Mock Modular Forms and Class Numbers of Quadratic Forms

Transcription

1 Mock Modular Forms and Class Numbers of Quadratic Forms I n a u g u r a l - D i s s e r t a t i o n zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakultät der Universität zu Köln vorgelegt von Michael Helmut Mertens aus Viersen Köln, 04

2 Berichterstatter: Prof. Dr. Kathrin Bringmann Prof. Dr. Sander Zwegers Tag der mündlichen Prüfung: 4. Juni 04

3 Gewidmet meinen Großvätern: meiner Mutter Vater in liebevollem Andenken und meines Vaters Vater zum 80. Geburtstag.

4 Kurzzusammenfassung Die vorliegende Arbeit befasst sich im Wesentlichen mit der Frage nach möglichen Rekursionsbeziehungen zwischen den Fourier-Koeffizienten einer bestimmten Klasse von Mock-Modulformen. Die prominentesten Beispiele solcher Fourier-Koeffizienten sind die Hurwitz-Klassenzahlen binärer quadratischer Formen, für die einige Rekursionen bereits lange bekannt sind. Als Beispiele sind hier unter anderen die Kronecker-Hurwitz-Klassenzahlrelationen sowie die Eichler-Selberg-Spurformel für Hecke-Operatoren auf Räumen von Spitzenformen anzuführen. Im Jahre 975 vermutete H. Cohen nun eine unendliche Serie von solchen Klassenzahlrelationen, die eng verwandt sind mit der erwähnten Eichler-Selberg- Spurformel. In dieser Arbeit beweise ich Cohen s Vermutung, sowie einige ähnliche Formeln für Klassenzahlen mit Hilfe wichtiger Resultate aus der Theorie der Mock-Modulformen. Mittels einer anderen Methode zeige ich schließlich, dass derlei Rekursionsbeziehungen ein generelles Phänomen für Fourier-Koeffizienten von Mock-Thetafunktionen und Mock-Modulformen vom Gewicht 3 darstellen. Als Spezial-fälle erhält man aus diesem Resultat alternative Beweise für die oben erwähnten Klassenzahlrelationen. Abstract This thesis deals with the question for possible recurrence relations among Fourier coefficients of a certain class of mock modular forms. The most prominent examples of such Fourier coefficients are the Hurwitz class numbers of binary quadratic forms, which satisfy many well-known recurrence relations. As examples one should mention the Kronecker-Hurwitz class number relations and the famous Eichler-Selberg trace formula for Hecke operators on spaces of cusp forms. In 975, H. Cohen conjectured an infinite family of such class number relations which are intimately related to the aforementioned Eichler-Selberg trace formula. In this thesis, I prove Cohen s conjecture and other similar class number formulas using important results from the theory of mock modular forms. By applying a different method I prove at the end that such recurrence relations are a quite general phenomenon for Fourier coefficients of mock theta functions and mock modular forms of weight 3. As special cases, one gets an alternative proof for the aforementioned class number relations.

5 Contents I Introduction 5 I. History I. Scope of this Thesis I.3 Acknowledgements II Modular Forms and Generalizations II. Elliptic Modular Forms II.. Definition and Examples II.. Operators on Modular Forms II. Jacobi Forms II.3 Harmonic Maaß-Forms and Mock Modular Forms II.4 Appell-Lerch sums IIICohen s Conjecture 5 III. Some Preliminaries III.. The Gamma function III.. Preparatory Lemmas III. The Proof IV Other Class Number Relations 39 IV. Completions IV. The Results V Relations for Fourier Coefficients of Mock Modular Forms 47 V. Holomorphic Projection V. Mock Modular Forms of weight V.. Hypergeometric series V.. Calculations V..3 Proof of Theorem V V.3 Mock Theta Functions V.3. Calculations V.4 Examples V.4. Trace Formulas V.4. Class Number Relations V.5 Conclusion

6 CONTENTS 4 References 77 Nomenclature 78 Index of Names 8 Glossary 83 Erklärung 83 Curriculum Vitae 84

7 Chapter I Introduction I. History One classical problem in number theory is to calculate class numbers of various objects, such as quadratic forms, number fields, genera of lattices, quaternion algebras, and many more. It occurs in many different interesting questions, e.g. in elementary number theory, where it is a vital tool to answer the question which integers can be represented by a given quadratic form, or in algebraic number theory, where it measures, how far the ring of integers of a number field, or more generally a maximal order in a division algebra over Q, is from being a principal ideal domain. One of the first systematic treatments of class numbers of binary integral quadratic forms is given in C.F. Gauß masterwork, his treatment Disquisitiones Arithmeticae from 80 [6]. In Chapter 5 he introduces so-called reduction theory which means, in a more modern manner, to distinguish certain standard representatives of equivalence classes of quadratic forms: Binary integral quadratic forms can be viewed as matrices of the form a b Q =, a, b, c Z. b c The number D := det Q is called the discriminant of the form Q. Now the modular group SL Z acts on the set Q D of quadratic forms with fixed discriminant D via Q, γ γ tr Qγ. For convenience we shall assume our forms to be primitive, i.e. gcda, b, c =. If we view a quadratic form as a function Q : Z Z, x, y x x, yq = ax + bxy + cy, y we may regard the action of SL Z as an orientation-preserving base change of the lattice Z, which motivates the notion of properly equivalent forms, i.e. quadratic forms in the same SL Z-orbit. The number of inequivalent forms of fixed discriminant D is called the class number hd. Gauß reduction theory can be formulated as follows: 5

8 CHAPTER I. INTRODUCTION 6 Theorem I.. C.F. Gauß, 80. Let D < 0. Then each SL Z-orbit of Q D contains exactly one reduced form. A primitive form Q = a b c b is called reduced if b a c b > 0 if a = c or b = a. In principle, this gives an explicit method to enumerate all reduced forms of given negative discriminant and thus determine the class number. Another approach towards calculating class numbers was made by L. Dirichlet about 40 years after the appearance of the Disquisitiones. In 839 he found a closed formula for class numbers of primitive binary quadratic forms [44, pp and pp ] using methods of complex analysis. Theorem I.. L. Dirichlet, 839. Let D be a fundamental discriminant i.e. either D is square-free or D = 4m with m, 3 mod4 and m squarefree, χ D = D the Kronecker symbol and define the Dirichlet L-series to the character χ D as the analytic continuation of Ls, χ D := χ D nn s. n= For D < 0 define w 3 = 6, w 4 = 4 and w D = for D < 4. Then it holds that hd = w D D L, χ D π A detailed proof is given in [6]. Note that the w D is exactly the order of the unit group of Z Q D. From a computational point of view, Dirichlet s class number formula is not really an improvement toward reduction theory, since evaluation of L-functions is not so easy. A faster way to produce tables of class numbers was introduced by L. Kronecker [38] and A. Hurwitz [3, 33]. They found a somewhat surprising recurrence relation for so called Hurwitz class numbers. The Hurwitz class number Hn is slight modification of the regular class number, I.. if n = 0, h n/f Hn = if n 0, 3 mod4 and n > 0, w f n/f n 0 otherwise with w d as in Theorem I... The value H0 = is merely for convenience. Kronecker and Hurwitz relate this quantity to certain divisor sums: They prove the identity I.. H4n s + λ n = σ n, s Z

