a-numbers of curves in Artin–Schreier covers

Booher, Jeremy; Cais, Bryden

doi:10.2140/ant.2020.14.593

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$a$-numbers of curves in Artin–Schreier covers

Jeremy Booher and Bryden Cais

Vol. 14 (2020), No. 3, 587–641

DOI: 10.2140/ant.2020.14.593

Abstract

Let $π : Y \to X$ be a branched $Z ∕ p Z$ -cover of smooth, projective, geometrically connected curves over a perfect field of characteristic $p > 0$ . We investigate the relationship between the $a$ -numbers of $Y$ and $X$ and the ramification of the map $π$ . This is analogous to the relationship between the genus (respectively $p$ -rank) of $Y$ and $X$ given the Riemann–Hurwitz (respectively Deuring–Shafarevich) formula. Except in special situations, the $a$ -number of $Y$ is not determined by the $a$ -number of $X$ and the ramification of the cover, so we instead give bounds on the $a$ -number of $Y$ . We provide examples showing our bounds are sharp. The bounds come from a detailed analysis of the kernel of the Cartier operator.

Keywords

$a$-numbers, Artin–Schreier covers, arithmetic geometry, covers of curves, invariants of curves

Mathematical Subject Classification 2010

Primary: 14G17

Secondary: 11G20, 14H40

Milestones

Received: 26 July 2018

Revised: 2 September 2019

Accepted: 7 October 2019

Published: 1 June 2020

Authors

Jeremy Booher
School of Mathematics and Statistics University of Canterbury Christchurch New Zealand

Bryden Cais
Department of Mathematics University of Arizona Tucson, AZ United States

Vol. 14, No. 3, 2020