Notices of the International Congress of Chinese Mathematicians

Volume 4 (2016)

Number 2

The Riemann hypothesis over finite fields: from Weil to the present day

Pages: 14 – 52

DOI: http://dx.doi.org/10.4310/ICCM.2016.v4.n2.a4

Author

James S. Milne (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)

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.

2010 Mathematics Subject Classification

01A60, 11-03, 14-03

Full Text (PDF format)