Expander graphs, gonality and variation of Galois representations

Jordan Ellenberg, Chris Hall, Emmanuel Kowalski

Published 2010-08-22, updated 2011-09-28Version 4

We show that families of coverings of an algebraic curve where the associated Cayley-Schreier graphs form an expander family exhibit strong forms of geometric (genus and gonality) growth. Combining this general result with finiteness statements for rational points under such conditions, we derive results concerning the variation of Galois representations in one-parameter families of abelian varieties.