#### Vol. 12, No. 5, 2018

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Correspondences without a core

### Raju Krishnamoorthy

Vol. 12 (2018), No. 5, 1173–1214
##### 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 ${\mathsc{G}}_{gen}$ together with a large group of “algebraic” automorphisms $A$. The graph ${\mathsc{G}}_{gen}$ measures the “generic dynamics” of the correspondence. We construct specialization maps ${\mathsc{G}}_{gen}\to {\mathsc{G}}_{phys}$ 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.

##### Keywords
Shimura curves, special points, correspondences, dynamics
##### Mathematical Subject Classification 2010
Primary: 14G35
Secondary: 05C25, 14H05, 37P55