Expander graphs, gonality and variation of Galois representations

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.

32 pages; v4: changes mostly in exposition

References in corpus (4)

• Growth in finite simple groups of Lie type of bounded rank
• Expansion in $S{L}_d(O_K/I)$, $I$ square-free
• Generalization of Selberg's 3/16 Theorem and Affine Sieve
• Un critère d'indépendance pour une famille de représentations l-adiques

Cited by in corpus (10)

• Growth in finite simple groups of Lie type of bounded rank
• Small Height and Infinite Non-Abelian Extensions
• l-independence for Compatible Systems of (mod l) Representations
• Super-approximation, I: p-adic semisimple case
• Super-approximation, II: the p-adic and bounded power of square-free integers cases
• Expander Graphs in Pure and Applied Mathematics
• Level structures on abelian varieties, Kodaira dimensions, and Lang's conjecture
• Growth in groups: ideas and perspectives
• A combinatorial Li-Yau inequality and rational points on curves
• An effective open image theorem for abelian varieties