pdf Except for the page numbering, this is essentially the same as the published version.

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

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 as:

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.

Clarification

On p.43 I write "For many decades, it was believed that algebraic equivalence and numerical equivalence coincide". In January 2022, I asked David Mumford about this. He responded: "I don't think many people thought that hom=alg. Severi's ideas were in very bad repute in the 60's. And also, I believe Grothendieck was susceptible to sweeping conjectures before he dug more deeply into some area."

History

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.