p-torsion of Genus Two Curves Over Prime Fields of Characteristic p

TL;DR

This paper investigates the structure of the p-torsion subgroup of Jacobians of genus two curves over prime fields with complex multiplication, revealing it is either trivial or of order p.

Contribution

It proves that the p-Sylow subgroup of such Jacobians is always either trivial or cyclic of order p, providing new insight into their torsion structure.

Findings

p-Sylow subgroup is either trivial or of order p

Results apply to Jacobians with complex multiplication

Advances understanding of torsion in genus two curves

Abstract

Consider the Jacobian of a hyperelliptic genus two curve defined over a prime field of characteristic p and with complex multiplication. In this paper we show that the p-Sylow subgroup of the Jacobian is either trivial or of order p.