9 CHAPTER I. INTRODUCTION 7 where I..3 λ k n := d n min d, n k d is the k-th power minimal-divisor sum and I..4 σ k n := d n d k is the usual k-th power divisor sum. Many other such relations have been discovered since: M. Eichler found in 955 that for all odd n the identity I..5 Hn s + λ n = 3 σ n s Z holds [8]. Another source of such relations is the famous Eichler-Selberg trace formula [9, 0,, 45] which can be stated as follows for precise definitions see Chapter II: Theorem I..3 M. Eichler, A. Selberg, 956. Let n be a natural number and k 4 be an even number. Then the trace of the nth Hecke operator T n on the space of cusp forms S k on SL Z is given by trace T k n = s Z g k s, nh4n s λ k n, where g k s, n is the coefficient of Xk in the Taylor expansion of sx + nx. Since the spaces S 4 and S 6 are 0-dimensional, we get the following class number relations from Theorem I..3: I..6 s nh4n s + λ 3 n = 0 I..7 s Z s 4 3ns + n H4n s + λ 5 n = 0. s Z In 975, H. Cohen [4] and D. Zagier [30, 50] made yet another approach towards understanding class numbers. Cohen considered a slight generalization of the Hurwitz class number which is motivated by Dirichlet s class number formula see [4, Definition.,.]: Let for r, n N { hr, n := r r!n r Lr, χ r π r r n, if r n, mod 4 0, otherwise

10 CHAPTER I. INTRODUCTION 8 with Ls, χ and χ D defined as in Theorem I.. and define ζ r, if n = 0 Hr, n := h r, n, if r n 0, mod 4 and n > 0 f f n 0, otherwise. Cohen defines the generating function of these numbers I..8 H r τ := Hr, nq n, n=0 τ H, q := e πiτ and shows the following result cf. [4, Theorem 3.]: Theorem I..4 H. Cohen, 975. For r the function H r in I..8 defines a modular form of weight r + on Γ 04. He proves this by writing H r as a linear combination of Eisenstein series of appropriate weight. Zagier on the other hand looked at the generating function I..9 H τ := H τ = Hnq n, τ H, q := e πiτ, n=0 which turns out not to be a modular form. But using an idea of Hecke cf. [9, ] and analytic continuation he finds that there is a modular completion of H cf. [30, Chapter,Theorem ]. Therefore let I..0 Rτ = + i 6π i τ ϑz dz z + τ 3 with ϑ as in Example II..4 iv. Theorem I..5 D. Zagier, 976. Define the non-holomorphic function Ĥ τ := H τ + Rτ. Then this function transforms like a modular form of weight 3 on Γ 04. In later years it was recognized that Ĥ belongs to a certain class of non-holomorphic modular forms, the so-called harmonic Maaß forms see Section II.3. These functions occured during the research on S. Ramanujan s mock theta functions: In his last letter to Hardy, Ramanujan introduced 7 functions in form of q-series which all had similar asymptotic behaviour as modular forms, but did not have a nice modular transformation property. He called these functions, which he defined in this very vague way, mock theta functions. Many attempts were made

11 CHAPTER I. INTRODUCTION 9 throughout the 0th century to put these mock theta functions into an appropriate context and give a mathematically precise definition of them. Finally, by work of S. Zwegers in his Ph.D. thesis [55], J.H. Bruinier and J. Funke in [], and K. Bringmann and K. Ono in [7, 8], a proper setting for the mock theta-functions or more generally mock modular forms was obtained. Many interesting generating functions arising from combinatorics turn out to be mock modular forms, such as the rank generating functions see e.g. [7] and [53] for partitions. Quite recently, in [6], K. Bringmann and J. Lovejoy related ranks of so called overpartitions again to Hurwitz class numbers. I. Scope of this Thesis In this thesis, I shall focus on relations among Fourier coefficients of mock modular forms similar to the Kronecker-Hurwitz formula I.. or the Eichler-Selberg trace formula Theorem I..3. For this we give a brief account of the needed facts about elliptic modular forms, Jacobi forms, and harmonic Maaß forms/mock modular forms as well as Appell-Lerch sums in Chapter II. The first goal is to give a detailed proof of a conjecture of Henri Cohen from 975. Based on his and Zagier s work on class numbers and special values of L-functions as well as computer experiments, he conjectured in [4] that the following should be true. Conjecture I.. H. Cohen, 975. Let I.. S4 τ, X := Hn s sx + nx + λ k+ nx k qn. n=0 n odd s Z s n Then the coefficient of X l in the formal power series S4 τ, X is a holomorphic modular form of weight l + on Γ 0 4. The first proof of this is given in [40]. Here, I will recall this proof in greater detail. Its basic idea is as follows. The coefficient in question can essentially be realized as a so-called Rankin-Cohen bracket see Definition II..8 of the functions H and ϑ plus a minimal divisor power sum. One can add non-holomorphic terms to each of these terms to make them transform like modular forms of the correct weight, and then it just remains to show that the non-holomorphic corrections cancel each other. This will be the content of Chapter III. This idea has also been used by Bringmann and Kane [4] to prove other class number relations which were conjectured by Bloom et al. in [0]. In Chapter IV we shall recall and slightly extend their results. Chapter V is dedicated to a different approach towards the previous results, namely holomorphic projection, which gives the striking observation, that k=0

12 CHAPTER I. INTRODUCTION 0 - loosely speaking - for every mock theta function and every mock modular form of weight 3 there are class number type relations among the Fourier coefficients. I.3 Acknowledgements There are several people who deserve to be mentioned in the acknowledgements of my Ph.D. thesis for various reason. Nonetheless, some choices have to be made or this would just take too long. According to St. Augustine, one should first think of those whom one forgets, thus I assure everyone who supported me during the last years through my studies and Ph.D. time but whose name does not show up here of my thankfulness right at the start. This thesis would not have been possible without the guidance and encouragement of my advisor, Prof. Dr. Kathrin Bringmann. For the last two years, she was always willing and able to help me when I got stuck at some point. For this I owe her deepest gratitude. I would also like to thank Prof. Dr. Sander Zwegers for agreeing to be the second advisor for my thesis. For the financial support I thank the Deutsche Forschungsgemeinschaft and Prof. Dr. Guido Sweers as spokesman of the DFG Graduiertenkolleg 69 Globale Strkturen in Geometrie und Analysis at the Universität zu Köln. I would like to thank my former and current colleagues in the Graduiertenkolleg and the number theory working group for the nice and friendly atmosphere and the many helpful discussions. Among these people, I especially want to set apart Dr. Ben Kane, Dr. Larry Rolen, René Olivetto, and my office mate José Miguel Zapata Rolón. For their support and their great education I want to thank my former professors and colleagues at RWTH Aachen University. Three of them should be mentioned especially: For one the advisor of my Bachelor- and Master thesis, Prof. Dr. Gabriele Nebe, whose great education, effort, and support made it possible for me to achieve my fellowship at the Graduiertenkolleg here in Cologne. For second I would like to thank Prof. Dr. Aloys Krieg for first introducing me to the topic of modular forms. And for third, I would like to thank Dr. Markus Kirschmer, who co-supervised both of my former theses and had a great influence on the success of my studies. Furthermore, Dr. Wolf Jung deserves to be mentioned here, since he got me interested in studying mathematics in the first place. Zu guter Letzt möchte ich mich bei meiner Familie, besonders meinen Eltern und Großeltern, für die unermüdliche Unterstützung und Ermutigung in den vergangenen Jahren bedanken. Sie haben mir stets und auf vielfältige Art und Weise den Rücken frei gehalten, was mir sowohl mein Studium als auch die Arbeit an dieser Dissertation extrem erleichtert hat.

