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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
Abstract

Suppose X is a smooth projective connected curve defined over an algebraically closed field of characteristic p > 0 and B X is a finite, possibly empty, set of points. Booher and Cais determined a lower bound for the a-number of a p-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 p-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.

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