  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.
