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 \(X \rightarrow X/G\). We apply our method to determine the \(\mathrm{PSL}(2, \mathbb{F}_\ell)\)-module structure of the space of holomorphic differentials of the reduction of the modular curve \(X(\ell)\) modulo \(p\) when \(p\) and are distinct odd primes and the action of \(\mathrm{PSL}(2, \mathbb{F}_\ell)\) on this reduction is not tamely ramified. This provides some non-trivial congruences modulo appropriate maximal ideals
[Pdf file]