Hilbert modular Eisenstein congruences of local origin

Fretwell, Dan and Roberts, Jenny (2026) Hilbert modular Eisenstein congruences of local origin. Journal of Number Theory, 280. pp. 861-896. ISSN 0022-314X

Abstract

Let F be an arbitrary totally real field. Under standard conditions we prove the existence of certain Eisenstein congruences between parallel weight k Hilbert eigenforms of level MP and Hilbert Eisenstein series of level M, for arbitrary ideal and prime ideal P not dividing 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.