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

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