The Hilbert symbol in the Hodge standard conjecture

TL;DR

This paper investigates the Hodge standard conjecture for certain varieties over finite fields with CM liftings, providing partial verification of the conjecture modulo 4 through intersection form analysis.

Contribution

It introduces a novel approach combining $ ext{l}$-adic and $p$-adic Hodge theory to analyze the intersection form and Hilbert symbol in the context of the Hodge standard conjecture.

Findings

Signature matches the conjecture modulo 4

Discriminant determined via $ ext{l}$-adic methods

Hilbert symbol computed using $p$-adic Hodge theory

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 $4$. 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.