arXiv 1901.006042019-01-03math.NT

有限素域上椭圆曲线上点数分布的若干性质

Some properties of the distribution of the numbers of points on elliptic curves over a finite prime field

Saiying He, James Mc Laughlin

备注：: 16 pages
发表信息：: Bulletin of the Australian Mathematics Society 75 (2007), no. 1, 135--149
DOI：: 10.1017/S0004972700039034

摘要：

令$p \geq 5$是素数，对于$a, b \in \mathbb{F}_{p}$，令$E_{a,b}$表示具有方程$y^2=x^3+a\,x + b$的$\mathbb{F}_{p}$上的椭圆曲线。像往常一样定义的跟踪Frobenius $a_{p,\,a,\,b}$由\begin{equation*} \#E_{a,b}(\mathbb{F}_{p}) = p+1 -a_{p,\,a,\,b}. \end{equation*}我们使用基本的事实指数和已知的结果有关的二元二次型在有限域上评估总和$\sum_{t\in\mathbb{F}_{p}} a_{p,\, t,\, b}$，$\sum _{t \in \mathbb{F}_{p}} a_{p,\,a,\, t}$，$ \sum_{t=0}^{p-1}a_{p,\,t,\,b}^{2}$，$ \sum_{t=0}^{p-1}a_{p,\,a,\,t}^{2}$和$ \sum_{t=0}^{p-1}a_{p,\,t,\,b}^{3}$为素数$p$在各种同余类。作为我们结果的一个例子，我们证明了如下：设$p \equiv 5$ $($ mod 6 $)$是素数，设$b \in \mathbb{F}_{p}^{*}$ .然后是\begin{equation*} \sum_{t=0}^{p-1}a_{p,\,t,\,b}^{3}= -p\left((p-2)\left(\frac{-2}{p}\right) +2p\right)\left(\frac{b}{p}\right). \end{equation*}

英文摘要：

Let $p \geq 5$ be a prime and for $a, b \in \mathbb{F}_{p}$, let $E_{a,b}$ denote the elliptic curve over $\mathbb{F}_{p}$ with equation $y^2=x^3+a\,x + b$. As usual define the trace of Frobenius $a_{p,\,a,\,b}$ by \begin{equation*} \#E_{a,b}(\mathbb{F}_{p}) = p+1 -a_{p,\,a,\,b}. \end{equation*} We use elementary facts about exponential sums and known results about binary quadratic forms over finite fields to evaluate the sums $\sum_{t\in\mathbb{F}_{p}} a_{p,\, t,\, b}$, $\sum _{t \in \mathbb{F}_{p}} a_{p,\,a,\, t}$, $ \sum_{t=0}^{p-1}a_{p,\,t,\,b}^{2}$, $ \sum_{t=0}^{p-1}a_{p,\,a,\,t}^{2}$ and $ \sum_{t=0}^{p-1}a_{p,\,t,\,b}^{3}$ for primes $p$ in various congruence classes. As an example of our results, we prove the following: Let $p \equiv 5$ $($mod 6$)$ be prime and let $b \in \mathbb{F}_{p}^{*}$. Then \begin{equation*} \sum_{t=0}^{p-1}a_{p,\,t,\,b}^{3}= -p\left((p-2)\left(\frac{-2}{p}\right) +2p\right)\left(\frac{b}{p}\right). \end{equation*}

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