The Hilbert symbol in the Hodge standard conjecture
Giuseppe Ancona
Université de Strasbourg et CNRS, Strasbourg, FranceAdriano Marmora
Université de Strasbourg et CNRS, Strasbourg, France

Abstract
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 . This amounts to determining the discriminant and the Hilbert symbol of the intersection product. The first is obtained by -adic arguments whereas the second needs a careful computation in -adic Hodge theory.
Cite this article
Giuseppe Ancona, Adriano Marmora, The Hilbert symbol in the Hodge standard conjecture. Comment. Math. Helv. (2025), published online first
DOI 10.4171/CMH/591