2 elements , and 2 are permuted by conjugation by . Thus, at least one is xed by conjugation by . We may relabel to be one of the xed ones so that = . This implies that h ; i is an order 78 4 subgroup of G, and thus corresponds to a real cubic extension of Q. That means that the original sextic polynomial factors over this cubic eld, which would have to be in the list of elds unrami ed outside S on page 85. We may check which of the sextics in Table 6.4 factors over the cubics listed on page 85, and if so then we conclude L = L. This accounts for our notation for the 17 cubic extensions of Q(i) (up to conjugacy) listed in Table 6.5. Also, Table 6.4 shows the cubic elds kj named in Table 6.1 over which these sextic polynomials factor. The splitting eld of the K4 sextic contain the splitting elds 



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