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:

$K_i(X)\otimes 𝐐=0,i>0.$

It is named after Aleksei Nikolaevich Parshin and Alexander Beilinson.

Finite fields

The conjecture holds if $dimX=0$ by Quillen's computation of the K-groups of finite fields, showing in particular that they are finite groups.

Curves

The conjecture holds if $dimX=1$ 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 $dimX=1$.