13 Chapter II Modular Forms and Generalizations In this chapter I give a short exposé on the types of modular and automorphic objects that are going to be used in the rest of this thesis. We fix some notation: The letter τ will always denote a variable living on the complex upper half-plane H := {τ C Imτ > 0} while z, u, v may represent arbitrary complex variables. For brevity, we define x := Reτ, y := Imτ, and q := e πiτ. By Γ SL Z we usually denote the group II.0. Γ 0 N := {γ SL Z c 0 mod N}, for some N N. An element of such a group is denoted by a b γ :=. c d Sometimes, we shall also need the following special subgroups of SL Z. II.0. II.0.3 Γ N := {γ Γ 0 N a d mod N} ΓN := {γ SL Z γ I }, where I n denotes the n n unity matrix. Note that ΓN is the kernel of the canonical epimorphism SL Z SL Z/NZ and therefore a normal subgroup of SL Z. It is called the principal congruence subgroup of level N, and every subgroup of SL Z containing ΓN is called a congruence subgroup. II. Elliptic Modular Forms Here I give a short summary of the necessary theory of elliptic modular forms. The main references are [37] and [, Chapter ] for modular forms of integral weight, half-integral weight is treated e.g. in [47].

14 CHAPTER II. MODULAR FORMS AND GENERALIZATIONS II.. Definition and Examples Throughout this subsection, fix k Z and Γ SL Z if k Z assume Γ Γ 0 4. It is well-known that Γ acts on the upper half-plane via Möbius transformations, γ, τ γ.τ := aτ + b cτ + d for γ = a c d b Γ. For a function f : H C and γ Γ, we define the weight k slash operator as { cτ + d k f aτ+b, if k Z cτ+d f k γτ = k εd cτ + d f aτ+b, if k + Z, c d where m n denotes the extended Legendre symbol in the sense of [47], τ denotes the principal branch of the square root i.e. π < arg τ π, and {, if d mod4 ε d := i, if d mod4.. Now let P Q := Q { }. Defining a :=, a +b := 0, and := for 0 d a, b, d 0 one sees that SL Z acts transitively on P Q. Therefore there are only finitely many Γ-orbits on P Q. cτ+d Definition II... i A cusp of Γ is a coset representative of Γ\P Q. ii A one-periodic holomorphic function f : H C is said to be meromorphic resp. holomorphic at i if the function ˆf : Ė C with fτ = ˆfe πiτ cf. [4, Satz VI..4] has a meromorphic resp. holomorphic continuation onto E, where E := {z C z < } denotes the unit disk and Ė := E \ {0}. iii Let c P Q and γ 0 SL Z such that γ 0.c = i. Then a holomorphic function f : H C with f k γτ = fτ for some k Z and all τ H and γ Γ SL Z is said to be meromorphic resp. holomorphic at c, if the function k dτ w fγ0.w dw with w := γ 0.τ is holomorphic at i. Definition II... A function f : H C is called a modular form of weight k Z on Γ = Γ 0N or of level N N and character χ for χ a Dirichlet character modulo N if the following conditions are met: i f is holomorphic on H. ii f is invariant under the weight k slash operator, i.e. for all γ = a b c d Γ and τ H we have f k γτ = χdfτ.

15 CHAPTER II. MODULAR FORMS AND GENERALIZATIONS 3 iii f is holomorphic at the cusps of Γ. If we replace iii by iii f has at most a pole in every cusp of Γ, then we call f a weakly holomorphic modular form. A modular form that vanishes at every cusp of Γ is called a cusp form. The C- vector space of modular forms resp. cusp forms resp. weakly holomorphic modular forms of weight k on Γ = Γ 0 N with character χ is denoted by M k N, χ resp. S k N, χ resp. M! k N, χ. For other groups Γ we write analogously M kγ etc. Due to the fact that a weakly holomorphic modular form f is per definitionem one-periodic, it has a Fourier expansion around of the form fτ = where m 0 Z and again, q := e πiτ. n=m 0 a f nq n Remark II..3. Clearly it holds that products of modular forms are again modular forms which turns the graded vector space M Γ := k N 0 M k into a graded C-algebra. Example II..4. i Let k 4 be an even integer and let G k τ := mτ + n k. m,n Z m,n 0,0 These so-called Eisenstein series are absolutely convergent and therefore define holomorphic functions which are easily seen to be modular forms of weight k on SL Z cf. [37, Chapter III, ]. They have the following Fourier expansion, E k τ := ζk G kτ = k B k σ k nq n, where ζs is the Riemann ζ-function, B k is the kth Bernoulli number cf. [4, p. 03], and σ k is defined as in I..4. ii In the case k =, the series G is only conditionally convergent and gives, fixing a certain order of summation, a Fourier development E τ = 4 n= σ nq n, n=

16 CHAPTER II. MODULAR FORMS AND GENERALIZATIONS 4 which is perfectly convergent. This function defines a so-called quasimodular form of weight, see [] for details. The transformation under SL Z is given by II.. E aτ + b cτ + d = cτ + d E τ 6i ccτ + d. π transforms like a mod- The non-holomorphic function Êτ := E τ 3 πy ular form of weight under SL Z. iii The discriminant function τ := E3 4 τ E 6 τ 78 is a cusp form of weight on SL Z since the only cusp of SL Z is i and since the constant term of the Fourier expansion of vanishes, vanishes at that cusp. iv The Dedekind η-function defined by ητ := q 4 q n is a modular form of weight with multiplier system on SL Z which satisfies η 4 τ = τ. For details, see e.g. [37, Chapter III, 6]. n= v The ϑ-series of the lattice Z defined by ϑτ := n Z q n is a modular form of weight on Γ 04. vi More generally, for N N and s N and χ an even character modulo N of conductor F with 4sF N, the theta series II.. ϑ s,χ τ := n Z χnq sn is a modular form of weight on Γ 4N. For an odd character χ, the theta series II..3 θ s,χ τ := n Z χnnq sn is a cusp form of weight 3 on the same group. An important fact about modular forms is, that they are in a sense quite rare. As B. Mazur put it: Modular forms are functions on the complex plane that are inordinately symmetric. They satisfy so many internal symmetries that their mere existence seem like accidents. But they do exist. quoted from

