Holomorphic automorphic forms and cohomology

TL;DR

This paper explores the relationship between holomorphic automorphic forms of various weights and their associated cohomology classes, extending classical Eichler theory to non-integer weights without growth restrictions.

Contribution

It generalizes the Eichler integral to real weights, characterizes the image in cohomology, and introduces new tools involving boundary germs and harmonic lifts.

Findings

Generalized Eichler integral injects into cohomology for non-integer weights

Characterization of cusp forms and automorphic forms without growth conditions

Cohomology with boundary germ coefficients distinguishes all forms

Abstract

We investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least 2 this correspondence is given by the Eichler integral. Knopp generalized this to real weights. We show that for weights that are not an integer at least 2 the generalized Eichler integral gives an injection into the first cohomology group with values in a module of holomorphic functions, and characterize the image. We impose no condition on the growth of the automorphic forms at the cusps. For real weights that are not an integer at least 2 we similarly characterize the space of cusp forms and the space of entire automorphic forms. We give a relation between the cohomology classes attached to holomorphic automorphic forms of real weight and the existence of harmonic lifts. A tool in establishing these results is the relation to cohomology groups with values in modules of "analytic boundary germs", which are represented by harmonic functions on subsets of the upper half-plane. Even for positive integral weights cohomology with these coefficients can distinguish all holomorphic automorphic forms, unlike the classical Eichler theory.