Correspondences without a core

Krishnamoorthy, Raju

doi:10.2140/ant.2018.12.1173

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Correspondences without a core

Raju Krishnamoorthy

Vol. 12 (2018), No. 5, 1173–1214

DOI: 10.2140/ant.2018.12.1173

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 $G_{gen}$ together with a large group of “algebraic” automorphisms $A$ . The graph $G_{gen}$ measures the “generic dynamics” of the correspondence. We construct specialization maps $G_{gen} \to G_{phys}$ 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

Milestones

Received: 10 May 2017

Revised: 16 January 2018

Accepted: 29 March 2018

Published: 31 July 2018

Authors

Raju Krishnamoorthy
Freie Universität Berlin Germany

Vol. 12, No. 5, 2018