Parshin's conjecture

In mathematics, more specifically in algebraic geometry, Parshin's conjecture (also referred to as the Beilinson–Parshin conjecture) states that for any smooth projective variety X defined over a finite field, the higher algebraic K-groups vanish up to torsion:^[1]

${\displaystyle K_{i}(X)\otimes \mathbf {Q} =0,\ \,i>0.}$

It is named after Aleksei Nikolaevich Parshin and Alexander Beilinson.

Finite fields

The conjecture holds if ${\displaystyle dim\ X=0}$ by Quillen's computation of the K-groups of finite fields,^[2] showing in particular that they are finite groups.

Curves

The conjecture holds if ${\displaystyle dim\ X=1}$ by the proof of Corollary 3.2.3 of Harder.^[3] Additionally, by Quillen's finite generation result^[4] (proving the Bass conjecture for the K-groups in this case) it follows that the K-groups are finite if ${\displaystyle dim\ X=1}$.