Hodge Index Theorem for Arithmetic Divisors with Degenerate Green Functions

As we indicated in our paper [9], the standard arithmetic Chow groups introduced by Gillet-Soulé [3] are rather restricted to consider arithmetic analogues of geometric problems. In this note, we would like to propose a suitable extension of the arithmetic Chow group of codimension one, in which the Hodge index theorem still holds as in papers [1], [7] and [14]. Let X → Spec(Z) be a regular arithmetic variety with d = dimXQ. As we defined in [9], ĈH p D(X) is a group, consisting of pairs (Z, g) with cycles Z of codimension p on X and currents g of type (p−1, p−1) on X(C), modulo arithmetical rational equivalence. It seems