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In the context of arithmetic surfaces, J.-B. Bost defined a generalized Arithmetic Chow Group (ACG) using the Sobolev space $L_1^{2}$. 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. |