Talks

## Torsion subvarieties and Betti map

32
Given an algebraic family of abelian varieties, i.e. a morphism of algebraic varieties ${\mathcal A}\to S$ whose generic fibers are abelian varieties, and a section $s: S\to{\mathcal A}$, we are interested in the subvarieties $X\subset S$ where $s$ takes a torsion value. These varieties are called torsion varieties. We prove that, outside special cases easy to classify, there are only finitely many torsion hypersurfaces. A main tool in the proof is represented by the so called 'Betti map', i.e. the 'logarithm' of the section. We provide some results on the rank of this map, and show some concrete applications. This research has been done in collaboration with Y. Andr\'e, D. Masser and U. Zannier