p-groups, p-rank, and semistable reduction of coverings of curves

Yang, Yu

doi:10.2140/ant.2024.18.281

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$p$-groups, $p$-rank, and semistable reduction of coverings of curves

Yu Yang

Vol. 18 (2024), No. 2, 281–317

DOI: 10.2140/ant.2024.18.281

Abstract

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.

Keywords

$p$-rank, semistable reduction, pointed semistable curve, pointed semistable covering

Mathematical Subject Classification

Primary: 14E20, 14G17

Secondary: 14G20, 14H30

Milestones

Received: 18 April 2022

Revised: 22 September 2022

Accepted: 13 April 2023

Published: 6 February 2024

Authors

Yu Yang
Research Institute for Mathematical Sciences Kyoto University Kyoto Japan

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Vol. 18, No. 2, 2024