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

Frauke M. Bleher, Adam Wood

Published 2021-10-20, updated 2022-06-14Version 2

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. In this paper we show that for $m > 1$, the decomposition of $\mathrm{H}^0(X,\Omega_X^{\otimes m})$ into a direct sum of indecomposable $kG$-modules is uniquely determined by the divisor class of a canonical divisor of $X/G$ together with the lower ramification groups and the fundamental characters of the closed points of $X$ that are ramified in the cover $X\to X/G$. This extends to arbitrary $m > 1$ the $m = 1$ case treated by the first author with T. Chinburg and A. Kontogeorgis. We discuss some applications to congruences between modular forms in characteristic $0$, to the tangent space of the global deformation functor associated to $(X,G)$, and to the $kG$-module structure of Riemann-Roch spaces associated to divisors on $X$.