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arXiv 1208.0919math.NT
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关于椭圆曲线上点的个数

Averages of the number of points on elliptic curves

Greg Martin, Paul Pollack, Ethan Smith

备注：
Corrected definition of K*(N)
发表信息：
Algebra and Number Theory, 8(4): 813--836, 2014
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摘要：

如果$E$是定义在$\mathbb Q$上的椭圆曲线，并且$p$是$E$的良好约化的素数，则令$E(\mathbb F_p)$表示约化曲线上模$p$的点集。通过设置$M_E(N):= \#\{p: \#E(\mathbb F_p)= N\}$定义算术函数$M_E(N)$。最近，David和第三作者研究了椭圆曲线$E$的某些“盒子”上的$M_E(N)$的平均值。假设一个关于短间隔素数的合理猜想，他们证明了以下结果：对于奇数$N$，对于一个显式给定的函数$K^{\ast}(N)$，$M_E(N)$在一个具有足够大边的盒子上的平均值是$\sim \frac{K^{\ast}(N)}{\log{N}}$。函数$K^{\ast}(N)$有些特殊：定义为素数除以$N$的乘积，乍一看它类似于乘法函数。但进一步的研究表明，它不是，所以不能直接研究其性质的乘法数论的常用工具。在本文中，我们克服了这些困难，并证明了一些统计结果关于$K^{\ast}(N)$。例如，我们确定的平均值$K^{\ast}(N)$在所有$N$，奇$N$和总理$N$，我们表明，$K^{\ast}(N)$有一个分布函数。我们还解释了我们的结果如何与现有的定理和代数乘法性质的$\# E(\mathbb F_p)$，如Koblitz猜想。

英文摘要：

If $E$ is an elliptic curve defined over $\mathbb Q$ and $p$ is a prime of good reduction for $E$, let $E(\mathbb F_p)$ denote the set of points on the reduced curve modulo $p$. Define an arithmetic function $M_E(N)$ by setting $M_E(N):= \#\{p: \#E(\mathbb F_p)= N\}$. Recently, David and the third author studied the average of $M_E(N)$ over certain "boxes" of elliptic curves $E$. Assuming a plausible conjecture about primes in short intervals, they showed the following: for odd $N$, the average of $M_E(N)$ over a box with sufficiently large sides is $\sim \frac{K^{\ast}(N)}{\log{N}}$ for an explicitly-given function $K^{\ast}(N)$. The function $K^{\ast}(N)$ is somewhat peculiar: defined as a product over the primes dividing $N$, it resembles a multiplicative function at first glance. But further inspection reveals that it is not, and so one cannot directly investigate its properties by the usual tools of multiplicative number theory. In this paper, we overcome these difficulties and prove a number of statistical results about $K^{\ast}(N)$. For example, we determine the mean value of $K^{\ast}(N)$ over all $N$, odd $N$ and prime $N$, and we show that $K^{\ast}(N)$ has a distribution function. We also explain how our results relate to existing theorems and conjectures on the multiplicative properties of $\# E(\mathbb F_p)$, such as Koblitz's conjecture.

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