# Correspondences without a Core

## TL;DR

This paper investigates the properties of correspondences between curves without a core, especially étale correspondences, constructing infinite graphs to analyze their dynamics and generalizing existing theorems on étale orbits.

## Contribution

It introduces a framework for studying correspondences without a core using infinite graphs and automorphisms, extending understanding of their dynamics and orbit structures.

## Key findings

- Constructed an infinite graph $\
- Generalized a theorem of Hallouin and Perret on étale orbits.
- Established connections between algebraic automorphisms and graph dynamics.

## Abstract

We study the formal properties of correspondences of curves without a core, focusing on the case of \'{e}tale 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 \'{e}tale 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.