On the Picard numbers of abelian varieties in arbitrary characteristic

In this talk, I will introduce and study the set R_g^{(p)} of Picard numbers of abelian varieties of dimension g over algebraically closed fields of characteristic p \geq 0 . The main questions that I will address are:
- Which positive integers occur as the Picard number of an abelian variety of fixed dimension?
- Does the Picard number impact the structure of an abelian variety? If yes, how?

Many of the results that I will discuss hold in any characteristic, but there are a few exceptions and extra twists in positive characteristic, as one might expect. The picture over fields of characteristic zero is completely described, while the situation in positive characteristic still presents some questions yet to be answered.