2017/12/13 by Sanath K. Devalapurkar, John Halliday, Devalapurkar, Sanath +1
Computer Science · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1712.04921
openalex publication_date 2017/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a principally polarized abelian variety A of dimension g over an algebraically closed field k of characteristic p, the p torsion A[p] is a finite flat p-torsion group scheme of rank p2g. There are exactly 2g possible group schemes that can occur as some such A[p]. In this paper, we study which group schemes can occur as J[p], where J is the Jacobian of a hyperelliptic curve defined over \mathbbFp. We do this by computing explicit formulae for the action of Frobenius and its dual on the de Rham cohomology of a hyperelliptic curve with respect to a given basis. A theorem of Oda's in [Oda69] allows us to relate these actions to the p-torsion structure of the Jacobian. Using these formulae and the computer algebra system Magma, we affirmatively resolve questions of Glass and Pries in [Cor05] on whether certain group schemes of rank p8 and p10 can occur as J[p] of a hyperelliptic curve of genus 4 and 5 respectively.