On Constant Terms of Eisenstein Series

TL;DR

This paper derives formulas for the constant terms of Hilbert modular Eisenstein series at all cusps, linking them to special values of Hecke L-series, with applications to the Brumer-Stark conjecture and units.

Contribution

It extends previous work by providing explicit formulas for constant terms of Eisenstein series at all cusps, applicable to broader classes and arithmetic problems.

Findings

Explicit formulas for constant terms at all cusps

Connection to special values of Hecke L-series

Applications to Brumer-Stark conjecture and units

Abstract

We calculate the constant terms of certain Hilbert modular Eisenstein series at all cusps. Our formula relates these constant terms to special values of Hecke $L$-series. This builds on previous work of Ozawa, in which a restricted class of Eisenstein series were studied. Our results have direct arithmetic applications---in separate work we apply these formulas to prove the Brumer-Stark conjecture away from $p=2$ and to give an exact analytic formula for Brumer-Stark units.