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

Abstract Let be the Fermat cubic curve over . In 2002, Schoen proved that the group is infinite for all primes . We show that is infinite for all prime numbers . This gives the first example of a smooth projective variety over such that is infinite for all but at most finitely many primes . A key tool is a recent theorem of Farb–Kisin–Wolfson, whose proof uses the prismatic cohomology of Bhatt–Scholze.