Congruence conditions for the mod $\lambda$ values of the Fourier coefficients of classical eigenforms

TL;DR

This paper classifies when Fourier coefficients of classical eigenforms satisfy specific congruences modulo primes, linking these to Galois representations and extending prior work, with applications to elliptic curves.

Contribution

It provides a comprehensive classification of congruence conditions for Fourier coefficients in terms of Galois representation images, extending Swinnerton-Dyer's work.

Findings

Certain congruences are more often implied by prime p congruences when a_p(f) ≡ 0 mod λ.

The classification connects Fourier coefficient congruences to the image of mod λ Galois representations.

Examples from weight 2 newforms illustrate the theoretical results.

Abstract

We classify all instances of the condition $a_{p}(f) \equiv x \bmod \lambda$ being related to a congruence on the prime $p$, where $a_{p}(f)$ denotes the $p$th Fourier coefficient of a classical normalised cuspidal eigenform $f$ and $\lambda$ is a prime in the number field generated by the Fourier coefficients of $f$. This classification is done in terms of the (projective) image of the mod $\lambda$ Galois representation associated with $f$ and extends work by Swinnerton-Dyer. We highlight that for $x = 0$, this condition is more often implied by a congruence on the prime $p$ than the general value of $a_{p}(f) \bmod \lambda$. Finally, we illustrate various instances of these congruences through examples from the setting of weight 2 newforms attached to rational elliptic curves.