%0 Journal Article %T Stable models of Hecke operators %A Jan Vonk %J Mathematics %D 2015 %I arXiv %X 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. %U http://arxiv.org/abs/1501.03488v2