Expander graphs, gonality and variation of Galois representations

TL;DR

This paper demonstrates that families of algebraic curve coverings with expander Cayley-Schreier graphs lead to significant geometric growth and influence the variation of Galois representations in abelian varieties.

Contribution

It establishes a link between expander graph properties and geometric growth, and applies this to study Galois representation variation in algebraic families.

Findings

Families with expander Cayley-Schreier graphs show strong geometric growth.

Results imply finiteness of rational points under certain conditions.

Provides new insights into Galois representation variation in algebraic geometry.

Abstract

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.