2009/11/30 by Ricardo Menares
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Algebraic geometry #Context (archaeology) #Intersection (aeronautics) #Jacobian matrix and determinant #Modular curve #Modular form #Number theory #Projection (relational algebra) #math.AG #math.NT #msc:11F32 #msc:11G18 #msc:14G40
paper · pdf · doi:10.1007/s00229-011-0455-8
published as Manuscripta Mathematica, 136(3), 501-543 (2011) · 38 pages. Minor corrections
openalex publication_date 2011/04/18 · arxiv created 2013/03/06 · arxiv updated 2015/03/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
In the context of arithmetic surfaces, Bost defined a generalized Arithmetic Chow Group (ACG) using the Sobolev space L21. We study the behavior of these groups under pull-back and push-forward and we prove a projection formula. We use these results to define an action of the Hecke operators on the ACG of modular curves and to show that they are self-adjoint with respect to the arithmetic intersection product. The decomposition of the ACG in eigencomponents which follows allows us to define new numerical invariants, which are refined versions of the self-intersection of the dualizing sheaf. Using the Gross-Zagier formula and a calculation due independently to Bost and Kuehn we compute these invariants in terms of special values of L series. On the other hand, we obtain a proof of the fact that Hecke correspondences acting on the Jacobian of the modular curves are self-adjoint with respect to the Néron-Tate height pairing.