The Hilbert symbol in the Hodge standard conjecture

  • Giuseppe Ancona

    Université de Strasbourg et CNRS, Strasbourg, France
  • Adriano Marmora

    Université de Strasbourg et CNRS, Strasbourg, France
The Hilbert symbol in the Hodge standard conjecture cover
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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