Time to Pick Up Our Heads and Look Inland

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/152483/1/lob10342_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/152483/2/lob10342.pd

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

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.Comment: Added references and Corollary 1.5. 17 page