Colloquium: "Ordinary Primes for Abelian Varieties"

Abstract: Given a system of polynomial equations with integral coefficients, a natural question (which can not have a complete answer)  is how the number of solutions n(p) of the mod p congruences varies with the prime p.  This problem motivated Gronthendieck's construction of l-adic etale cohomology, and Deligne's proof of the Weil conjecture in 1980. Grothendieck's construction associates a convex polygon to each prime p, known as the Newton polygon.

The question of how the Newton polygon varies with the prime p is also a natural question and one that cannot be answered in general, but a conjecture of Serre predicts that the set of ordinary primes (that is, those for which the Newton polygon is as low as possible) has positive density. This conjecture is known to be true for abelian varieties with complex multiplication (Shimura--Tanayama, 1971), elliptic curves (Serre, 1977), abelian surfaces (Ogus, 1982), abelian varieties with trivial endomorphisms (Pink, 1983) and in few other instances. 

We will discuss some of the ideas behind these results and recent progress for abelian varieties with non-trivial endomorphisms, including the case of those with almost complex multiplication by an abelian number field.

This talk is based on joint work in progress with Victoria Cantoral-Farfan, Wanlin Li, Rachel Pries, and Yunqing Tang.

A tea will be served in room 200 at 3:30pm.

Host: Wanlin Li