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

Abstract

Let $p\ge 5$ 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 $y^2=x^3+ax+b$. 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 $b\in {\mathbb{F}}_p^{\ast }$. 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*}

16 pages