A note on the spaces of Eisenstein series on general congruence subgroups

TL;DR

This paper introduces a new approach to studying spectral Eisenstein series on congruence subgroups, linking their properties to principal congruence subgroups and providing explicit bases with algebraic Fourier coefficients.

Contribution

It develops a novel method using Hecke's theory to analyze spectral Eisenstein series, constructing explicit bases and connecting to classical theorems like Eichler-Shimura.

Findings

Spectral Eisenstein series at s=0 form a basis for Eisenstein series space.

Fourier coefficients of basis elements are in cyclotomic fields.

Provides a simple proof of the Eichler-Shimura isomorphism.

Abstract

This article proposes a new approach to studying the spectral Eisenstein series of weight $k$ on a congruence subgroup of $\text{SL}_2(\mathbb{Z})$ using Hecke's theory of Eisenstein series for the principal congruence subgroups. Our method provides a gateway to analytic and arithmetic properties of the spectral Eisenstein series using corresponding results for the principal congruence subgroup. We show that the specializations of the weight $k$ spectral Eisenstein series at $s = 0$ give rise to a basis for the space of Eisenstein series on a general congruence subgroup, and the Fourier coefficients of the basis elements lie in a cyclotomic number field. Our philosophy also yields an explicit basis parameterized by cusps for the space of Eisenstein series with a nebentypus character. We utilize the spectral basis for the space of Eisenstein series to provide a simple proof of the Eichler-Shimura isomorphism theorem for the entire space of modular forms.