Stable models of Hecke operators

2015/01/14 by Jan Vonk, Vonk, Jan
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematics #Number Theory (math.NT) #Pure mathematics #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #math.AG #math.NT

paper · pdf · doi:10.48550/arxiv.1501.03488

19 pages, comments and suggestions are warmly welcomed; Updated version with some added material

openalex publication_date 2015/01/14 · arxiv created 2015/05/18 · arxiv updated 2015/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Groupes de Monodromie en Géométrie Algébrique
• On modular representations of ( Q/Q) arising from modular forms
• Théorie de Hodge : III
• The irreducibility of the space of curves of given genus

Related

• The Frobenius and monodromy operators for curves and abelian varieties
• Quaternions and Kudla's matching princple
• A general framework for the analytic Langlands correspondence
• Hecke orbits on Shimura varieties of Hodge type
• On the cohomology of certain non-compact Shimura varieties (with an appendix by Robert Kottwitz)