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arXiv · 1704.00335

Correspondences without a Core

Raju Krishnamoorthy
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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.

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Raju Krishnamoorthy. 2017-05-12. Correspondences without a Core. https://doi.org/10.2140/ant.2018.12.1173

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