Congruences between Hilbert modular forms of weight $2$, and special values of their $L$-functions

TL;DR

This paper explores how congruences between Hilbert cusp forms and Eisenstein series of weight 2 lead to congruences between algebraic parts of their L-function values, extending Vatsal's earlier results.

Contribution

It generalizes Vatsal's result by establishing a link between Fourier coefficient congruences and special L-value congruences for Hilbert modular forms.

Findings

Established congruences between algebraic parts of L-values

Extended Vatsal's results to Hilbert modular forms of weight 2

Linked Fourier coefficient congruences to L-value congruences

Abstract

The purpose of this paper is to show how a congruence between (the Fourier coefficients of) a Hilbert cusp form and a Hilbert Eisenstein series of parallel weight $2$ gives rise to congruences between algebraic parts of critical values of their $L$-functions. This is a generalization of a result of V. Vatsal.