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:

$K_i(X) \otimes \mathbf Q = 0 \ \, i > 0.$

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^[1] has shown that the K-groups are finitely generated, while Harder's computations^[2] show that the groups are torsion. The two results together thus show Parshin's conjecture for curves.

References

1. see Grayson, Dan (1982). "Algebraic K-theory, Part I (Oberwolfach, 1980)" (PDF). Lecture Notes in Math. 966. Berlin, New York: Springer. |chapter= ignored (help)
2. Harder, Günter (1977). "Die Kohomologie S-arithmetischer Gruppen über Funktionenkörpern". Invent. Math. 42: 135–175. doi:10.1007/bf01389786.

• Kahn, Bruno (2005). "Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry". In Friedlander, Eric; Grayson, Daniel. Handbook of K-Theory I. Springer. pp. 351–428., see Conj. 51