17 CHAPTER II. MODULAR FORMS AND GENERALIZATIONS 5 Theorem II..5. For every congruence subgroup Γ, the space M k Γ and therefore S k Γ are finite-dimensional. Moreover, we have the dimension formulas 0, if k < 0 or k odd dim M k SL Z = k, if k 0 and k mod +, if k 0 and k mod, k dim S k SL Z = dim M k for k Z and { 0, if k < 0 dim M k Γ 0 4 = k +, if k 0, 0, if k 4 dim S k Γ 0 4 = k, if k > and k / Z, if k > and k Z k for k + Z Z. A proof of this can be found in [7, Chapter 3]. For computational purposes one has the following result called the Sturm bound or Hecke bound due to J. Sturm see [36, Theorem 3.3]. Theorem II..6 J. Sturm, 987. Let Γ SL Z be a congruence subgroup of SL Z of index M and f M k Γ with Fourier series fτ = n=0 a fnq n where there is an m 0 0 such that a f n = 0 for all n m 0. If then f is identically zero. m 0 > M k, This means that in order to decide equality of two modular forms it suffices to compare the first Fourier coefficients up to a certain bound. For special groups, there are are explicit formulas for their indices in SL Z. Proposition II..7. For every N N we have [SL Z : Γ 0 N] = N +, p p N [SL Z : Γ N] = N p, p N where the product is taken over the primes dividing N.

18 CHAPTER II. MODULAR FORMS AND GENERALIZATIONS 6 For a proof, see for example [7, pp. 3f]. Since in many cases arithmetic functions such as divisor sums occur as Fourier coefficients of modular forms, the finite dimension of the space of modular forms is the source of a vast collection of striking identities among these functions. First examples are the Hurwitz identities for divisor sums which A. Hurwitz discovered in his dissertation [3]: Since the spaces M 8 SL Z and M 0 SL Z are onedimensional, it must hold that E 4 = E 8 and E 4 E 6 = E 0 since the constant terms of all the Fourier series is. This yields the famous Hurwitz identities for divisor power sums n σ 7 n = σ 3 n + 0 σ 3 mσ 3 n m, m= n σ 9 n = σ 5 n 0σ 3 n σ 3 mσ 5 n m, which are extremely difficult to prove without using the theory of modular forms or of elliptic functions. II.. m= Operators on Modular Forms As we already remarked, products of modular forms are again modular forms. It turns out that by involving derivatives of modular forms which themselves are not modular forms but rather quasi-modular forms, see [, section 5.3] one can define a different product on the algebra of modular forms. Definition II..8. Let f, g be smooth functions defined on the upper half plane and k, l R >0, ν N 0. Then we define the νth Rankin-Cohen bracket of f and g as [f, g] ν = k + ν l + ν r D r fd s g s r r+s=ν where for non-integral entries we define m Γm + := s Γs + Γm s +. Here, the letter Γ denotes the usual Gamma function. Furthermore, we set D t = d. πi dt Proposition II..9 Theorem 7. in [4]. Let f, g be not necessarily holomorphic modular forms of weights k and l respectively on the same group Γ. Then [f, g] ν is modular of weight k + l + ν on Γ. Since each Rankin-Cohen bracket is obviously a bilinear operator on M Γ, it can as well be regarded as a product on this algebra. For ν > 0, it is not associative and it is commutative if and only if ν is even. The bracket [, ] 0 coincides with

19 CHAPTER II. MODULAR FORMS AND GENERALIZATIONS 7 the usual product, while the bracket [, ] defines a Lie-bracket on M Γ. This gives M Γ the structure of a so called Poisson-algebra cf. [, p. 53]. Some important unary operators on modular forms are defined as follows. Definition II..0. Let f : H C be a not necessarily holomorphic, but - periodic function with Fourier expansion fτ = n Z a f n, yq n. Then we define for N N and χ a generalized Dirichlet character modulo N the operators f UNτ := a f Nn, y II..4 q n, N n Z II..5 II..6 II..7 f V Nτ := fnτ, f S N,r τ := n Z n r mod N a f n, yq n f χτ := n Z a f n, yχnq n. sieving operator, If f transforms like a modular form of level M, then each of the above operators sends f to a modular form of in general higher level. Under certain conditions, the operator UN can also preserve or even reduce the level. Note that all these operators can be realized by extending the definition of the slash operator to GL Q +, the group of matrices over Q with positive determinant. In particular, we have the following. Proposition II... Let f : H C a function transforming like a modular form of weight k and character χ on Γ 0 M. Then it holds that i f UN and f V N transform like modular forms of the same weight and character on Γ 0 NM. ii If N M then f UN also keeps the same level as f and if N M and χ is a character modulo M N then the level of f UN reduces to M N. iii For f M k 4 with Fourier coefficient a f n = 0 for all n mod 4 we have f U4 M k. iv For N = p a prime and r 0 mod p, the sieving operator S p,r changes the group to Γ 0 lcmm, p Γ p. v Let ψ be a character modulo m, then f ψ transforms like a modular form on Γ 0 Mm with character χψ.

20 CHAPTER II. MODULAR FORMS AND GENERALIZATIONS 8 Proof. Assertions i-iii are contained in Lemmas and 4 of [39], for v we refer the reader to Proposition.8 in [4]. Claim iv is mentioned e.g. in [4], but not proven, so we give a short prove here. It is clear that where ζ p := e πi p l p 0 l p 0 f S p,r τ = p l = 0 N ζp rl f l p τ 0 is a primitive pth root of unity. For a c d b Γ 0M we have that a b l a l p = c b + l l a d c p p p c d 0 c d + lc, p thus the group Γ 0 lcmm, p Γ p is a subgroup of Γ 0 M which is normalized by. This implies iv. Definition II... Let k Z and f M k Γ 0 N with Fourier expansion fτ = m=0 a fmq m and let n N. Then the nth Hecke operator of level N is defined by. f T n k Nτ = m=0 Remark II..3. The operator T n kn to S k Γ 0 N. II. Jacobi Forms d gcdm,n gcdd,n= mn d k a f d qm. maps M k Γ 0 N to M k Γ 0 N and S k Γ 0 N Jacobi forms appear in many different contexts of modular forms. They are more or less an amalgam of modular forms and elliptic functions. First examples, namely the Jacobi Theta function, were already studied by C.G.J. Jacobi in the 9th century, special cases of these even go back to L. Euler, but it was not until 985, when the first systematic treatment of their theory appeared. This treatment [] by M. Eichler and D. Zagier is still the standard reference for the theory of Jacobi forms. Here, we only need some basic properties. Definition II... Let φ : C H C be a holomorphic function and k, m N 0. We call φ a Jacobi form of weight k and index m on Γ, if the following conditions hold. i For all γ = a c d b Γ and z, τ C H it holds that φ k,m γz, τ := cτ + d k cz πim z e cτ+d φ cτ + d, aτ + b = φz, τ, cτ + d

