Period functions of half-integral weight modular forms

Abstract

In this paper, we study the Eichler cohomology associated with half-integral weight cusp forms using the Dedekind eta function η(z) and the theta function θ(z). We prove that η-multiplication (resp. θ-multiplication) gives an isomorphism between the space of cusp forms of a half-integral weight and the cohomology group associated with the space ηP (resp. θP). We also show that there is an isomorphism between the direct sum of two spaces of cusp forms of half-integral weights and the cohomology group. © Société Arithmétique de Bordeaux, 2015.