On congruence modules related to Hilbert Eisenstein series

TL;DR

This paper extends the study of congruence modules from elliptic to Hilbert Eisenstein series, involving construction, control theorems, and explicit computations in the Hilbert modular forms setting.

Contribution

It generalizes Ohta's work to Hilbert modular forms, providing new constructions, control theorems, and computations of congruence modules in this broader context.

Findings

Constructed adelic Hilbert Eisenstein series with explicit constant terms

Proved a control theorem for ordinary $\Lambda$-adic Hilbert modular forms

Computed congruence modules related to Hilbert Eisenstein series

Abstract

We generalize the work of Ohta on the congruence modules attached to elliptic Eisenstein series to the setting of Hilbert modular forms. Our work involves three parts. In the first part, we construct Eisenstein series adelically and compute their constant terms by computing local integrals. In the second part, we prove a control theorem for one-variable ordinary $\Lambda$-adic Hilbert modular forms following Hida's work on the space of multivariable ordinary $\Lambda$-adic Hilbert cusp forms. In part three, we compute congruence modules related to Hilbert Eisenstein series through an analog of Ohta's methods.