The Jacquet-Langlands correspondence, Eisenstein congruences, and integral L-values in weight 2

TL;DR

This paper extends classical congruence results relating Fourier coefficients and L-values of weight 2 elliptic modular forms to more general settings using the Jacquet-Langlands correspondence, including nonsquare levels and Hilbert modular forms.

Contribution

It generalizes Mazur's congruence results from prime level elliptic modular forms to nonsquare levels and Hilbert modular forms using the Jacquet-Langlands correspondence.

Findings

Established congruences for Fourier coefficients and L-values in broader settings.

Extended classical results to Hilbert modular forms.

Demonstrated the applicability of Jacquet-Langlands in new contexts.

Abstract

We use the Jacquet-Langlands correspondence to generalize well-known congruence results of Mazur on Fourier coefficients and L-values of elliptic modular forms for prime level in weight 2 both to nonsquare level and to Hilbert modular forms.