MATH Seminar

Title: Expander families and variation of Galois representations
Seminar: Algebra
Speaker: Chris Hall of University of Wyoming
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2013-10-02 at 4:00PM
Venue: W306
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Abstract: Given a pair of curves U,V over the complex numbers, one can associate to a finite unramified map $V \to U$ a finite Cayley-Schreier graph. In this talk we consider families of maps $V_i \to U$ indexed by a parameter i such that the family of associated graphs is an expander family. As we will explain, the expander hypothesis has remarkable geometric implications, e.g. the set of $V_i$ such that the gonality of $V_i$ is less than your favorite positive number N is finite. We will also explain some of the arithmetic implications, e.g. for all but finitely many $V_i$, there are only many points on $V_i$ defined over some extension of K of degree at most N. As one an application, we can derive results on the variation of Galois representations in a one-parameter family of abelian varieties defined over a number field.