Non-ordinary curves with a Prym variety of low p-rank
arXiv: Number Theory(2017)
摘要
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 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.
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关键词
Curve, Jacobian, Prym variety, Abelian variety, p-Rank, Supersingular, Moduli space, Kummer surface
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