Inventiones mathematicae

, Volume 23, Issue 1, pp 73–77 | Cite as

Some consequences of the Riemann hypothesis for varieties over finite fields

  • Nicholas M. Katz
  • William Messing


We deduce from Deligne's form of the Riemann hypothesis and the hard Lefschetz theoreminl-adic cohomology the corresponding facts for any “reasonable” cohomology theory, in particular for crystalline cohomology, and give some applications to algebraic cycles.


Finite Field Riemann Hypothesis Cohomology Theory Algebraic Cycle Crystalline Cohomology 
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  1. 1.
    Deligne, P.:Lectures at the International Geometry Colloquium. July 4–11, 1973 Cambridge,EnglandGoogle Scholar
  2. 2.
    Berthelot, P.: Le théorème de Lefschetz ≪Faible≫ en Cohomologie Cristalline. To appearGoogle Scholar
  3. 3.
    Kleiman, S.: Algebraic Cycles and the Weil Conjectures, in Dix Exposés sur la Cohomologie des Schemas, pp. 359–386. Amsterdam: North-Holland 1969Google Scholar

Copyright information

© Springer-Verlag 1974

Authors and Affiliations

  • Nicholas M. Katz
    • 1
  • William Messing
    • 2
  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsMITCambridgeUSA

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