Correspondences without a core

Abstract

We study the formal properties of correspondences of curves without a core, focusing on the case of étale correspondences. The motivating examples come from Hecke correspondences of Shimura curves. Given a correspondence without a core, we construct an infinite graph ${\mathcal{G}}_{gen}$ together with a large group of “algebraic” automorphisms $A$. The graph ${\mathcal{G}}_{gen}$ measures the “generic dynamics” of the correspondence. We construct specialization maps ${\mathcal{G}}_{gen}\to {\mathcal{G}}_{phys}$ to the “physical dynamics” of the correspondence. Motivated by the abstract structure of the supersingular locus, we also prove results on the number of bounded étale orbits, in particular generalizing a recent theorem of Hallouin and Perret. We use a variety of techniques: Galois theory, the theory of groups acting on infinite graphs, and finite group schemes.