Constant terms of Eisenstein series over a totally real field

TL;DR

This paper generalizes the computation of constant terms of Eisenstein series to totally real fields, providing explicit formulas at all cusps in terms of Hecke L-functions, extending prior work over the rationals.

Contribution

It computes constant terms of Eisenstein series over totally real fields at all cusps, generalizing previous results limited to rational numbers and specific cusps.

Findings

Explicit formulas for constant terms at all cusps.

Connection to Hecke L-functions.

Extension of prior rational field results.

Abstract

In this paper, we compute constant terms of Eisenstein series defined over a totally real field, at various cusps. In his paper published in 2003, M. Ohta computed the constant terms of Eisenstein series of weight two over the field of rational numbers, at all equivalence classes of cusps. As for Eisenstein series defined over a totally real field, S. Dasgupta, H. Darmon and R. Pollack calculated the constant terms at particular (not all) equivalence classes of cusps in 2011. We compute constant terms of Eisenstein series defined over a general totally real field at all equivalence classes of cusps, and describe them explicitly in terms of Hecke $L$-functions. This investigation is motivated by M. Ohta's work on congruence modules related to Eisenstein series defined over the field of rational numbers.