Weierstrass mock modular forms and elliptic curves

TL;DR

This paper explores how Weierstrass mock modular forms relate to elliptic curves, revealing their role in encoding central L-values and derivatives, and providing new p-adic formulas for associated modular forms.

Contribution

It introduces a novel connection between Weierstrass mock modular forms and elliptic curve L-values, including a theta lift construction and p-adic formulas for modular forms.

Findings

Mock modular forms encode central L-values and derivatives.

A theta lift produces harmonic Maass forms with Fourier coefficients related to L-value vanishing.

p-adic formulas for weight 2 newforms are derived using Hecke algebra actions.

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 $E/\mathbb{Q}$. We show that mock modular forms which arise from Weierstrass $\zeta$-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 H\"ovel, 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.