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:Conjecture 51 in {{cite book|title=Handbook of K-Theory I|year=2005|publisher=Springer|pages=351–428|author=Kahn, Bruno|editor1=Friedlander, Eric |editor2=Grayson, Daniel |chapter=Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry}}
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It is named after Aleksei Nikolaevich Parshin and Alexander Beilinson.
Finite fields
The conjecture holds if by Quillen's computation of the K-groups of finite fields,{{cite journal|last=Quillen|first=Daniel|title=On the cohomology and K-theory of the general linear groups over a finite field|journal=Ann. of Math.|volume=96|year=1972|pages=552–586}} showing in particular that they are finite groups.
Curves
The conjecture holds if by the proof of Corollary 3.2.3 of Harder.{{cite journal|last=Harder|first=Günter|title=Die Kohomologie S-arithmetischer Gruppen über Funktionenkörpern|journal=Invent. Math.|year=1977|volume=42|pages=135–175|doi=10.1007/bf01389786|bibcode=1977InMat..42..135H }}
Additionally, by Quillen's finite generation result{{cite book|last=Grayson|first=Dan|title=Algebraic K-theory, Part I (Oberwolfach, 1980)|chapter=Finite generation of K-groups of a curve over a finite field (after Daniel Quillen)|url=http://www.math.uiuc.edu/~dan/Papers/FiniteGeneration.pdf|year=1982|volume=966|series=Lecture Notes in Math.|publisher=Springer|location=Berlin, New York}} (proving the Bass conjecture for the K-groups in this case) it follows that the K-groups are finite if .
References
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