Galois structure of the holomorphic differentials of curves

Let $X$ be a smooth projective geometrically irreducible curve over a perfect field $k$ of positive characteristic $p$. Suppose $G$ is a finite group acting faithfully on $X$ such that $G$ has non-trivial cyclic Sylow $p$-subgroups. We show that the decomposition of the space of holomorphic differentials of $X$ into a direct sum of indecomposable $k[G]$-modules is uniquely determined by the lower ramification groups and the fundamental characters of closed points of $X$ that are ramified in the cover $X\to X\mathrm{/}G$. We apply our method to determine the $\mathrm{P}\mathrm{S}\mathrm{L}(2,{\mathbb{F}}_{\mathrm{\ell }})$-module structure of the space of holomorphic differentials of the reduction of the modular curve $\mathcal{X}(\mathrm{\ell })$ modulo $p$ when $p$ and $\mathrm{\ell }$ are distinct odd primes and the action of $\mathrm{P}\mathrm{S}\mathrm{L}(2,{\mathbb{F}}_{\mathrm{\ell }})$ on this reduction is not tamely ramified. This provides some non-trivial congruences modulo appropriate maximal ideals containing $p$ between modular forms arising from isotypic components with respect to the action of $\mathrm{P}\mathrm{S}\mathrm{L}(2,{\mathbb{F}}_{\mathrm{\ell }})$ on $\mathcal{X}(\mathrm{\ell })$.