On congruences mod ${\mathfrak p}^m$ between eigenforms and their attached Galois representations

TL;DR

This paper investigates congruences mod prime powers between eigenforms and their associated Galois representations, especially when forms have different weights, providing necessary and sufficient conditions for such congruences.

Contribution

It extends the understanding of congruences between eigenforms mod prime powers, especially in the challenging case of differing weights, and links these to Galois representations.

Findings

Established necessary and sufficient conditions for eigenform congruences mod ${rak p}^m$ with different weights.

Generalized Sturm's theorem to handle congruences in the equal weight case.

Clarified the connection between eigenform congruences and modular Galois representations.

Abstract

Given a prime $p$ and cusp forms $f_1$ and $f_2$ on some $\Gamma_1(N)$ that are eigenforms outside $Np$ and have coefficients in the ring of integers of some number field $K$, we consider the problem of deciding whether $f_1$ and $f_2$ have the same eigenvalues mod ${\mathfrak p}^m$ (where ${\mathfrak p}$ is a fixed prime of $K$ over $p$) for Hecke operators $T_{\ell}$ at all primes $\ell\nmid Np$. When the weights of the forms are equal the problem is easily solved via an easy generalization of a theorem of Sturm. Thus, the main challenge in the analysis is the case where the forms have different weights. Here, we prove a number of necessary and sufficient conditions for the existence of congruences mod ${\mathfrak p}^m$ in the above sense. The prime motivation for this study is the connection to modular mod ${\mathfrak p}^m$ Galois representations, and we also explain this connection.