Realizing Artin–Schreier covers with minimal a-numbers in positive characteristic

Abney-McPeek, Fiona; Berg, Hugo; Booher, Jeremy; Choi, Sun Mee; Fukala, Viktor; Marinov, Miroslav; Müller, Theo; Narkiewicz, Paweł; Pries, Rachel; Xu, Nancy; Yuan, Andrew

doi:10.2140/involve.2022.15.559

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Realizing Artin–Schreier covers with minimal $a$-numbers in positive characteristic

Fiona Abney-McPeek, Hugo Berg, Jeremy Booher, Sun Mee Choi, Viktor Fukala, Miroslav Marinov, Theo Müller, Paweł Narkiewicz, Rachel Pries, Nancy Xu and Andrew Yuan

Vol. 15 (2022), No. 4, 559–590

DOI: 10.2140/involve.2022.15.559

Abstract

Suppose $X$ is a smooth projective connected curve defined over an algebraically closed field of characteristic $p > 0$ and $B \subset X$ is a finite, possibly empty, set of points. Booher and Cais determined a lower bound for the $a$ -number of a $ℤ ∕ p ℤ$ -cover of $X$ with branch locus $B$ . For odd primes $p$ , in most cases it is not known if this lower bound is realized. In this note, when $X$ is ordinary, we use formal patching to reduce that question to a computational question about $a$ -numbers of $ℤ ∕ p ℤ$ -covers of the affine line. As an application, when $p = 3$ or $p = 5$ , for any ordinary curve $X$ and any choice of $B$ , we prove that the lower bound is realized for Artin–Schreier covers of $X$ with branch locus $B$ .

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Keywords

Artin–Schreier cover, characteristic-$p$, Cartier operator, $p$-rank, $p$-torsion, formal patching, wild ramification, $a$-number, curve, finite field, Jacobian, arithmetic geometry

Mathematical Subject Classification

Primary: 11G20, 11T06, 14D15, 14H40, 15A04

Secondary: 11C08, 14G17, 14H30, 15B33

Milestones

Received: 31 March 2020

Revised: 15 February 2021

Accepted: 4 January 2022

Published: 7 January 2023

Communicated by Kenneth S. Berenhaut

Authors

Fiona Abney-McPeek
Harvard College Cambridge, MA United States

Hugo Berg
Department of Computer Science University of Oxford United Kingdom

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

Sun Mee Choi
Massachusetts Institute of Technology Cambridge, MA United States

Viktor Fukala
Department of Computer Science ETH Zürich Switzerland

Miroslav Marinov
PROMYS Europe Mathematical Institute University of Oxford United Kingdom

Theo Müller
Institut für Mathematik Humboldt-Universität zu Berlin Germany

Paweł Narkiewicz
PROMYS Europe Mathematical Institute University of Oxford United Kingdom

Rachel Pries
Department of Mathematics Colorado State University Fort Collins, CO United States

Nancy Xu
Princeton University Princeton, NJ United States

Andrew Yuan
Brown University Providence, RI United States

Vol. 15, No. 4, 2022