摘要翻译：
我们讨论了有限域上曲线的Riemann假设的Enrico Bombieri的证明。重新表述，它指出定义在有限域$\f_q$上的曲线$\c$上的点数为$q+O(\sqrt{q}）$。第一个证明是由Andr\'eWeil在1942年给出的。这个证明使用了$\c\times\c$上的因子交集，使得到目前为止对原始黎曼假设的应用不成功，因为$\spec\z\times\spec\z=\spec\z$是一维的。斯捷潘诺夫于1969年发现了一种新的证明方法。这种方法在1973年被Bombieri大大简化和推广。Bombieri的方法使用$\c\times\c$上的函数，再次排除了直接转换为原始黎曼假设的证明。然而，$\c\times\c$上的两个坐标具有不同的作用，一个坐标扮演多项式变量的几何角色，另一个坐标扮演多项式系数的算术角色。$\c$的Frobenius自同构作用于$\c\乘以\c$的几何坐标。在最后一节中，我们对Nevanlinna理论如何提供一个在几何坐标上带有Frobenius作用的二维的$\spec\z\times\spec\z$模型提出了一些建议。
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英文标题：
《The Riemann Hypothesis for Function Fields over a Finite Field》
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作者：
Machiel van Frankenhuijsen
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最新提交年份：
2008
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分类信息：

一级分类：Mathematics        数学
二级分类：Number Theory        数论
分类描述：Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
素数，丢番图方程，解析数论，代数数论，算术几何，伽罗瓦理论
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一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  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.
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PDF链接：
https://arxiv.org/pdf/0806.0044