• Peter Roquette
Part of the Lecture Notes in Mathematics book series (LNM, volume 2222)


With the appearance of Weil’s above mentioned three books, the RHp was settled and our story comes to an end. But the mathematical development inspired by this or that item of our story persists and is still present. From the numerous literature in this direction I will mention here three papers only:


  1. [Bom74]
    E. Bombieri, Counting points on curves over finite fields (d’apres S.A. Stepanov), Sem. Bourbaki 1972/1973. Expose No. 430, Lecture Notes in Mathematics, vol. 383 (Springer, Berlin, 1974), pp. 234–241Google Scholar
  2. [FJ08]
    M.D. Fried, M. Jarden, Field Arithmetic, 3rd edn. (Springer, Berlin, 2008), xiv+792 pp. Revised edn. by Moshe Jarden (2008)Google Scholar
  3. [FK88]
    E. Freitag, R. Kiehl, Étale Cohomology and the Weil Conjecture. With a Historical Introduction by J. A. Dieudonné (Springer, Berlin, 1988)CrossRefGoogle Scholar
  4. [Gro58]
    A. Grothendieck, Sur une note de Mattuck-Tate. J. Reine Angew. Math. 200, 208–215 (1958)MathSciNetzbMATHGoogle Scholar
  5. [Mil16]
    J.S. Milne, The Riemann hypothesis over finite fields - from Weil to the present day, in The Legacy of Bernhard Riemann After One Hundred and Fifty Years, vol. II (International Press Somerville, MA, 2016), pp. 487–565; (Higher Education Press, Beijing, 2016)MathSciNetCrossRefGoogle Scholar
  6. [MT58]
    A. Mattuck, J. Tate, On the inequality of Castelnuovo-Severi. Abh. Math. Semin. Univ. Hamburg 22, 295–299 (1958)MathSciNetCrossRefGoogle Scholar
  7. [OS16]
    F. Oort, N. Schappacher, Early history of the Riemann hypothesis in positive characteristic, in The Legacy of Bernhard Riemann After One Hundred and Fifty Years, vol. II (International Press, Somerville, MA, 2016), pp. 595–631; (Higher Education Press, Beijing, 2016)Google Scholar
  8. [Ste69]
    S.A. Stepanov, Über die Anzahl der Punkte einer hyperelliptischen Kurve über einem einfachen endlichen Körper. Izv. Akad. Nauk SSSR, Ser. Mat. 33, 1171–1181 (1969)Google Scholar

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Authors and Affiliations

  • Peter Roquette
    • 1
  1. 1.Mathematical InstituteHeidelberg UniversityHeidelbergGermany

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