Paweł Borówka

## Title:

Non-simple principally polarised abelian varieties

## Abstract:

We will start by recalling Poincare Reducibility Theorems for abelian varieties that state that an abelian variety is simple or isogenous to a product of abelian varieties of smaller dimensions. Then, we will improve the statement in the following way. Let $Is^g_D$ be the locus in the moduli space of principally polarised abelian varieties of dimension g that contain an abelian subvariety of at most half dimension and restricted polarisation of type D. Then $Is^g_D$ is an irreducible subvariety in the moduli space and the locus of non-simple abelian varieties is the countable union of all $Is^g_D$. Moreover, in the similar way to Humbert, we will produce equations for $Is^g_D$ in the Siegel upper half-space. If time permits, we will show some applications of the theory to Jacobians of curves and Pryms. The talk is based on the results included in the preprint 'Non-simple principally polarised abelian varieties'