Non-ordinary curves with a prym variety of low p-rank

Celik, Turku Ozlum and Elias, Yara and Güneş, Burçin and Newton, Rachel and Ozman, Ekin and Pries, Rachel and Thomas, Lara (2018) Non-ordinary curves with a prym variety of low p-rank. In: Bouw, Irene I. and Ozman, Ekin and Johnson-Leung, Jennifer and Newton, Rachel, (eds.) Women in Numbers Europe II: Contributions to Number Theory and Arithmetic Geometry. Association for Women in Mathematics Series, 11. Springer Cham, pp. 117-158. ISBN 978-3-319-74997-6 (Print) 978-3-319-74998-3 (Online)

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If π : Y → X is an unramified double cover of a smooth curve of genus g, then the Prym variety P π is a principally polarized abelian variety of dimension g − 1. When X is defined over an algebraically closed field k of characteristic p, it is not known in general which p-ranks can occur for P π under restrictions on the p-rank of X. In this paper, when X is a non-hyperelliptic curve of genus g = 3, we analyze the relationship between the Hasse-Witt matrices of X and P π . As an application, when p ≡ 5 mod  6, we prove that there exists a curve X of genus 3 and p-rank f = 3 having an unramified double cover π : Y → X for which P π has p-rank 0 (and is thus supersingular); for 3 ≤ p ≤ 19, we verify the same for each 0 ≤ f ≤ 3. Using theoretical results about p-rank stratifications of moduli spaces, we prove, for small p and arbitrary g ≥ 3, that there exists an unramified double cover π : Y → X such that both X and P π have small p-rank.
Item Type: Book Section / Chapter
Uncontrolled Keywords: Abelian variety; Curve; Jacobian; Kummer surface; Moduli space; p-Rank; Prym variety; Supersingular
Divisions: Faculty of Engineering and Natural Sciences
Depositing User: Burçin Güneş
Date Deposited: 26 Jul 2023 12:19
Last Modified: 26 Jul 2023 12:19

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