## Algebra & Number Theory

### $a$-numbers of curves in Artin–Schreier covers

#### Abstract

Let $π:Y→X$ be a branched $Z∕pZ$-cover of smooth, projective, geometrically connected curves over a perfect field of characteristic $p>0$. We investigate the relationship between the $a$-numbers of $Y$ and $X$ and the ramification of the map $π$. This is analogous to the relationship between the genus (respectively $p$-rank) of $Y$ and $X$ given the Riemann–Hurwitz (respectively Deuring–Shafarevich) formula. Except in special situations, the $a$-number of $Y$ is not determined by the $a$-number of $X$ and the ramification of the cover, so we instead give bounds on the $a$-number of $Y$. We provide examples showing our bounds are sharp. The bounds come from a detailed analysis of the kernel of the Cartier operator.

#### Article information

Source
Algebra Number Theory, Volume 14, Number 3 (2020), 587-641.

Dates
Revised: 2 September 2019
Accepted: 7 October 2019
First available in Project Euclid: 2 July 2020

https://projecteuclid.org/euclid.ant/1593655266

Digital Object Identifier
doi:10.2140/ant.2020.14.587

Mathematical Reviews number (MathSciNet)
MR4113776

#### Citation

Booher, Jeremy; Cais, Bryden. $a$-numbers of curves in Artin–Schreier covers. Algebra Number Theory 14 (2020), no. 3, 587--641. doi:10.2140/ant.2020.14.587. https://projecteuclid.org/euclid.ant/1593655266

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