abstract: Let S be a surface, V be the Q-vector space of divisors on S modulo numerical equivalence and d be the dimension of V. The intersection product defines a non degenerate quadratic form on V. The Hodge index theorem says that it is of signature (1,d-1). In the Sixties Grothendieck conjectured a generalization of this statement to cycles of any codimension on a variety of any dimension. In characteristic zero this conjecture is a consequence of Hodge theory but in positive characteristic almost nothing is known. Instead of studying these quadratic forms at the archimedean place we will study them at p-adic places. It turns out that this question is more tractable, thanks to p-adic Hodge Theory. Moreover, using classical product formulas on quadratic forms, the p-adic result will give non-trivial informations on the archimedean place. For instance, we will prove the original conjecture for abelian fourfolds.