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

Let $E$ be the Fermat cubic curve over $\overline{Q}$. In 2002, Schoen proved that the group $C{H}^2\left(E^3\right)/\ell$ is infinite for all primes $\ell \equiv 1\left(\mathrm{mod}3\right)$. We show that $C{H}^2\left(E^3\right)/\ell$ is infinite for all prime numbers $\ell >5$. This gives the first example of a smooth projective variety $X$ over $\overline{Q}$ such that $C{H}^2\left(X\right)/\ell$ 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.