Equivariant primitives of Eisenstein series for congruence subgroups

TL;DR

This paper investigates the structure of equivariant primitives of Eisenstein series for congruence subgroups, establishing their equivalence to non-holomorphic Eisenstein series and providing explicit formulas, especially for genus zero cases.

Contribution

It characterizes equivariant primitives of Eisenstein series for congruence subgroups and derives explicit formulas, extending known results for the full modular group.

Findings

Equivariant primitives are precisely the non-holomorphic Eisenstein series.

Closed formulas generalize existing results for the full modular group.

In genus zero cases, Eisenstein series of weight two are expressed as single-valued logarithms.

Abstract

We study equivariant primitives of Eisenstein series for principal congruence subgroups and show that they are precisely the corresponding non-holomorphic Eisenstein series. We present closed formulas that naturally generalise existing results for the full modular group. We also focus on Eisenstein series of weight two in the case where the modular curve has genus zero. We show that in those cases the non-holomorphic Eisenstein series of weight two can be written as single-valued logarithms whose argument is a rational function of the Hauptmodul.