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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
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Abstract |
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Suppose is a smooth projective connected curve defined over an algebraically closed field of characteristic and is a finite, possibly empty, set of points. Booher and Cais determined a lower bound for the -number of a -cover of with branch locus . For odd primes , in most cases it is not known if this lower bound is realized. In this note, when is ordinary, we use formal patching to reduce that question to a computational question about -numbers of -covers of the affine line. As an application, when or , for any ordinary curve and any choice of , we prove that the lower bound is realized for Artin–Schreier covers of with branch locus . |
Keywords
Artin–Schreier cover, characteristic-$p$, Cartier operator,
$p$-rank, $p$-torsion, formal patching, wild ramification,
$a$-number, curve, finite field, Jacobian, arithmetic
geometry
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Mathematical Subject Classification
Primary: 11G20, 11T06, 14D15, 14H40, 15A04
Secondary: 11C08, 14G17, 14H30, 15B33
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Milestones
Received: 31 March 2020
Revised: 15 February 2021
Accepted: 4 January 2022
Published: 7 January 2023
Communicated by Kenneth S. Berenhaut
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Authors |
| Fiona Abney-McPeek | |
| Harvard College Cambridge, MA United States |
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| Hugo Berg | |
| Department of Computer Science University of Oxford United Kingdom |
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| Jeremy Booher | |
| School of Mathematics and
Statistics University of Canterbury Christchurch New Zealand |
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| Sun Mee Choi | |
| Massachusetts Institute of
Technology Cambridge, MA United States |
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| Viktor Fukala | |
| Department of Computer Science ETH Zürich Switzerland |
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| Miroslav Marinov | |
| PROMYS Europe Mathematical Institute University of Oxford United Kingdom |
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| Theo Müller | |
| Institut für Mathematik Humboldt-Universität zu Berlin Germany |
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| Paweł Narkiewicz | |
| PROMYS Europe Mathematical Institute University of Oxford United Kingdom |
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| Rachel Pries | |
| Department of Mathematics Colorado State University Fort Collins, CO United States |
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| Nancy Xu | |
| Princeton University Princeton, NJ United States |
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| Andrew Yuan | |
| Brown University Providence, RI United States |
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