Examples for the standard conjecture of Hodge type

TL;DR

The paper addresses the existence and construction of abelian varieties over finite fields that satisfy the standard conjecture of Hodge type while possessing Tate classes not Lefschetz and not liftable to characteristic zero. It combines Honda–Tate theory, CM-motive decomposition and Ancona’s rank-2 criterion to produce infinitely many examples for all primes $p$ and even dimensions $g\ge 4$, organized around mildly exotic behaviour. A key methodological innovation is the Weil-number minimality analysis and a Galois-module framework for multiplicative relations among Frobenius eigenvalues, enabling a classification of exotic summands and their liftability. The results significantly expand the known instances where the standard conjectures hold in positive characteristic and provide a concrete strategy to generate further mildly exotic abelian varieties with prescribed CM-type and endomorphism structures.

Abstract

For each prime number $p$ and each integer $g \geqslant 5$, we construct infinitely many abelian varieties of dimension $g$ over $\overline{\mathbb{F}}_p$ satisfying the standard conjecture of Hodge type. The main tool is a recent theorem of Ancona on certain rank $2$ motives. These varieties are constructed explicitly through Honda-Tate theory. Moreover, they have Tate classes that are not generated by divisors nor liftable to characteristic zero. Also, we prove a result towards a classification of simple abelian varieties for which the result of Ancona can be applied to. Along the way, we prove results of independent interest about Honda-Tate theory and about multiplicative relations between algebraic integers.

Theorems & Definitions (152)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Conjecture 2.2: Standard conjecture of Hodge type Gro68
• Definition 2.3: Lefschetz cycles
• Proposition 2.4
• Definition 2.5: Tate classes and potential Tate classes
• Definition 2.6: Algebraic classes