Galois structure of the holomorphic differentials of curves

TL;DR

This paper investigates the structure of holomorphic differentials on algebraic curves with group actions in positive characteristic, revealing how ramification data determines module decompositions and applying this to modular curves to find new congruences.

Contribution

It provides a method to determine the Galois module structure of differentials using ramification data and applies it to modular curves with non-tame group actions.

Findings

Decomposition of differentials is uniquely determined by ramification data.

Explicit structure of differentials for reductions of modular curves is obtained.

New congruences between modular forms are established.

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 $X\to X/G$. 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 $\mathcal{X}(\ell)$ modulo $p$ when $p$ and $\ell$ 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 containing $p$ between modular forms arising from isotypic components with respect to the action of $\mathrm{PSL}(2,\mathbb{F}_\ell)$ on $\mathcal{X}(\ell)$.