Polarizations, torsors and theta groups

TL;DR

This paper investigates the existence of line bundles representing polarizations on abelian varieties over non-algebraically closed fields, providing criteria and examples where such line bundles do or do not exist.

Contribution

It offers a criterion based on the kernel of the polarization for the existence of such line bundles and constructs examples with negative answers in higher dimensions.

Findings

Existence criterion depends only on the kernel of the polarization.

Positive results for polarizations of odd or small even degree.

Counterexamples exist for dimensions g ≥ 7.

Abstract

Let $\lambda\colon A\rightarrow A^{\vee}$ be a polarization on an abelian variety over a field $k$. If $k$ is not algebraically closed, there might not exist an ample line bundle on $A$ defined over $k$ that represents $\lambda$. To remedy this, Poonen and Stoll have asked the following question: does there exist a line bundle on an $A$-torsor that represents $\lambda$? We give a criterion for the existence of such a torsor and line bundle which only depends on the kernel of $\lambda$. Using this criterion, we show that the answer to the question is yes when the polarization has odd or small even degree. On the other hand, we show that for every $g\geq 7$, there exists a polarized $g$-dimensional abelian variety for which the answer to the question is no.