MATH Seminar

Title: Abelian varieties with big monodromy
Seminar: Algebra and Number Theory
Speaker: David Zureick-Brown of Emory University
Contact: TBA
Date: 2011-10-13 at 3:00PM
Venue: MSC E406
Download Flyer
Abstract: Serre proved in 1972 that the image of the adelic Galois representation associated to an elliptic curve E without complex multiplication has open image; moreover, he also proved that for an elliptic curve over Q the index of the image is always divisible by 2 (and in particular never surjective). More recently, Greicius in his thesis gave criteria for surjectivity and gave an explicit example of an elliptic curve E over a number field K with surjective adelic representation. Soon after, Zywina, building on earlier work of Duke, Jones, and others, proved that the adelic image `random' elliptic curve is maximal. In this talk I will explain recent work with David Zywina in which we generalize these theorems and prove that a random abelian variety in a family with big monodromy has maximal image of Galois. I'll explain the analytic and geometric techniques used in previous work and the new geometric ideas -- in particular, Nori's method of semistable approximation-- needed to generalized to higher dimension.