Parshin’s conjecture revisited

Abstract

Parshin’s conjecture states that Ki(X)ℚ = 0 for i > 0 and X smooth and projective over a finite field Fq. The purpose of this paper is to break up Parshin’s conjecture into several independent statements, in the hope that each of them is easier to attack individually. If CHn(X,i) is Bloch’s higher Chow group of cycles of relative dimension n, then in view of Ki(X)ℚ ≅ ⊕n CHn(X,i)ℚ, Parshin’s conjecture is equivalent to Conjecture P(n) for all n, stating that CHn(X,i)ℚ = 0 for i > 0, and all smooth and projective X. We show, assuming resolution of singularities, that Conjecture P(n) is equivalent to the conjunction of three conjectures A(n), B(n) and C(n), and give several equivalent versions of these conjectures.