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
together with a large group of “algebraic” automorphisms
. The
graph
measures the “generic dynamics” of the correspondence. We construct specialization
maps
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.
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