21 CHAPTER II. MODULAR FORMS AND GENERALIZATIONS 9 ii For all λ, µ Z we have φ m λ, µz, τ := e πimλ τ+λz φz + λτ + µ, τ = φz, τ, iii φ has a Fourier development cn, rq n ζ r, ζ := e πiz, n,r Z with cn, r = 0 for n < r 4m. We call z the elliptic and τ the modular variable of φ. This definition has been extended to include Jacobi forms of half-integral weight and index as well as Jacobi forms with several elliptic and modular variables then indexed by symmetric matrices or lattices, see for example [3], [49], [54], and the references therein. The probably most popular and best-known example of a Jacobi form is the Jacobi Theta function, II.. which has weight and index. Θv; τ := q ν ν +Z e πiνv+, Remark II... In the case of integral weight and index, the slash operators from Definition II.. i and ii define a group action of the Jacobi group Γ Z on the space of holomorphic functions C H C. Recall that for elliptic modular forms we could construct new examples of modular forms having possibly different level by extending the definition of the slash operator to SL Q. Naively trying this here unfortunately yields some trouble, since for λ, µ Q the elliptic transformation property ii in Definition II.. does not define a group action anymore. The proper extension of the slash operator for Jacobi forms is rather given by the next theorem see Theorem.4, []. Theorem II..3. The set G J := { γ, X, ζ γ SL R, X R, ζ C, ζ = } is a group via the multiplication law γ, X, ζγ, X, ζ = γγ, Xγ + X, ζζ e πi det Xγ X. Throughout this thesis a vector shall always be understood as row vector, if not stated otherwise

22 CHAPTER II. MODULAR FORMS AND GENERALIZATIONS 0 This group acts on the space of holomorphic functions φ : C H C via φ a c d b,λ, µ, ζz, τ = ζ m cτ + d k e πim «cz+λτ+µ +λ cτ+d τ+λz+λµ z φ cτ + d, aτ + b. cτ + d The above result now implies a handy theorem that we make use of later on. Theorem II..4 Theorem.3, []. Let φ be a Jacobi form on Γ of weight k N and index m N and let λ and µ be rational numbers. Then the function fτ := e πiλmτ φλτ + µ, τ is a modular form of weight k on the group { } a b Γ a λ + cµ, bλ + d µ, mcµ + d aλµ bλ Z. c d II.3 Harmonic Maaß-Forms and Mock Modular Forms The first one to study non-holomorphic modular forms systematically was H. Maaß. The notion of harmonic weak Maaß forms was first introduced by J.H. Bruinier and J. Funke in []. A survey of their work and its connection to number theory is given in [43] and also in [5]. For k Z let us first introduce the weight k hyperbolic Laplacian II.3. k := y x + y + iky x + i. y Definition II.3.. We call a smooth function f : H C a harmonic weak Maaß 3 form of weight k Z on Γ = Γ 0N with character χ if the following conditions are met.. f transforms like a modular form of weight k, i.e. f k γ = χdf for all γ = a b c d Γ.. f lies in the kernel of the hyperbolic Laplacian, i.e. k f f grows at most linearly exponentially at the cusps of Γ. The vector space of harmonic Maaß forms of weight k on the group Γ 0 N and character χ is denoted by H k N, χ resp. H k Γ for other groups Γ. From the Γ-equivariance it is plain that a harmonic Maaß form possesses a Fourier expansion, and from the fact that it is annihilated by k as well as the growth condition it is not hard to see the following. 3 in the literature the spelling Maass form is more common

23 CHAPTER II. MODULAR FORMS AND GENERALIZATIONS Lemma II.3. [43], Lemma 7.. Let f be a harmonic weak Maaß form of weight k. Then f has a canonical splitting into II.3. fτ = f + τ + 4πy k k c f 0 + f τ, where for some m 0, n 0 Z we have the Fourier expansions and f τ = f + τ = n=m 0 c + f nqn c f nnk Γ k; 4πnyq n. n=n 0 n 0 As usually we set q := e πiτ and II.3.3 Γα; x := t α e t dt, x > 0, denotes the incomplete Gamma function. This motivates the following definition. x Definition II.3.3. i The functions f + resp. 4πy k c k f 0+f τ in Lemma II.3. are referred to as the holomorphic resp. non-holomorphic part of the harmonic Maaß form f. ii The holomorphic part of a harmonic weak Maaß form of weight k is called a mock modular form of weight k. The non-holomorphic part of a harmonic Maaß form is now associated to a weakly holomorphic modular form. Proposition II.3.4 [], Proposition 3.. Define for k Z the operator ξ k := iy k τ. Then the mapping H k N, χ M! k N, χ, f ξ kf is well-defined and surjective with kernel M k! N, χ. With the notation from Lemma II.3. we have ξ k fτ = 4π k n=n 0 c f nqn.

24 CHAPTER II. MODULAR FORMS AND GENERALIZATIONS Definition II.3.5. Let f be a harmonic Maaß form of weight k on Γ. i We call the function ξ k f the shadow of f or the mock modular form f +. ii A mock modular form of weight, whose shadow is a linear combination of weight 3 theta functions as in II..3, is called a mock theta function. iii The preimage of M k Γ resp. S k Γ under ξ k is denoted by M k Γ resp. S k Γ. In many cases applications, mock modular forms do not occur on their own, but often combined with regular modular forms. This motivates the following definition. Definition II.3.6. Let f be a mock modular form of weight k and g be a holomorphic modular form of weight l. i The product f g is called a mixed mock modular form of weight k, l. ii More generally, the νth Rankin-Cohen bracket [f, g] ν of f and g is called a mixed mock modular form of weight k, l and degree ν. II.4 Appell-Lerch sums As already mentioned in the introduction, there were originally 7 examples of mock theta functions that Ramanujan introduced in his deathbed letter to Hardy. Further examples were discovered later for example in Ramanujan s Lost Notebook. The first consistent framework for all the mock theta functions of Ramanujan was given by S. Zwegers in his 00 Ph.D. thesis [55] written under the direction of D. Zagier. Actually, Zwegers found 3 different frameworks which the mock theta functions of Ramanujan fit into: Appell-Lerch sums, indefinite theta functions, and Fourier coefficients of meromorphic Jacobi forms. In this section, we shall give a brief account of the most important facts about Appell-Lerch sums as can be found in [55, 56] that we use in Chapter III. Definition II.4.. For τ H and u, v C \ Z Zτ we define the level l not to be confuse with the level of a modular form or group Appell-Lerch sum by the expression A l u, v = A l u, v; τ := e πilu n Z ln q l nn+ e πinv e πiu q n. This function is holomorphic for all u, v, τ where it is defined, but does not quite transform nicely under modular inversion, see e.g. Proposition.5 in [55] for the case of level. But by adding a certain non-holomorphic, but real-analytic

25 CHAPTER II. MODULAR FORMS AND GENERALIZATIONS 3 function which has the same modular defect, we can complete this sum to transform like a Jacobi form. This function is defined as II.4. II.4. Ru; τ := ν +Z t Et := { sgnν E ν + Im u } y ν q ν e πiνu, y e πu du = sgnt β t, II βx := u e πu du, x where for the second equality in II.4. we refer to [55, Lemma.7]. The function R itself satisfies several functional equations that we will need later. Proposition II.4. [55], Proposition.9. The function R fulfills the elliptic transformation properties i Ru + ; τ = Ru; τ ii Ru; τ + e πiu πiτ Ru + τ; τ = e πiu πi τ 4 iii R u = Ru. In the proof of Theorem III.., the following observation is vital. It has already been mentioned in [9] without proof, so we give one here. Proposition II.4.3. The function R lies in the kernel of the renormalized Heat operator D τ + Du, hence II.4.4 Du R = D τr. Proof. We set Z ν = ν / q ν e πiνu for ν + Z and abbreviate { Υ ν := sgnν E ν + Im u } y. y With this we get D τ Ru, τ = = ν +Z ν +Z [ πi e πν+im u y y [ Im u yπ y y Im u + ν + Im u iy y i y ν e πν+im u y y + ν Υ ν ]Z ν Υ ν ν ] Z ν

