The Problem

In this report we want to discuss a project that was carried through as part of the REU on rational points on elliptic curves that was run at the Department of Mathematics, University of Utah, during the Summer of 2003. Even though we include some statements and proofs, there are no new results in what follows.

The broad objective of this project was to study the distribution of the number of rational points of elliptic curves over the finite field $\ensuremath{\mathbb{Z}}_p = \ensuremath{\mathbb{Z}}/(p\ensuremath{\mathbb{Z}})$. One of the participants, Jenise Smalley, wrote the software and produced the data that led this experiment.

Let $p\geq 5$ be a prime number that we will fix for the rest of this note. All elliptic curves over $\ensuremath{\mathbb{Z}}_p$ can be written in Weierstrass form

$$C: y^2 = x^3 + b x +c b,c \in \ensuremath{\mathbb{Z}}_p.$$ (1)

We denote by $\char93 C(\ensuremath{\mathbb{Z}}_p) = \char93 \{ (x,y) \in \ensuremath{\mathbb{Z}}_p^2 : y^2 = x^3 + b x +c\}$, the number of rational points on the curve $C$.

By taking all possible values¹ of $b$ and $c$ in $\ensuremath{\mathbb{Z}}_p$ we can compute all the possible values of $\char93 C(\ensuremath{\mathbb{Z}}_p)$. Figure 1 shows the histogram corresponding to the distribution of these values.

Figure 1: Histogram for $p=5$

On Figure 1 the first column on the left corresponds to the number of rational points; the second column is the number of curves with the given number of points.

A first observation is that only certain numbers appear as number of points on elliptic curves: all numbers between $2$ and $10$. Notice that $\char93 \ensuremath{\mathbb{Z}}_5^2 = 25$ so the actual range of points is much smaller than it could have been. This has a ``simple'' explanation, after Hasse's Theorem that states that for all $p$ and all elliptic curves $C$

$$\vert p+1 - \char93 C(\ensuremath{\mathbb{Z}}_p)\vert < 2 \sqrt{p}.$$

Thus, for $p=5$, $\vert 6-\char93 C(\ensuremath{\mathbb{Z}}_p)\vert< 4.47$, so that $1.53< \char93 C(\ensuremath{\mathbb{Z}}_p) < 10.47$. The fact that every number in this range is realized by some curve follows from the work of Honda.

Our next observation is that the histogram is symmetric with respect to the number $p+1=6$. We may think that this is a ``feature'' of the case $p=5$, but as you can see in Figures 2 and 3, this turns out to be the case in general².

Figure 2: Histogram for $p=37$

Figure 3: Histogram fro $p=89$

In what follows we will, first, prove that all histograms are symmetric with respect to $p+1$. Then, we will explain why this is so.