Eichler cohomology in general weights using spectral theory

TL;DR

This paper develops a spectral theory-based approach to Eichler cohomology for modular forms of positive real weights, providing new proofs and extending results to vector-valued forms, except at weight 1.

Contribution

It introduces a novel spectral theory method to establish a perfect pairing between modular forms and Eichler cohomology for all positive weights except 1, extending previous theorems.

Findings

Established a perfect pairing for all positive weights except 1.

Provided a new proof of a 2010 theorem by Knopp and Mawi.

Extended results to vector-valued modular forms.

Abstract

In this paper, we construct a pairing between modular forms of positive real weight and elements of certain Eichler cohomology groups that were introduced by Knopp in 1974. We use spectral theory of automorphic forms to show that this pairing is perfect for all positive weights except 1. The approach in this paper gives a new proof of a theorem by Knopp and Mawi from 2010 for all real weights excluding 1 and also a version of this theorem for vector-valued modular forms.