26 CHAPTER II. MODULAR FORMS AND GENERALIZATIONS 4 and II.4.5 D u Ru; τ = ν +Z = ν +Z [ u πi e πν+im y y y νυ ν ]Z ν i y [ e πν+imu y y νυ ν ]Z ν, yπ hence Du Ru; τ = πi ν +Z = ν +Z and the assertion is proven. [ e πν+im u y y yπ 4πy ν + Im u y ] +νe πν+imu y y y Z ν iy [ e πν+imu y y νυ ν ]νz ν yπ ν +Z [ Im u yπ y Theorem II.4.4 Theorem., [56]. Define ν e πν+imu y y + ν Υ ν ]Z ν iy II.4.6  l u, v; τ := A l u, v; τ + i l e πiku Θ k=0 v + kτ + l ; lτ R lu v kτ l ; lτ. with Θ as in II... Then the following equations hold true. S Âl u, v = Âu, v. E Âlu+λ τ+µ, v+λ τ+µ = lλ +µ e πiulλ λ e πivλ q l λ λ λ  l u, v for λ i, µ i Z. M Âl u, v ; aτ+b lu +uv = cτ + deπic cτ+d  cτ+d cτ+d cτ+d l u, v; τ for γ = a c d b SL Z. Another way to summarize E and M is to say that Âl transforms like a Jacobi form of weight and index l 0.

27 Chapter III Cohen s Conjecture In this chapter, we give a proof of Conjecture I.. following [40], but in greater detail. For this, we need some preparatory calculations which we give in Section III., the proof itself is subject of Section III.. III. Some Preliminaries III.. The Gamma function Since many of the calculations in this and the following chapters involve identities about the Gamma function, we recall them in this very short subsection. As a reference, see e.g. [5, Chapter IV.]. For Res > 0, the Gamma function is defined by the integral Γs := 0 e t t s dt. It has an meromorphic continuation to the entire complex plane with simple poles in s = n, n N 0, with residue n. It satisfies the following functional n! equations wherever all the expressions make sense, III.. III.. III..3 Γs + = sγs, π ΓsΓ s = sinπs, Γs = s s s + Γ Γ. π Equation III.. dates back to Euler and is sometimes called the reflection formula. It yields for example immediately the special value Γ = π. Equation III..3 is known as Legendre s duplication formula. 5

28 CHAPTER III. COHEN S CONJECTURE 6 III.. Preparatory Lemmas We first rewrite Cohen s conjecture I.. in a matter such that we see, which kinds of modular objects we are dealing with. Remark III... The coefficient of X ν in I.. is given by III..4 c ν [H, ϑ]ν τ [H, ϑ] ν τ + + Λ ν+,odd τ, π where c ν = ν! Γν+ and III..5 Λ l,odd τ := λ l n + q n+ n=0 with λ l as in I..3. The coefficient of X ν+ is identically 0. Proof. From [4, p. 83] we see that the first part of I.. equals πν!f odd ν ϑ, H X ν Γν +, ν!πiν ν=0 where for smooth functions f, g on H we define Fν odd f, gτ := Fν f, gτ F ν f, gτ + with F ν as in [4, Theorem 7.]. Using that in general for smooth functions f, g we have ν!πi ν [f, g] ν = F ν f, g we see that this equals c ν [H, ϑ]ν τ [H, ϑ] ν τ + X ν ν=0 which implies our claim. Thus each of the coefficients has to parts, one including the class number generating function and one including the function Λ l, which is sometimes called a mock Eisenstein series. Lemma III... For odd k N, the function Λ k,odd can be written as a linear combination of derivatives of Appell-Lerch sums, more precisely where we define Λ k,odd = Dk va odd 0, τ + ; τ, A odd u, v; τ :=e πiu n Z n odd n q nn+ e πinv e πiu q n = A u, v; τ A u, v + ; τ.

29 CHAPTER III. COHEN S CONJECTURE 7 Proof. First we remark that the right-hand side of the identity to be shown is actually well-defined because as a function of u, A u, v; τ has simple poles in Zτ + Z cf. [55, Proposition.4] which cancel out if the sum is only taken over odd integers. Thus the equation actually makes sense. Then we write Λ k,odd as a q-series Λ k,odd τ = minl +, m + k q l+m+ = = l=0 m=0 l + k q l+l++r + l=0 r= l + k q l+ l=0 l + k q l+ q l+ r + l=0 r= = l + k q l+ q + l+ l=0 = l + k q l+ ql+ ql+ l=0 = l + k q l+ + q l+ +l+ q l+ l=0 and compare this to D k v Aodd 0, τ +, τ = n Z n odd = n Z = n=0 n=0 n k qn +n q n n + kqn+ +n+ q n+ n + kqn+ +n+ q n+ + n= l + k q l+ l=0 l + k q l+ l=0 n + kq n+ + n+ q n+ +n+ n+ = n + kqn+ + n kqn+ q n+ q n+ n=0 n=0 +n+ + q n+ = n + kqn+ q n+ =Λ k,odd

30 CHAPTER III. COHEN S CONJECTURE 8 From this lemma we immediately see that the function c ν [Ĥ III..6, ϑ] ντ [Ĥ, ϑ] ντ + + Dv ν+ Â odd 0, τ + ; τ transforms like a modular form on some subgroup of Γ 0 4. Due to the polynomial growth of the Fourier coefficients it is plain that this function doesn t explode near the cusps. Thus in order to prove Conjecture I.. it is enough to make sure that the mentioned subgroup actually equals Γ 0 4 and to look at the non-holomorphic parts III..7 c ν [R, ϑ]ν τ [R, ϑ] ν τ + and III..8 i 4 ν+ l=0 l=0 ν + l l D l v R τ ; τdν l+ v Θτ + ; τ k+ k + l Dv l R τ ; τdν l+ v Θτ + ; τ l and show that these cancel each other. From this we see the necessity to investigate the derivatives of Θ and R evaluated at the torsion points ±τ + ; τ and ±τ + ; τ. First we observe the following: Remark III..3. Assume that Dv rθτ+; τ = f rτ and Du rr τ ; τ = g r τ for some functions f r, g r. From the definitions we immediately see that Dv r πi Θv; τ + = e 4 Dv rθv; τ and Dr πi uru; τ + = e 4 Du r Ru; τ. Then we see by shifting τ τ +, we get Dv r πi Θτ + ; τ = e 4 D r v Θ τ + + ; τ + = e πi 4 fr τ + and similarly Du r πi R τ ; τ = e 4 gr τ +. Hence it suffices to show only one of the identities in question. Lemma III..4. For r N 0 one has III..9 and Dv r Θ τ + ; τ r r = q 4 r s D τ s s ϑτ s=0 III..0 Dv r Θτ + ; τ = iq /4 r s=0 r r s Dτ s s ϑ τ +

