Parshin’s conjecture for smooth projective schemes over finite fields

Prove that for every smooth projective scheme Y over a finite field, the rational higher Chow groups CH^r(Y,j)⊗_Z Q vanish for all integers r and all positive integers j.

Background

Parshin’s conjecture is used to eliminate the uniquely divisible summands that occur in the paper’s structure theorem for higher Chow groups over finite fields. Under the conjecture, the remaining groups in the relevant ranges become finite or vanish.

The paper notes that the conjecture is known for smooth proper curves over finite fields, but it is stated in general for smooth projective schemes.

References

Let $Y$ be a smooth projective scheme over a finite field. Then $CHr(Y,j)\otimes_\ZQ=0$ for every $r\inZ$ and every $j>0$.

— Divisibility and torsion in higher Chow groups over arithmetic fields  (2609.11178 - Hiranouchi et al., 10 Sep 2026) in Conjecture 3.1, Section 3, subsection “Finite fields”