The Riemann Hypothesis over Finite Fields: from Weil to the present day --- J.S. Milne   Top
Expository Notes
A Primer of Commutative Algebra
Motives---Grothendieck's Dream
What is a Shimura Variety?
Introduction to Shimura Varieties
Shimura Varieties and Moduli
Tannakian Categories
The Work of Tate


The statement of the Riemann hypothesis makes sense for all global fields, not just the rational numbers. For function fields, it has a natural restatement in terms of the associated curve. Weil's work on the Riemann hypothesis for curves over finite fields led him to state his famous "Weil conjectures", which drove much of the progress in algebraic and arithmetic geometry in the following decades. In this article, I describe Weil's work and some of the ensuing progress.


  1. Weil's work in the 1940s and 1950s
  2. Weil cohomology
  3. The standard conjectures
  4. Motives
  5. Deligne's proof of the Riemann hypothesis over finite fields
  6. The Hasse-Weil zeta function

Published in:

The Legacy of Bernhard Riemann after One Hundred and Fifty Years (Lizhen Ji, Frans Oort, Shing-Tung Yau Editors), ALM 35, 2015, pp.487-565.
Reprinted in the Notices of the International Congress of Chinese Mathematicians (ICCM Notices), Vol. 4, No. 2 (2016), pp. 14-52.

First posted 02.07.15, 65 pages.
Final version 19.07.15. Two errors and some misprints fixed; slightly expanded; 67 pages.
Current version 02.09.15. Added comma; other very minor edits.
Current version 14.09.15. Added translation of letter of Weil. Except for the page numbering, this is essentially the same as the published version.