Torsion points on Jacobian varieties via Anderson's p-adic soliton theory

TL;DR

This paper advances Anderson's $p$-adic soliton theory to study torsion points on Jacobian varieties, providing stronger intersection results with the theta divisor for broader classes of curves and introducing a new connecting map.

Contribution

It extends Anderson's $p$-adic soliton framework and introduces a novel map linking the $p$-adic loop group with the formal group, leading to improved torsion point results.

Findings

Stronger intersection results between theta divisor and torsion points.

New examples illustrating the extended theory.

A novel map connecting $p$-adic loop group and formal group.

Abstract

Anderson introduced a $p$-adic version of soliton theory. He then applied it to the Jacobian variety of a cyclic quotient of a Fermat curve and showed that torsion points of certain prime order lay outside of the theta divisor. In this paper, we evolve his theory further. As an application, we get a stronger result on the intersection of the theta divisor and torsion points on the Jacobian variety for more general curves. New examples are discussed as well. A key new ingredient is a map connecting the $p$-adic loop group and the formal group.