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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 $B\subset X$ is a finite, possibly empty, set of points. Booher and Cais determined a lower bound for the $a$-number of a $ℤ∕pℤ$-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 $ℤ∕pℤ$-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