Elliptic curves and Fourier coefficients of meromorphic modular forms

Abstract: We discuss several congruences satisfied by the coefficients of meromorphic modular forms, or equivalently, $p$-adic behaviors of meromorphic modular forms under the $U_p$ operator, that are summarized from numerical experiments, connecting meromorphic modular forms to symmetric powers of elliptic curves. We also provide heuristic explanations for these congruences as well as prove some of them using hypergeometric functions and the Borcherds--Shimura lift.