Equivariant primitives of Eisenstein series for congruence subgroups

TL;DR

This work extends Brown's theory of equivariant primitives from the full modular group to principal congruence subgroups $\Gamma(N)$, showing that equivariant primitives of Eisenstein series are exactly the corresponding non-holomorphic Eisenstein series and providing closed formulas. It develops two spanning sets of Eisenstein series, analyzes their $L$-series via Clausen values and Bernoulli polynomials, and computes Eisenstein cocycles through the Eichler–Shimura framework. The main result is a complete description of equivariant primitives for $\Gamma(N)$, along with explicit expressions for weight-two cases on genus-zero curves, where primitives reduce to single-valued logarithms of the Hauptmodul. These findings illuminate the interplay between modular geometry, non-holomorphic Eisenstein series, and real-analytic equivariant structures, with potential extensions to higher-length iterated integrals and higher-genus modular curves.

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.

Theorems & Definitions (20)

• Definition 1
• Definition 2
• Proposition 1
• proof
• Theorem 1
• proof
• Theorem 2: Eichler-Shimura isomorphism
• Theorem 3
• proof
• Theorem 4