Correspondences in Arakelov geometry. Applications to Hecke operators on modular curves

Details Correspondences in Arakelov geometry. Applications to Hecke operators on modular curves Correspondences in Arakelov geometry. Applications to Hecke operators on modular curves

2009-11-03

arxiv.org/abs/0911.0546v2

Mathematics

Number Theory

38 pages. Grammar corrections, added references

Scientific paper

In the context of arithmetic surfaces, J.-B. Bost defined a generalized Arithmetic Chow Group (ACG) using the Sobolev space L^2_1. 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 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 to U. Kuehn we compute these invariants in terms of special values of L series.

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