The Hilbert symbol in the Hodge standard conjecture

TL;DR

This work proves that for varieties over finite fields admitting a CM lifting, the Hodge standard conjecture signature holds modulo $4$. The authors separate the problem into a discriminant calculation using $ ext{ell}$-adic methods and a Hilbert-symbol calculation via $p$-adic Hodge theory, ultimately reducing to CM-simple motives and their CM-quadratic forms over $p$-adic fields. A central achievement is constructing and analyzing $p$-adic periods $oldsymbol{ extlambda}$, relating them to Lubin--Tate periods, and showing $oldsymbol{ extlambda}oldsymbol{ extlambda}^*$ is not a norm from $F$ to $F_0$, which drives the key local-global obstruction vanishing. The results yield that the negative part of the intersection form is divisible by $4$, with concrete corollaries for abelian varieties and products of CM K3 surfaces, highlighting a deep link between CM lifting, $p$-adic period theory, and the Hodge standard conjecture in positive characteristic.

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.

Theorems & Definitions (107)

• Definition 1.1
• Conjecture 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Conjecture 3.1
• Theorem 3.2
• Remark 3.3
• Lemma 3.4