Stable models of Hecke operators

TL;DR

This paper studies the geometry of correspondences between curves over non-Archimedean fields, providing stable models of Hecke operators and extending the theory of canonical subgroups with new geometric insights.

Contribution

It proves that correspondences over non-Archimedean fields have potentially stable reduction, generalizing previous results, and explicitly constructs stable models of Hecke operators on Shimura curves.

Findings

Correspondences have potentially stable reduction over non-Archimedean fields.

Explicit stable models of Hecke operators on quaternionic Shimura curves are constructed.

Generalization of the geometric theory of canonical subgroups by Goren and Kassaei.

Abstract

We investigate the geometry of correspondences between curves, and prove that correspondences over a non-Archimedean valued field have potentially stable reduction, generalising and strengthening results of Coleman and Liu. This yields a concrete description of the operator on the cohomology of the generic fibres arising from linearisation of the correspondence, via the weight-monodromy filtration and Picard-Lefschetz theory. We explicitly determine stable models of Hecke operators on various quaternionic Shimura curves, and prove a generalisation of the geometric theory of canonical subgroups by Goren and Kassaei.