Giuseppe Ancona: On the standard conjecture of Hodge type for abelian fourfolds

Let S be a surface and V be the Q-vector space of divisors on S modulo numerical equivalence. The intersection product defines a non degenerate quadratic form on V. We know since the Thirties that it is of signature (1,n). 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 an easy 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. Moreover, using a classical product formula on quadratic forms, the p-adic result will give us non-trivial informations on the archimedean place. For instance, we will prove the original conjecture for abelian fourfolds.