Generalized Serre–Tate theory

Abstract In late sixties Serre and Tate developed a theory of ordinary abelian varieties

(or p-divisible groups) over perfect fields of characteristic p, and of their canonical lifts. In particular they proved that any canonical lift of an ordinary abelian variety over a finite field has complex multiplication. In the last six years we developed a generalized Serre–Tate theory: we considered abelian varieties with some families of cycles. In more technical words, we considered universal abelian schemes over the different integral canonical models of Shimura varieties of Hodge type, we mentioned in the last talk.

As part of this theory, we obtained among other things Shimura-ordinary abelian varieties (or p-divisible groups), and Shimura-canonical lifts. The most technical and beautiful part of the whole theory is related to the following problem (implied by the Hodge conjecture): a Shimura-canonical lift of a Shimura-ordinary abelian variety over a finite field has complex multiplication. We report on our solution to this problem, while presenting the similarities between the classical and generalized Serre–Tate theory; we will review briefly what complex multiplication is. Again, for the proof, we rely heavily on Satake’s work mentioned in the last talk.

We assume just familiarity with finite fields (and their algebraic closures) and reductive groups.