Galois structure of the holomorphic differentials of curves

Abstract

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⟶X/G. We apply our method to determine the PSL(2,Fℓ)-module structure of the space of holomorphic differentials of the reduction of the modular curve X(ℓ) modulo p when p and ℓ are distinct odd primes and the action of PSL(2,Fℓ) 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 PSL(2,Fℓ) on X(ℓ).