Weierstrass mock modular forms and elliptic curves

Abstract

Mock modular forms, which give the theoretical framework for Ramanujan's enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves E/Q. We show that mock modular forms which arise from Weierstrass ζ -functions encode the central L-values and L-derivatives which occur in the Birch and Swinnerton-Dyer Conjecture. By defining a theta lift using a kernel recently studied by Hövel, we obtain canonical weight 1/2 harmonic Maass forms whose Fourier coefficients encode the vanishing of these values for the quadratic twists of E. We employ results of Bruinier and the third author, which builds on seminal work of Gross, Kohnen, Shimura, Waldspurger, and Zagier. We also obtain p-adic formulas for the corresponding weight 2 newform using the action of the Hecke algebra on the Weierstrass mock modular form.

Algebraic formulas for the coefficients of mock theta functions and Weyl vectors of Borcherds products

“…Further, in [13], Ono and the first author used a variant of the Kudla-Millson theta lift to find a finite algebraic formula for the partition function p(n) in terms of traces of CM-values of a certain non-holomorphic modular function. Recently, a similar theta lift was used in [4] to prove a refinement of a theorem of [12] connecting the vanishing of the central derivative of the twisted L-function of an even weight newform and the rationality of some coefficient of the holomorphic part of a half-integral weight harmonic Maass form. This so-called Millson theta lift, which maps weight 0 to weight 1/2 harmonic Maass forms, was studied in great detail by Alfes-Neumann in her thesis [3], and by AlfesNeumann and the second author in [5].…”

A proof of the Thompson moonshine conjecture

“…Note, however, that the Moonshine phenomenon we prove in this paper is not directly related to this generalized moonshine considered by Queen, but more reminiscent of the following. In 2011, Eguchi et al [19] observed connections like the ones between the dimensions of irreducible representations of the Monster group and coefficients of the modular function J for the largest Mathieu group M 24 and a certain weight 1 2 mock theta function. Cheng et al [12,13] generalized this to Moonshine for groups associated with the 23 Niemeier lattices, the non-isometric even unimodular root lattices in dimension 24, which has become known as the Umbral Moonshine Conjecture.…”

“…Definition 2.1 A vector space V is called a superspace, if it is equipped with a Z/2Z-grading V = V (0) ⊕ V (1) , where V (0) is called the even and V (1) is called the odd part of V . For an endomorphism α of V respecting this grading, i.e., α(V (i) ) ⊆ V (i) , we define its supertrace to be strace(α) = trace α| V (0) − trace α| V (1) . Now let G be a finite group and (V, ρ) a representation of G. If the G-module V admits a decomposition into an even and odd part as above which is compatible with the G-action, we call V a G-supermodule.…”

“…It took until the first decade of the twenty-first century before work by Zwegers [43], Bruinier and Funke [7] and Bringmann and Ono [5,6] established the "right" framework for these enigmatic functions of Ramanujan's, namely that of harmonic Maaß forms. Since then, there have been many applications of harmonic Maaß forms both in various fields of pure mathematics, see for instance [1,4,9,16], among many others, and mathematical physics, especially in regard to quantum black holes and wall crossing [15] as well as Mathieu and Umbral Moonshine [12,13,18,23]. For a general overview on the subject, we refer the reader to [31,42].…”

“…Author details 1 Department of Mathematics, Princeton University, Fine Hall, Washington Rd., Princeton, NJ 08544, USA, 2 Department of Mathematics and Computer Science, Emory University, 400 Dowman Drive, Atlanta, GA 30322, USA.…”