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

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 $p=3$.