Atkin and Swinnerton-Dyer congruences for meromorphic modular forms

TL;DR

This paper extends Atkin and Swinnerton-Dyer congruences to meromorphic modular forms, revealing new p-adic recurrence relations and exploring their implications, especially at CM points, with explicit examples and partial proofs.

Contribution

It generalizes ASD congruences to meromorphic forms and connects them with Scholl's congruences and fibers of elliptic curves, including explicit CM point analysis.

Findings

ASD congruences extend to meromorphic modular forms.

p-adic recurrence relations relate to Scholl's congruences and elliptic curve fibers.

At CM points, relations simplify to 2-term recurrences and support conjectures about meromorphic modular forms.

Abstract

In the 1970's, Atkin and Swinnerton-Dyer conjectured that Fourier coefficients of holomorphic modular cusp forms on noncongruence subgroups of $\text{SL}_2(\mathbb{Z})$ satisfy certain $p$-adic recurrence relations which are analogous to Hecke's recurrence relations for congrunece subgroups. In 1985, this was proven in seminal work of Scholl and it was recently extended to weakly holomorphic modular forms by Kazalicki and Scholl. We show that Atkin and Swinnerton-Dyer type congruences extend to the setting of meromorphic modular forms and that the $p$-adic recurrence relations arise from Scholl's congruences in addition to a contribution of fibers of universal elliptic curves at the poles. Moreover, when the poles are located at CM points, we exploit the CM structure to reduce these $p$-adic recurrence relations to $2$-term relations and we give explicit examples. Using this framework, we partially prove conjectures that certain meromorphic modular forms are magnetic.