Torsion points on hyperelliptic Jacobians via Anderson's $p$-adic soliton theory

TL;DR

This paper demonstrates that certain torsion points do not lie on the theta divisor in the Jacobian of specific hyperelliptic curves, using Anderson's $p$-adic soliton theory, extending previous results on Fermat curves.

Contribution

It introduces a novel application of Anderson's $p$-adic soliton theory to hyperelliptic Jacobians, showing non-existence of torsion points on the theta divisor for these curves.

Findings

Torsion points of certain orders are not on the theta divisor.

Extension of Anderson's method from Fermat curves to hyperelliptic curves.

Provides new insights into the structure of hyperelliptic Jacobians.

Abstract

We show that torsion points of certain orders are not on a theta divisor in the Jacobian variety of a hyperelliptic curve given by the equation $y^2=x^{2g+1}+x$ with $g \geq 2$. The proof employs a method of Anderson who proved an analogous result for a cyclic quotient of a Fermat curve of prime degree.