### Abstract

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.

### Contents

- Weil's work in the 1940s and 1950s
- Weil cohomology
- The standard conjectures
- Motives
- Deligne's proof of the Riemann hypothesis over finite fields
- 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.

It will be reprinted in the Notices of the International Congress
of Chinese Mathematicians (December 2016 issue).

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.