Correspondences in Arakelov geometry and applications to the case of Hecke operators on modular curves

Abstract

In the context of arithmetic surfaces, 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 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 Kühn 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.where Eis is the space spanned by compactified irreducible components of fibers, and (1.5) J ∼ = J 0 (N )(Q) ⊗ R as real vector spaces. It is possible to choose the isomorphism in (1.5) such that the Hecke actions on both sides are compatible (Corollary 4.11). Hence we may further decomposewhere f runs through a basis of S 2 Γ 0 (N ) consisting of eigenforms.Let ω be the dualising sheaf on X 0 (N ). By taking the divisor induced by a section of this sheaf with an appropriate choice of L 2 1 -Green function (cf. Section 4.4), we obtain a classω ∈ CH(N ). Following the decompositions (1.4) and (1.6) we can then writê ω =ω Eis +ω J ,ω J = fω f . Functoriality in L 2