  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
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
N\'eron-Tate height pairing.
