Vol. 14, No. 3, 2020

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

Jeremy Booher and Bryden Cais

Vol. 14 (2020), No. 3, 587–641
Abstract

Let π : Y X be a branched ZpZ-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