We prove various explicit formulas concerning $p$-rank of $p$-coverings of pointed semistable curves over discrete valuation rings. In particular, we obtain a full generalization of Raynaud’s formula for $p$-rank of fibers over nonmarked smooth closed points in the case of arbitrary closed points. As an application, for abelian $p$-coverings, we give an affirmative answer to an open problem concerning boundedness of $p$-rank asked by Saïdi more than twenty years ago.

中文翻译：

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p 群、p 秩和曲线覆盖的半稳定约简

我们证明了各种显式公式$p$-等级为$p$-离散估值环上尖头半稳定曲线的覆盖。特别是，我们获得了雷诺公式的完整推广$p$-在任意闭合点的情况下，纤维在无标记平滑闭合点上的等级。作为一个应用程序，对于阿贝尔$p$-coverings，我们对有关有界性的开放问题给出了肯定的答案$p$——萨伊迪二十多年前提出的排名问题。