A Generalization of Eichler Integrals and Certain Local Systems over Spin Riemann Surfaces

Abstract

Let $(X,K_X^{1/2})$ be a spin Riemann surface – a Riemann surface together with a halfcanonical bundle (4.1) – of genus $\ge 2$. By using infinite dimensional representations of the fundamental group of $X$, we obtain many local systems on $X$, which taken together define a resolution of the halfcanonical ring of $X$ and indicate a non-abelian theory of abelian integrals on $X$. The work has a root in a study of the complex structure on the Fricke moduli space [14].