Eichler integrals and generalized second order Eisenstein series
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 vectorvalued modular forms, which we here formulate in classical terms.
 Publication:

arXiv eprints
 Pub Date:
 March 2022
 DOI:
 10.48550/arXiv.2203.15462
 arXiv:
 arXiv:2203.15462
 Bibcode:
 2022arXiv220315462A
 Keywords:

 Mathematics  Number Theory;
 11F11;
 11F30;
 11F75