Rationality of the periods of Eisenstein series

TL;DR

This paper proves that the spectral Eisenstein series on congruence subgroups have images defined over cyclotomic fields, generalizes invariants related to quadratic fields, and offers an elementary derivation of Fourier expansions.

Contribution

It extends previous results to broader classes of Eisenstein series and provides new insights into their algebraic and analytical properties.

Findings

Spectral Eisenstein series images are over cyclotomic fields.

Generalization of invariants linked to imaginary quadratic fields.

Elementary derivation of Fourier expansion of Maass Eisenstein series.

Abstract

The article generalizes an observation of Zagier and Gangl to show that the image of the spectral Eisenstein series on a general congruence subgroup of $\text{SL}_2(\mathbb{Z})$, under the Eichler-Shimura isomorphism, is defined over a cyclotomic number field. We use the same technique to generalize an invariant attached to imaginary quadratic fields in connection with the polylogarithm conjecture on the special values of $L$-functions. Our treatment also provides an elementary derivation of the Fourier expansion of the Maass Eisenstein series on congruence subgroups presented as a power series.