31 CHAPTER III. COHEN S CONJECTURE 9 Proof. The proof is just a simple calculation: Obviously it holds that Dv r Θv; τ = ν r q ν e πiνv+. ν +Z Therefore D r v Θτ +, τ = ν r q ν+ν e πiν ν +Z = q 4 n + r q n+ n Z [ r r = q 4 n s ] r s q n s n Z s=0 r r = q 4 r s n s q n s s=0 n Z }{{} =0 for s odd r r = q 4 r s n s q n s s=0 r = q 4 s=0 r s n Z r s D s τϑτ By Remark III..3 one gets the result for plugging in τ +, τ. Lemma III..5. The following identities are true: III.. III.. III..3 III..4 R τ ; τ = iq 4 R τ ; τ = q 4 D u R τ ; τ = + i 4π q 4 D u R τ ; τ = + i 4π q 4 i τ τ i ϑz dz i z + τ 3 q 4 ϑ z + z + τ 3 dz + q 4. Proof. As above, we only prove III.. and III..3. Equation III.. follows easily from the transformation properties of R as given in Proposition II.4. the numbers above the equality signs give the used transformation: R τ ; τ ii = e πiτ+ πiτ e πiτ+ πiτ R τ ; τ iii = iq 4 + R τ + ; τ i = iq 4 R τ ; τ,

32 CHAPTER III. COHEN S CONJECTURE 30 which gives III... Let us now turn to III..3. We have by II.4.5 that e 4πny sgnn 4yπ D u R τ ; τ = iq 4 n Z n + β4n y q n, with β as in II.4.3 and sgn0 :=. By partial integration we get for real t 0 that βt = π t e πt π Γ III..5 ; πt with Γα; x as in II.3.3. We further observe the following τ H, n N: III..6 i τ From here we calculate D u R τ, τ =i e πinz dz = i iz + τ 3 ν +Z e πin x+it y y + t 3 = ie πinx+πny = iq n πn = iq n πnγ 4yπ e 4πν y { ν sgnν sgn =iq 4 n Z n + =iq 4 =iq 4 n Z n Z\{0} 4yπ e 4πn y dt e πny+t y 4πny y + t 3 e u u 3 du ; 4πny. ν β 4 ν } y q ν +ν { sgn n + 4yπ e 4πny sgnn i q 4 β0 + iq 4 }{{} 4yπ = } sgnn β4n y n + π sgnnnγ ; 4πn y q n β4n y q n q n

33 CHAPTER III. COHEN S CONJECTURE 3 =iq 4 n Z\{0} i q 4 + iq 4 4yπ π n τ i e πin z iz + τ 3/dz i πn = + i 4π q 4 i τ ϑz z + τ 3 i q 4, which is what we claimed. Remark III..6. From III..6 we can also deduce another representation of the function R: Rτ = 8π y + 4 π n= nγ ; 4πn y q n. With this, we can modify III..8 in the following way. Lemma III..7. For all ν N 0 it holds true that ν+ ν + µ µ µ=0 =q 4 and µ=0 ν ν λ λ=0 µ=0 ν+ ν + µ µ = iq 4 where D µ v R τ ; τ D ν µ+ v Θ τ + ; τ [ Dµ v R τ ; τ + ν ν λ [ λ=0 µ=0 b ν,µ,λ := b ν,µ,λ Dµ v b ν,µ,λ ν µ λ D λ τ ϑτ ν µ λ + µ + D v µ R τ ; τdν µ+ v Θτ + ; τ ν µ λ + R τ ; τ + µ + ν µ λ Dτ λ ϑτ +, Dv µ+ R τ ; τ] ] Dv µ+ R τ ; τ ν +! µ!λ!ν µ λ +! = ν +. µ, λ, ν µ λ +

34 CHAPTER III. COHEN S CONJECTURE 3 Proof. Again by Remark III..3 it is enough to prove the first claim. For simplicity, we omit the arguments of the functions in the calculation. We obtain ν+ ν + µ D v µ RDν µ+ v Θ = µ=0 ν ν + µ=0 µ µ D µ v RDν µ+ v Θ ν µ=0 ν + Dv µ+ RDv ν µ Θ µ + ν µ III..9 ν ν + ν µ + = q 4 ν µ λ+ Dv µ RDτ λ ϑ µ λ µ=0 λ=0 }{{} =b ν,µ,λ =q 4 ν µ ν ν + ν µ µ + λ }{{} = ν µ λ+ b µ+ ν,µ,λ µ=0 λ=0 ν µ ν µ=0 λ=0 [ Dµ v R + ν µ λ D µ+ ν µ λ + Dv µ+ R µ + Interchanging the sums gives the desired result. This yields the v RDτ λ ϑ ] b ν,µ,λ Corollary III..8. Up to the level, Conjecture I.. is true if the identity III..7 D λ τ Rτ = i 4 q 4 λ + holds true for all λ N 0. λ λ + µ=0 λ µ + µ + µ 4 ν µ λ D λ τ ϑ λ µ [ Dµ v R τ ; τ Dv µ+ R τ ; τ] Proof. Lemma III..7 gives us that Conjecture I.. holds true if the identity III..8 ν + c ν ν λ λ = i ν λ 4 q 4 µ=0 ν ν λ [ Dµ v R τ b ν,µ,λ ν µ λ Dτ ν λ Rτ ] ν µ λ + ; τ + Dv µ+ R τ µ + ; τ

35 CHAPTER III. COHEN S CONJECTURE 33 does as well. We can simplify this a little further: We have ν + c ν π ν = ν! λ ν λ Γ Γ ν + 3 ν + Γ Γ ν + ν λ + 3 λ! Γ λ + ν λ! ν πγ ν + 3 = λ Γ ν λ + 3 Γ λ + By Legendre s duplication formula III..3 we see that λ! = Γλ + = λ Γ λ + λ!, π Γ ν λ + 3 = π ν λ ν λ +!, ν λ +! Γ ν 3 = π ν ν +! ν +!. Thus we get that = ν µ λ b ν,µ,λ c ν+ ν ν λ λ!ν λ! ν! III..3ν λ! = ν! III..3 = ν λ! ν! ν λ Γν λ + 3Γ λ + ν +! πγ ν + 3 µ!λ!ν µ λ +! Γ ν + 3 ν µ Γ ν +! ν λ + 3 µ!ν µ λ +! 4 π ν λ ν λ+! ν µ ν λ+! ν +! π ν ν+! µ!ν µ λ +! 4 ν+! ν λ +! = ν λ + µ!ν µ λ +! ν µ λ ν λ + =. µ 4 and hence the corollary. 4 ν µ λ ν µ λ 4 Before we conclude this section, we take care that the completed coefficient in III..6 indeed transforms like a modular form on Γ 0 4. Remark III..9. Note that we already know by Proposition II.. the function c ν [Ĥ, ϑ] ντ [Ĥ, ϑ] ν τ + from III..6 transforms like a modular form of weight ν + on Γ 0 4.

