The Galois module structure of holomorphic poly-differentials and Riemann-Roch spaces

TL;DR

This paper characterizes the Galois module structure of holomorphic poly-differentials and Riemann-Roch spaces on algebraic curves over fields of positive characteristic, linking it to ramification data and divisors, with applications to deformation theory and modular forms.

Contribution

It extends the understanding of Galois module structures of holomorphic differentials to higher tensor powers, incorporating ramification data, and applies this to deformation spaces and modular forms in characteristic p.

Findings

Unique determination of module decomposition by divisor class and ramification data

Extension of known results from canonical divisors to higher tensor powers

Application to modular forms and deformation theory

Abstract

Suppose $X$ is a smooth projective geometrically irreducible curve over a perfect field $k$ of positive characteristic $p$. Let $G$ be a finite group acting faithfully on $X$ over $k$ such that $G$ has non-trivial, cyclic Sylow $p$-subgroups. If $E$ is a $G$-invariant Weil divisor on $X$ with $\mathrm{deg}(E)> 2g(X)-2$, we prove that the decomposition of $\mathrm{H}^0(X,\mathcal{O}_X(E))$ into a direct sum of indecomposable $kG$-modules is uniquely determined by the class of $E$ modulo $G$-invariant principal divisors, together with the ramification data of the cover $X\to X/G$. The latter is given by the lower ramification groups and the fundamental characters of the closed points of $X$ that are ramified in the cover. As a consequence, we obtain that if $m>1$ and $g(X)\ge 2$, then the $kG$-module structure of $\mathrm{H}^0(X,\Omega_X^{\otimes m})$ is uniquely determined by the class of a canonical divisor on $X/G$ modulo principal divisors, together with the ramification data of $X\to X/G$. This extends to arbitrary $m > 1$ the $m = 1$ case treated by the first author with T. Chinburg and A. Kontogeorgis. We discuss applications to the tangent space of the global deformation functor associated to $(X,G)$ and to congruences between prime level cusp forms in characteristic $0$. In particular, we complete the description of the precise $k\mathrm{PSL}(2,\mathbb{F}_\ell)$-module structure of all prime level $\ell$ cusp forms of even weight in characteristic $p=3$.