Konferenz: Symposium about Pure Mathematics


Distribution of Torsion Points in certain Families of Abelian Varieties

{{_Ltalk:R}} Prof. Dr. Philipp Habegger
Datum: 14.12.09   Zeit: 15.00 - 16.00   Raum: Y27H46

Abstract: Abelian varieties are connected, projective algebraic groups. That is, they are subsets of projective space defined by homogeneous polynomials and come with a group structure also defined by polynomial equations. The interplay between the geometry and the group structure of an abelian variety is an important theme in Diophantine and arithmetic geometry. For example, the Manin-Mumford Conjecture, now a Theorem of Raynaud, gives precise information on the closure (in the Zariski topology) of a set of torsion points of an abelian variety. A family of abelian varieties is, roughly said, a variety with a polynomial map to a base variety where the fibers are abelian varieties and such that the group structure varies algebraically.

In this talk I will begin by giving a basic overview of aforementioned objects. In the second half I will treat some new results on distribution of torsion points with "additional structure" for a specific family of abelian varieties. "Additional structure" refers to properties of the fiber containing the torsion points: e.g. one could demand that this abelian variety has additional endomorphisms or that it is isogenous to a fixed abelian variety. The proofs involve a mixture of arithmetic tools (such as height functions), geometric tools (such as intersection theory), and analytic tools (such as monodromy).