The Riemann Hypothesis over Finite Fields: From Weil to the Present Day
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: Weil cohomology (etale, crystalline); Grothendieck's standard conjectures; motives; Deligne's proof; Hasse-Weil zeta functions and Langlands functoriality.
Keywords
Cite
@article{arxiv.1509.00797,
title = {The Riemann Hypothesis over Finite Fields: From Weil to the Present Day},
author = {James Milne},
journal= {arXiv preprint arXiv:1509.00797},
year = {2021}
}
Comments
This is my contribution to the book "The legacy of Bernhard Riemann after one hundred and fifty years", edited by S.T. Yau et al