Hilbert modular Eisenstein congruences of local origin

TL;DR

This paper establishes the existence of Eisenstein congruences between Hilbert modular forms and Eisenstein series over totally real fields, linking these congruences to special values of Hecke L-functions and Euler factors.

Contribution

It proves the existence of Eisenstein congruences of local origin for Hilbert modular forms over arbitrary totally real fields, including cases with non-trivial Hecke characters.

Findings

Congruences relate to special values of Hecke L-functions.

Results include conditions for congruences to be satisfied by newforms.

General theorem on when such congruences can be realized by newforms.

Abstract

Let $F$ be an arbitrary totally real field. Under weak conditions we prove the existence of certain Eisenstein congruences between parallel weight $k \geq 3$ Hilbert eigenforms of level $\mathfrak{mp}$ and Hilbert Eisenstein series of level $\mathfrak{m}$, for arbitrary ideal $\mathfrak{m}$ and prime ideal $\mathfrak{p}\nmid \mathfrak{m}$ of $\mathcal{O}_F$. Such congruences have their moduli coming from special values of Hecke $L$-functions and their Euler factors, and our results allow for the eigenforms to have non-trivial Hecke character. After this, we consider the question of when such congruences can be satisfied by newforms, proving a general result about this.