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 , F l ) -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 , F l ) 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 , F l ) on X ( l ) .

中文翻译：

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曲线全纯微分的伽罗瓦结构

摘要 设 X 是在具有正特征 p 的完美域 k 上的平滑投影几何不可约曲线。假设 G 是一个忠实地作用于 X 的有限群，使得 G 具有非平凡的循环 Sylow p-子群。我们表明，将 X 的全纯微分空间分解为不可分解的 k [ G ] -模的直接和是由下分支群和在覆盖 X ⟶ 中分支的 X 的闭点的基本特征唯一确定的X/G。我们应用我们的方法来确定当 p 和 l 是不同的奇素数时模曲线 X ( l ) 模 p 的归约的全纯微分空间的 PSL ( 2 , F l ) -模结构和 PSL ( 2 , F l ) 对这种减少的影响并不温和。