Hilbert modular Eisenstein congruences of local origin

Abstract

Let F be an arbitrary totally real field. Under standard conditions we prove the existence of certain Eisenstein congruences between parallel weight $k\ge 3$ Hilbert eigenforms of level $\mathrm{mp}$ and Hilbert Eisenstein series of level $m$, for arbitrary ideal $m$ and prime ideal $p\nmid m$ of $O_F$. 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 general results about this.