We will begin with a historical survey of (a) the bilinear relations of the periods of a Riemann surface and (b)the question on the existence of enough meromorphic functions on quotients of a finite dimensional vector spaces by a co-compact lattice. These questions led to the notion of abelian varieties. We will discuss the classification problem for abelian varieties of a given dimension, first over the field of complex numbers, then over an arbitrary base field after extending the definition of abelian varieties to general fields. It turns out that the parameter spaces, or moduli spaces, of abelian varieties are themselves algebraic varieties. These moduli spaces have a lot of symmetries, known as Hecke symmetries.

The second half of the talk we will focus on the Hecke symmetries over a base field of positive characteristic p>0. There are more invariants for abelian varieties in characteristic p; they all come from the p-divisible group attached to abelian varieties. We will explore subvarieties of moduli spaces defined by p-adic invariants, and the close interaction of these subvarieties with Hecke symmetries.

### Penn Mathematics Colloquium

Wednesday, March 15, 2017 - 3:30pm

#### Ching-Li Chai

University of pennsylvania