Some Additional Experiments

Javier Fernandez

Department of Mathematics
University of Utah
Salt Lake City
UT 84112

jfernand@math.utah.edu

The purpose of this note is to report on a very simple experiment. Let $C$ be an elliptic curve over $\ensuremath{\mathbb{Q}}$. By Mordell's Theorem we know that, as a group, $C \simeq \ensuremath{\mathbb{Z}}^r \oplus T$, that is it contains infinitely many rational points if $r$, the rank, is positive.

Now, consider the reduction of the curve $C$ to $\ensuremath{\mathbb{Z}}_p = \ensuremath{\mathbb{Z}}/(p\ensuremath{\mathbb{Z}})$, and let $N_p = \char93 C(\ensuremath{\mathbb{Z}}_p)$. By Hasse's Theorem we know that

$$\vert p+1 - N_p \vert <2\sqrt{p}$$

so that, roughly speaking, $N_p \sim p+1$. In order to understand how $N_p$ differs from this ``expected value'' we will consider the ``density'' $D_p = \frac{N_p}{p}$.

Morally, at least, a curve with many rational points (that is, large $\char93 C(\ensuremath{\mathbb{Q}})$), is expected to have many points over $\ensuremath{\mathbb{Z}}_p$. We want to show this with a few examples. We will consider curves of the form

$$y^2 = x^3 + a x^2 + bx +c$$

with $a$, $b$ and $c$ as shown on Table 1

Table 1: Elliptic curves

Curve	$a$	$b$	$c$	rank
$C_0$	0	$1$	0	0
$C_1$	0	$13$	0	$1$
$C_2$	0	$14$	0	$2$
$C_3$	$337$	$337^2$	$337^3$	$3$
$C_{14}$	0	$402599774387690701016910427272483$	0	$14$

The following function, in some sense, measures how much $N_p$ deviates from the ``medium value'' $p$.

$$F(x) = \prod_{p \leq x, p\, prime} \frac{N_p}{p}$$

Figure 1 shows the graph of $F$ corresponding to the curves $C_0$ and $C_1$.

Figure 1: Graph of $F(x)$ for $C_0$ (below) and $C_1$ (above)

As you can see, there is a significant difference between the behavior of the two curves. As expected, $C_1$ seems to have more points than $C_0$, perhaps reflecting the fact its rank is $1$ while that of $C_0$ is 0. Actually, using Nagel-Lutz's Theorem it is easy to check that $C_0(\ensuremath{\mathbb{Q}}) \simeq \ensuremath{\mathbb{Z}}_2$.

Next we see what happens as we consider curves of higher rank. Figure 2 shows the graph of $F$ for $C_1$, $C_2$ and $C_3$.

Figure 2: Graphs of $F(x)$ for the curves $C_1$ (bottom), $C_2$ (middle) and $C_3$ (top)

We see that, again, all functions are, roughly, increasing and that the curves with higher rank have significantly bigger values of $F$. Notice the scale on the $F$-axis, compared to Figure 1.

Last, Figure 3 shows the graph of $F$ for $C_{14}$. Again, notice the scale on the $F$-axis and thus, how much bigger the values of $F$ are compared with those of the other curves.

Figure 3: Graph of $F(x)$ for the curve $C_{14}$

To conclude, let me mention that the function $(F(x))^{-1}$ of the elliptic curve $C$ is somehow related to the value of the $L$-function, $L(C,1)$. Thus, we see that for a curve of rank 0, the limiting value of $F(x)^{-1}$ is a non-zero number. On the other hand, for all the other curves (with rank $>0$) $F(x)^{-1}$ seems to approach to 0. In fact, the bigger the rank, the faster $F(x)^{-1}$ approaches 0. As you have guessed, this is the content of the Birch and Swinnerton-Dyer Conjecture that says that the order of the zero of $L(C,s)$ at $s=1$ should compute the rank of the curve $C$.