Elliptic curves and Fourier coefficients of meromorphic modular forms

TL;DR

This paper explores congruences and p-adic properties of coefficients of meromorphic modular forms, revealing connections to elliptic curves and Chowla--Selberg periods, supported by numerical experiments and theoretical proofs.

Contribution

It introduces new congruences for meromorphic modular form coefficients and links them to elliptic curves and p-adic periods, with proofs using hypergeometric functions and Borcherds--Shimura lift.

Findings

Congruences relate to symmetric powers of elliptic curves.

Connections to p-adic Chowla--Selberg periods in CM case.

Some congruences are proven using hypergeometric functions.

Abstract

We discuss several congruences satisfied by the coefficients of meromorphic modular forms, or equivalently, the $p$-adic behaviors of meromorphic modular forms under the $U_p$ operator, that are summarized from numerical experiments. In the generic case, we observe the connection to symmetric powers of elliptic curves, while in the CM case, we furthermore observe the connection to the $p$-adic analogue of the Chowla--Selberg periods. Along with the discussions, we will provide some heuristic explanations for these congruences as well as prove some of them using hypergeometric functions and the Borcherds--Shimura lift.