Eichler integrals and generalized second order Eisenstein series

TL;DR

This paper demonstrates that all Eichler integrals and generalized second order modular forms can be expressed as linear combinations of Eisenstein series, providing explicit Fourier expansions and a method for numerical evaluation.

Contribution

It establishes a linear combination representation of Eichler integrals using generalized Eisenstein series and develops a bootstrap method for numerical evaluation.

Findings

Fourier series expansions of generalized second order Eisenstein series are determined.

Tail estimates for these series are obtained via convexity bounds.

A bootstrap procedure for numerical evaluation of Eichler integrals is illustrated.

Abstract

We show that all Eichler integrals, and more generally all "generalized second order modular forms" can be expressed as linear combinations of corresponding generalized second order Eisenstein series with coefficients in classical modular forms. We determine the Fourier series expansions of generalized second order Eisenstein series in level one, and provide tail estimates via convexity bounds for additively twisted $\mathrm{L}$-functions. As an application, we illustrate a bootstrapping procedure that yields numerical evaluations of, for instance, Eichler integrals from merely the associated cocycle. The proof of our main results rests on a filtration argument that is largely rooted in previous work on vector-valued modular forms, which we here formulate in classical terms.