Automorphisms of families of abelian varieties

Complex abelian varieties and their automorphism groups are classical objects; we have a lot of information about their structure. However, if instead of an abelian variety one considers a (maybe degenerate) family of abelian varieties over a projective base and its fiberwise birational automorphism then the situation is much more vague. In particular, it is even not clear whether the birational automorphism of a family of abelian varieties is regularizable or not. I am going to speak about the criterion for regularizability of automorphisms of families which are indecomposable on a general fiber; i.e. such that all iterates of the automorphism do not preserve a decomposition of a general fiber into a product of two abelian varieties. The talk is based on a joint work in progress with C. Favre.