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arXiv · 2003.09028

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·Andrew Yuan
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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 $\mathbf{Z}/p \mathbf{Z}$-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 $\mathbf{Z}/p\mathbf{Z}$-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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Fiona Abney-McPeek, Hugo Berg, Jeremy Booher, Sun Mee Choi, Viktor Fukala, Miroslav Marinov, Theo Müller, Paweł Narkiewicz, Rachel Pries, Nancy Xu, Andrew Yuan. 2020-03-19. Realizing Artin-Schreier Covers with Minimal $a$-numbers in Positive Characteristic. https://doi.org/10.2140/involve.2022.15.559

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