The Riemann Hypothesis for Function Fields over a Finite Field
| dc.creator | van Frankenhuijsen, Machiel | |
| dc.date | 2008-05-31 | |
| dc.date | 2008-06-09 | |
| dc.date.accessioned | 2026-07-07T09:43:02Z | |
| dc.date.available | 2026-07-07T09:43:02Z | |
| dc.description | 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 André 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. | |
| dc.description | 30 pages, 2 figures all ø's are now \mathcal{O} | |
| dc.identifier | https://arxiv.org/abs/0806.0044 | |
| dc.identifier | http://arxiv.org/abs/0806.0044 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/162404 | |
| dc.subject | Number Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 11G20; 11R58, 14G15, 30D35 | |
| dc.title | The Riemann Hypothesis for Function Fields over a Finite Field | |
| dc.type | text |