Polarizations on abelian varieties are a central part of their theory, for example when constructing moduli spaces. Over an algebraically closed field, every polarization is represented by an ample line bundle on A, but this need not be true in general. To remedy this, Poonen and Stoll have asked: can every polarization be represented by a line bundle on some torsor under A? This question also arises naturally from the work of Alexeev on compactifying moduli of abelian varieties. In this talk, we will explain why the answer to this question is often yes and sometimes no. Along the way, we will encounter Mumford theta groups, negligible group cohomology and twisted cohomology of moduli spaces of abelian varieties.
