Period-like polynomials for $L$-series associated with half-integral weight cusp forms

TL;DR

This paper constructs a cohomology class for L-series of half-integral weight cusp forms and introduces a lift to integral weight forms, paralleling Eichler cohomology.

Contribution

It develops a new cohomology framework for half-integral weight cusp forms and defines a compatible lifting to integral weight forms.

Findings

Established a cohomology class for half-integral weight L-series

Defined a lift from half-integral to integral weight cusp forms

Demonstrated compatibility with associated L-series

Abstract

Given the L-series of a half-integral weight cusp form, we construct a cohomology class with coefficients in a finite dimensional vector space in a way that parallels the Eichler cohomology in the integral weight case. We also define a lift of half-integral weight cusp forms to integral weight modular forms that is compatible with the $L$-series of the respective forms.