Correspondences without a core

We study the formal properties of correspondences of curves without a core, focusing on the case of etale correspondences. The motivating examples come from Hecke correspondences of Shimura curves. Given a correspondence without a core, we construct an infinite graph G(gen) together with a large group of algebraic automorphisms A. The graph G(gen) measures the generic dynamics of the correspondence. We construct specialization maps G(gen) -> 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 etale 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.