Abelian varieties are higher-dimensional generalisations of elliptic curves and are ubiquitous in algebraic geometry and number theory. Central to their theory is the concept of a polarisation. If
A
is an abelian variety over an algebraically closed field, then every polarisation 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\DSEMIC in fact, this failure already occurs for Jacobians.

In 1999, Poonen and Stoll asked: can every polarisation be represented by a line bundle on some torsor under
A
? In this talk, I will expand on the background for this question and I will explain why the answer is often ``yes'', but not always. Along the way, we will encounter Mumford theta groups, Serre's notion of negligible group cohomology and moduli spaces of abelian varieties.