The Riemann Hypothesis for Function Fields over a Finite Field

TL;DR

This paper reviews Bombieri's proof of the Riemann hypothesis for function fields over finite fields, highlighting its methods, limitations, and potential connections to Nevanlinna theory for a broader geometric understanding.

Contribution

It explains Bombieri's novel approach using functions on the product of a curve with itself and explores how Nevanlinna theory might model the spectrum of integers in a two-dimensional setting.

Findings

Bombieri's proof confirms the Riemann hypothesis for function fields.

The method involves intersection theory on curves over finite fields.

Potential links to Nevanlinna theory suggest new geometric perspectives.

Abstract

We discuss Enrico Bombieri's proof of the Riemann hypothesis for curves over a finite field. Reformulated, it states that the number of points on a curve $\C$ defined over the finite field $\F_q$ is of the order $q+O(\sqrt{q})$. The first proof was given by Andr\'e Weil in 1942. This proof uses the intersection of divisors on $\C\times\C$, making the application to the original Riemann hypothesis so far unsuccessful, because $\spec\Z\times\spec\Z=\spec\Z$ is one-dimensional. A new method of proof was found in 1969 by S. A. Stepanov. This method was greatly simplified and generalized by Bombieri in 1973. Bombieri's method uses functions on $\C\times\C$, again precluding a direct translation to a proof of the original Riemann hypothesis. However, the two coordinates on $\C\times\C$ have different roles, one coordinate playing the geometric role of the variable of a polynomial, and the other coordinate the arithmetic role of the coefficients of this polynomial. The Frobenius automorphism of $\C$ acts on the geometric coordinate of $\C\times\C$. In the last section, we make some suggestions how Nevanlinna theory could provide a model of $\spec\Z\times\spec\Z$ that is two-dimensional and carries an action of Frobenius on the geometric coordinate.