Title: 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
$\mathcal{G}_{gen}\rightarrow\mathcal{G}_{phys}$ to the "physical dynamics" of
the correspondence. 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.