The Hilbert symbol in the Hodge standard conjecture

We study the Hodge standard conjecture for varieties over finite fields admitting a CM lifting, such as abelian varieties or products of K3 surfaces. For those varieties we show that the signature predicted by the conjecture holds true modulo 4 . This amounts to determining the discriminant and the Hilbert symbol of the intersection product. The first is obtained by \ell -adic arguments whereas the second needs a careful computation in p -adic Hodge theory.