Mock modular forms and class number relations

Abstract

Purpose: Almost 40 years ago, H. Cohen formulated a conjecture about the modularity of a certain infinite family of functions involving the generating function of the Hurwitz class numbers of binary quadratic forms. Methods: We use techniques from the theory of modular, mock modular, and Jacobi forms. Result: In this paper, we prove a slight improvement of Cohen's original conjecture. Conclusions: From our main result, we derive so far unknown recurrence relations for Hurwitz class numbers.

Poincaré square series of small weight

“…Remark 19. Mertens [11] has given other proofs of this and similar class number relations using mock modular forms. It seems likely that we can recover other class number relations (possibly some of the other relations of [11]) by studying the higher development coefficients (as defined in chapter 3 of [7]) of the realanalytic Jacobi Eisenstein series E * 2,1,0 (τ, z, s) in the same way that we have studied its zeroth development coefficient E * 2,1,0 (τ, 0, s) in this note, but we will not pursue that here.…”

Eichler–Selberg type identities for mixed mock modular forms

“…This function was historically the first fully understood example of a mock modular form, without the terminology having been introduced at the time [19]. In [27], the author used the above result as well as the properties of so called Appell-Lerch sums first extensively studied in this context by S. Zwegers in [35,36] to give a mock modular proof of an infinite family of class number relations for odd numbers n: and σ k (n) is the usual kth power divisor sum. The relation in (1.1) was first found by M. Eichler in [14], the one in (1.2) and infinitely many more had been conjectured by H. Cohen in [12].…”

“…The special results for the cases that we are interested in are worked out in Sections 5 and 6. As applications to our main result, we reprove the class number relations from [12,27], from the Eichler-Selberg trace formula, and generalizations of the ones from [6] in Section 7 in a more natural way than the method in [6,27].…”

On the trace formula for Hecke operators on congruence subgroups, II

“…While the explicit cusp form above seems not to have appeared in print before, the explicit generating series we get for odd coefficients was conjectured to be a cusp form by H. Cohen more than 30 years ago [2]. Although Cohen suggested that this modular form was related to the "trace form," whose coefficients are the traces of Hecke operators, the conjecture was only recently proved by different methods by Mertens [7]. In Corollary 1 we also compute explicitly the trace form for S k ( 0 (4), χ) if k is odd and χ is the nontrivial character modulo 4.…”