In mathematics, more specifically in algebraic geometry, Parshin's conjecture (also referred to as the BeilinsonâÂÂParshin conjecture) states that for any smooth projective variety X defined over a finite field, the higher algebraic K-groups vanish up to torsion:
It is named after Aleksei Nikolaevich Parshin and Alexander Beilinson.
The conjecture holds if by Quillen's computation of the K-groups of finite fields, showing in particular that they are finite groups.
The conjecture holds if by the proof of Corollary 3.2.3 of Harder. Additionally, by Quillen's finite generation result (proving the Bass conjecture for the K-groups in this case) it follows that the K-groups are finite if .