Torsion points on theta divisors

TL;DR

This paper establishes bounds on the number of torsion points on theta divisors of abelian varieties using representation theory, and explores specific cases with alternative methods.

Contribution

It introduces a new bound for torsion points on theta divisors based on irreducibility of a theta group representation, with additional approaches for the case n=2.

Findings

Derived a bound for n-torsion points on theta divisors

Presented two alternative methods for the case n=2

Utilized irreducibility of a theta group representation

Abstract

Using the irreducibility of a natural irreducible representation of the theta group of an ample line bundle on an abelian variety, we derive a bound for the number of $n$-torsion points that lie on a given theta divisor. We present also two alternate approaches to attacking the case $n=2$.