arXiv 0907.36642009-07math.NT

关于有限域扩张中代数曲线上点数的分布

On the Distribution of the Number of Points on Algebraic Curves in Extensions of Finite Fields

Omran Ahmadi, Igor E. Shparlinski

备注：: 14 pages

摘要：

设$\cC$是定义在$q$元素的有限域$\F_q$上的亏格$g \ge 1$的光滑绝对不可约曲线.设$# \cC(\F_{q^n})$为$\cC$上$\F_{q^n}$ -有理点的个数。在$\cC$的ζ-函数的根满足一定的乘法独立性条件下，得到了使得$(# \cC(\F_{q^n}) - q^n -1)/2gq^{n/2}$属于给定区间$\cI \subseteq [-1,1]$的$n =1, ..., N$个数的渐近公式.这可以被认为是Sato-Tate分布的类似物，它涵盖了当曲线$\E$被定义在$\Q$上并被认为是模连续素数$p$时的情况，尽管在我们的场景中分布函数是不同的。最近，E.科瓦尔斯基在统计设置。它对普通椭圆曲线是平凡满足的，我们也对一类自然的$g=2$曲线族建立了它。

英文摘要：

Let $\cC$ be a smooth absolutely irreducible curve of genus $g \ge 1$ defined over $\F_q$, the finite field of $q$ elements. Let $# \cC(\F_{q^n})$ be the number of $\F_{q^n}$-rational points on $\cC$. Under a certain multiplicative independence condition on the roots of the zeta-function of $\cC$, we derive an asymptotic formula for the number of $n =1, ..., N$ such that $(# \cC(\F_{q^n}) - q^n -1)/2gq^{n/2}$ belongs to a given interval $\cI \subseteq [-1,1]$. This can be considered as an analogue of the Sato-Tate distribution which covers the case when the curve $\E$ is defined over $\Q$ and considered modulo consecutive primes $p$, although in our scenario the distribution function is different. The above multiplicative independence condition has, recently, been considered by E. Kowalski in statistical settings. It is trivially satisfied for ordinary elliptic curves and we also establish it for a natural family of curves of genus $g=2$.

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