Hilbert modular Eisenstein congruences of local origin

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.

💡 Research Summary

The paper investigates Eisenstein congruences of “local origin’’ for Hilbert modular forms over an arbitrary totally real field (F). After recalling the classical Ramanujan congruence (\tau(n)\equiv\sigma_{11}(n)\pmod{691}) and its interpretation as a congruence between a cusp form and an Eisenstein series, the authors set out to generalise this phenomenon to parallel‑weight Hilbert eigenforms of weight (k\ge3) and level (\mathfrak{mp}), where (\mathfrak p) is a prime ideal not dividing the auxiliary ideal (\mathfrak m).

The first technical achievement is an explicit formula for the constant term of Hilbert Eisenstein series attached to a pair of narrow ray‑class characters (\eta,\psi). Using a generalisation of Ozawa’s method, they show that for the Eisenstein series (E_k(\eta,\psi)) the constant term at a cusp indexed by a narrow class (\lambda) is \