36 CHAPTER III. COHEN S CONJECTURE 34 Lemma III..0. The function Dv ν+  odd 0, τ + ; τ transforms like a modular form of weight ν + on Γ 0 4. Proof. It is plain from Theorem II.4.4 that the ν +st derivative of Â0, v; τ with respect to v has the modular transformation properties of a Jacobi form of weight ν + and index 0. By Theorem II..4, we therefore see that A ν τ := D ν+ v  0, τ+; τ transforms like a modular form of that weight on the group { Γ A := γ SL Z a + c Z and b + d } Z. The function that we are interested in is Dv ν+  0, τ + ; τ = l+ A ν ν Since one easily sees that 0 Γ Γ 0 A, we get that Dv ν+  0, τ + ; τ transforms nicely under Γ 0 4 and therefore, by Remark III..9, so does Dv ν+  odd 0, τ + ; τ III. The Proof Our proof for Conjecture I.. makes use of Corollary III..8. We show by induction that the identity stated there does indeed hold. The base case of the induction gives a new proof of Eichler s class number relation I..5, so we give this as proof of a separate theorem. Theorem III.. M. Eichler, 955. For odd numbers n N we have the class number relation Proof. Let Hn s + λ n = 3 σ n. s Z E odd τ = σ n + q n+, which is known to be a modular form of weight on Γ 0 4. Plugging in λ = 0 into III..7 gives us the equation Rτ = i [ 4 q 4 R τ ; τ + D v R ] τ ; τ. n=0

37 CHAPTER III. COHEN S CONJECTURE 35 This equality holds by Lemma III..5. Hence we know by Corollary III..8, Lemma III.., Remark III..9 and Lemma III..0 that H τϑτ H τ + ϑ τ + + Λ,odd τ is indeed a holomorphic modular form of weight on Γ 0 4 as well. By Theorem II..6 and Proposition II..7, a comparison of the first non-zero Fourier coefficient yields the result. The methods employed in [8] to prove this, are very different from these here. Eichler uses topological arguments about the action of Hecke operators on the Riemann surface associated to Γ 0 and counting of ideal classes in maximal orders of quaternion algebras. Now we can prove the main result of this chapter. Theorem III... Cohen s conjecture I.. is true. Moreover, for ν > 0 the coefficient of X ν in S4 τ; X see I.. is a cusp form. Proof. The base case of the induction in Theorem III... Thus suppose that III..7 holds for one λ N 0. Again omitting the arguments of the occurring R-derivatives for the sake of clearence of presentation, the induction hypothesis gives us Rτ = D τ Dτ λ Rτ { = i 4 q 4 λ λ 4 D λ+ τ + λ µ=0 µ=0 [ D τd µ v R + [ Dµ v R + ] λ µ + Dv µ+ R µ + ] λ µ + D τ Dv µ+ R µ + λ + µ 4 } λ µ λ + µ 4 λ µ Here we omitted again the argument τ ; τ of D v R again for the sake of clearence of presentation. The theorem of Schwarz now tells us that the partial derivatives interchange and therefore we have D τ D l u R τ ; τ = D l+ u R τ ; τ + D τ D l ur τ ; τ. According to Proposition II.4.3 this equals { λ i [ 4 q 4 λ+ 8 Dµ v R + 4 µ=0 λ µ µ + D µ+ v R + λ µ + µ + + D µ+ v R λ µ + Dv µ+3 R µ + ]

38 CHAPTER III. COHEN S CONJECTURE 36 } λ µ λ + µ 4 { [ = i ] λ+ 4 q 4 λ+ λ λ + R + λ + 3D vr + 4 µ µ= [ λ µ + 5 µ µ + Dv µ R 4µ λ µ + λ µ + 3 λ + µ + 3 λ µ + 3 µ µ + + µ + µ λ µ + λ µ + 3 [ λ λ + Dλ+ v R + ] } λ + λ + Dλ+3 v R. λ The last summand of the last equation simplifies to [ Dλ+ v R + ] λ + 3 Dλ+3 v R λ For the rest we see by elementary computations that λ + µ + λ µ + 5µ λ µ + λ µ + 3 = λ + 3 λ µ + λ µ + 3 µλ µ µ λ + λ + 3 λ + λ + 3 λ + 3 4λ + 0λ + 6 λ µ+ ] Dv + R = µ λ + 3 λ + λ + 3 = µ and in the same way that λ + λ + µ µ µ + λ µ + 3 λ + 3 =. µ + µ Thus in summary, we have that Dτ λ+ Rτ = i λ+ 4 q 4 λ+ + µ=0 λ µ + 3 µ + which proves Conjecture I... λ µ + 3 µ λ + 3 µ µ µ λ µ + λ µ + 3 λ µ+ [ 4 Dµ v R τ ; τ Dv µ+ R τ ; τ]

39 CHAPTER III. COHEN S CONJECTURE 37 The additional assertion that the modular forms we get from S4 are cusp forms except when ν = 0 can be seen by the fact that for smooth functions f, g : H C and real numbers k, l we have [f k γ, g l γ] ν = [f, g] ν k+l+ν for all γ SL R cf. [4, Theorem 7.]. Now for ν > 0 the function τ [H, ϑ] ν τ vanishes at the cusp i by construction and the intertwining property from above then yields that it has to vanish at all cusps. A similar intertwining property also holds for derivatives of index 0 Jacobi forms which is essentially shown in Lemma III..0. Since Dv ν+ A 0, τ + ; τ also vanishes at the cusp i if ν > 0, it has to vanish at all cusps, which finally proves the theorem. As an easy consequence from this theorem, we get some new class number relations. The first few are worked out below. Corollary III..3. By comparing the first few Fourier coefficients of the modular forms in Theorem III.. one finds for all odd n N the following class number relations 4s n H n s + λ 3 n = 0, s Z g 4 s, nh n s + λ 5 n = Y 4 x, y, z, t, s Z n=x +y +z +t g 6 s, nh n s + λ 7 n = Y 6 x, y, z, t, 3 s Z n=x +y +z +t g 8 s, nh n s + λ 9 n = Y 8 x, y, z, t 70 s Z n=x +y +z +t where g l n, s is the l-th Taylor coefficient of sx + nx and Y d x, y, z, t is a certain harmonic polynomial of degree d in 4 variables. Explicitly, we have and g 4 s, n = 6s 4 ns + n, g 6 s, n = 64s 6 80s 4 n + 4s n n 3, g 8 s, n = 56s 8 448s 6 n + 40s 4 n 40s n 3 + n 4, Y 4 x, y, z, t = x 4 6x y + y 4, Y 6 x, y, z, t = x 6 5x 4 y 0x 4 z + 30x y z + 5x z 4 5y z 4, Y 8 x, y, z, t = 3x x 6 y 490x 6 z + 63x 6 t 630x 4 y z 35x 4 y t +435x 4 z 4 630x 4 z t + 35x y z x y z t 66x z 6 +35x z 4 t 35t y z 4 + z 8. The first two of the above relations were already mentioned in [4].