2008/05/31 by Machiel van Frankenhuijsen, van Frankenhuijsen, Machiel
Mathematics · #11G20 #11R58 #14G15 #30D35 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11G20 #msc:11R58 #msc:14G15 #msc:30D35
paper · pdf · doi:10.48550/arxiv.0806.0044
30 pages, 2 figures all ø's are now \mathcal{O}
arxiv created 2008/06/09 · arxiv updated 2009/12/01
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 \Fq is of the order q+O(√(q)). The first proof was given by André Weil in 1942. This proof uses the intersection of divisors on \C×\C, making the application to the original Riemann hypothesis so far unsuccessful, because \spec\Z×\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×\C, again precluding a direct translation to a proof of the original Riemann hypothesis. However, the two coordinates on \C×\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×\C. In the last section, we make some suggestions how Nevanlinna theory could provide a model of \spec\Z×\spec\Z that is two-dimensional and carries an action of Frobenius on the geometric coordinate.