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