If A is an abelian variety over an algebraically closed field, then every polarization is represented by an ample line bundle on A. However, such a line bundle may not exist if the field is not algebraically closed, or when it is replaced by a more general base scheme; in fact, this failure already occurs for Jacobians. In 1999, Poonen and Stoll asked: can every polarization be represented by a line bundle on some torsor under A? In this talk, I will expand on the background for this question and give examples for which the answer to this question is no. Along the way, we will encounter Mumford theta groups, Serre's notion of negligible group cohomology and moduli spaces of abelian varieties.