Elliptic curves and Fourier coefficients of meromorphic modular forms

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.