  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.
