A $p$-adic cohomological approach to congruences of meromorphic modular forms

TL;DR

This paper introduces a $p$-adic cohomological framework to understand congruences between Fourier coefficients of meromorphic modular forms and Frobenius eigenvalues of elliptic curves, unifying their study via rigid cohomology and crystalline structures.

Contribution

It develops a novel $p$-adic cohomological approach that links modular form congruences with elliptic curve Frobenius actions through comparison theorems and cohomological sequences.

Findings

Established a cohomological interpretation of modular form congruences.

Connected Frobenius eigenvalues with $U_p$-operator actions on overconvergent forms.

Unified treatment for modular and Shimura curves with smooth integral models.

Abstract

We study congruences relating Fourier coefficients of meromorphic modular forms and Frobenius eigenvalues of elliptic curves corresponding to their poles. We develop a $p$-adic cohomological framework that interprets these congruences via the interaction between the rigid cohomology of modular curves and the crystalline structure of the associated elliptic curves. Using comparison theorems and the Gysin sequence, we relate the Frobenius actions in cohomology to the $U_p$-operator acting on spaces of overconvergent modular forms. Our approach applies uniformly to both modular curves and Shimura curves admitting smooth integral models over $\mathbb{Z}_p$.