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:
It is named after Aleksei Nikolaevich Parshin and Alexander Beilinson.
Points and curves
The conjecture holds for a finite field by Quillen's computations of the K-groups in this case. Secondly, for a smooth proper curve, Quillen has shown that the K-groups are finitely generated, while Harder's computations show that the groups are torsion. The two results together thus show Parshin's conjecture for curves.
References
Parshin's conjecture Wikipedia(Text) CC BY-SA