The Riemann Hypothesis over Finite Fields: From Weil to the Present Day

TL;DR

This paper reviews the development of the Riemann hypothesis for function fields over finite fields, highlighting key contributions from Weil to modern proofs and related conjectures in algebraic geometry.

Contribution

It provides a comprehensive overview of the historical and mathematical progress from Weil's initial work to Deligne's proof, emphasizing the evolution of related theories.

Findings

Weil's formulation of the Riemann hypothesis for curves over finite fields.

Development of Weil cohomology theories and their applications.

Deligne's proof of the Riemann hypothesis in this context.

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.