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

↑ see Grayson, Dan (1982). "Finite generation of K-groups of a curve over a finite field (after Daniel Quillen)". Algebraic K-theory, Part I (Oberwolfach, 1980) (PDF). Lecture Notes in Math. 966. Berlin, New York: Springer.
↑ 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

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[1] see Grayson, Dan (1982). "Finite generation of K-groups of a curve over a finite field (after Daniel Quillen)". Algebraic K-theory, Part I (Oberwolfach, 1980) (PDF). Lecture Notes in Math. 966. Berlin, New York: Springer.

[2] Harder, Günter (1977). "Die Kohomologie S-arithmetischer Gruppen über Funktionenkörpern". Invent. Math. 42: 135–175. doi:10.1007/bf01389786.