Géométrie Algébrique/algebraic Geometry Analytic Invariants in Arakelov Theory for Curves Invariants Analytiques Arakeloviens Pour Les Courbes

Arakelov theory for Riemann surfaces is based on two analytic invariants: the Green function and Faltings δ invariant. Both invariants are hard to compute and they are only known in a few cases (cf. [3],[1]). They are related by a formula of Faltings, which also involves the theta-function on the jacobian of the curve. In any case, it is an ineffective relation, because three of the four terms involved cannot be computed in general. We introduce a function ||J || on the jacobian of the curve, which gives a new relation between the Green function and the Faltings δ invariant. Our ||J || function is easily computable because it is defined in terms of derivatives of the theta-function.