Examples for the standard conjecture of Hodge type

Abstract

For each prime number $p$ and each integer $g⩾4$, 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 (2021). 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 (2021) can be applied to. Along the way, we prove results of independent interest about Honda–Tate theory and about multiplicative relations between algebraic integers.