Algebra & Number Theory

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

Jeremy Booher and Bryden Cais

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Let π:YX be a branched ZpZ-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

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

Received: 26 July 2018
Revised: 2 September 2019
Accepted: 7 October 2019
First available in Project Euclid: 2 July 2020

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Mathematical Reviews number (MathSciNet)

Primary: 14G17: Positive characteristic ground fields
Secondary: 11G20: Curves over finite and local fields [See also 14H25] 14H40: Jacobians, Prym varieties [See also 32G20]

$a$-numbers Artin–Schreier covers arithmetic geometry covers of curves invariants of curves


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.

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