On Spaces Associated with Invariant Divisors on Galois Covers of Riemann Surfaces and Their Applications

TL;DR

This paper studies Galois covers of Riemann surfaces, focusing on G-invariant divisors and their associated function spaces, generalizing classical formulas and providing new results for Abelian groups and genus zero cases.

Contribution

It generalizes the trace formula and Chevalley–Weil formula for G-invariant divisors on Galois covers, with new results for Abelian groups and genus zero surfaces.

Findings

Generalized trace formula for $q$-differentials

Extended Chevalley–Weil formula for G-invariant divisors

Characterization of non-special G-invariant divisors in specific cases

Abstract

Let $f:X \to S$ be a Galois cover of Riemann surfaces, with Galois group $G$. In this paper we analyze the $G$-invariant divisors on $X$, and their associated spaces of meromorphic functions, differentials, and $q$-differentials. We generalize the trace formula for non-trivial elements of $G$ on $q$-differentials, as well as the Chevalley--Weil Formula. When $G$ is Abelian or when the genus of $S$ is 0 we prove additional results, and we also determine the non-special $G$-invariant divisors when both conditions are satisfied.