Varieties over $\bar{\mathbb{Q}}$ with infinite Chow groups modulo almost all primes

Abstract

Let $E$ be the Fermat cubic curve over $\bar{\mathbb{Q}}$. In 2002, Schoen proved that the group $CH^2(E^3)/\ell$ is infinite for all primes $\ell\equiv 1\pmod 3$. We show that $CH^2(E^3)/\ell$ is infinite for all prime numbers $\ell> 5$. This gives the first example of a smooth projective variety $X$ over $\bar{\mathbb{Q}}$ such that $CH^2(X)/\ell$ is infinite for all but at most finitely many primes $\ell$. A key tool is a recent theorem of Farb--Kisin--Wolfson, whose proof uses the prismatic cohomology of Bhatt--Scholze.