The Chevalley-Weil Formula for Orbifold Curves

TL;DR

This paper extends the Chevalley-Weil formula to orbifold curves, providing a new tool for decomposing canonical representations in the context of ramified Galois covers, with applications to modular and Fermat curves.

Contribution

It proves an analogous Chevalley-Weil formula for orbifold curves and applies it to modular and Fermat curves, advancing the understanding of their canonical representations.

Findings

Derived a Chevalley-Weil formula for orbifold curves.

Decomposed canonical representations of modular and Fermat curves.

Provided explicit formulas for ramified Galois covers of orbifold curves.

Abstract

In the 1930s Chevalley and Weil gave a formula for decomposing the canonical representation on the space of differential forms of the Galois group of a ramified Galois cover of Riemann surfaces. In this article we prove an analogous Chevalley-Weil formula for ramified Galois covers of orbifold curves. We then specialize the formula to the case when the base orbifold curve is the (reduced) modular orbifold. As an application of this latter formula we decompose the canonical representations of modular curves of full, prime level and of Fermat curves of arbitrary exponent.