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SOME CASES OF OORT’S CONJECTURE ABOUT NEWTON POLYGONS OF CURVES

Part of: Zeta and $L$-functions: analytic theory Arithmetic problems. Diophantine geometry Curves Arithmetic algebraic geometry

Published online by Cambridge University Press:  02 December 2024

RACHEL PRIES Open the ORCID record for RACHEL PRIES [Opens in a new window]
RACHEL PRIES*
Affiliation:
Department of Mathematics Colorado State University Fort Collins CO 80523 United States
*
[email protected]
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Abstract

This paper contains a method to prove the existence of smooth curves in positive characteristic whose Jacobians have unusual Newton polygons. Using this method, I give a new proof that there exist supersingular curves of genus $4$ in every prime characteristic. More generally, the main result of the paper is that, for every $g \geq 4$ and prime p, every Newton polygon whose p-rank is at least $g-4$ occurs for a smooth curve of genus g in characteristic p. In addition, this method resolves some cases of Oort’s conjecture about Newton polygons of curves.

Keywords

CurveJacobianmoduli spaceNewton polygonsupersingular

MSC classification

Primary: 11G10: Abelian varieties of dimension $> 1$ 11G20: Curves over finite and local fields 11M38: Zeta and $L$-functions in characteristic $p$ 14H10: Families, moduli (algebraic) 14H40: Jacobians, Prym varieties
Secondary: 14G10: Zeta-functions and related questions (Birch-Swinnerton-Dyer conjecture) 14G15: Finite ground fields
Type
Article
Information
Nagoya Mathematical Journal , Volume 257 , March 2025 , pp. 93 - 103
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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