## Elliptic curves for high school students

I had to give a talk to high school students about some mathematical notion: I decided to tell them something about elliptic curves, but not the usual speech about cryptography, finite fields and the group law on a cubic curve…

Instead, I talked about the perhaps less known appearances of elliptic functions as solutions of classical ODEs (even if I don’t really know much about these myself). The simplest mechanical system whose motion is governed by an elliptic curve is the pendulum: the reason for this is that the ODE $\ddot{x} + \sin x = 0$ which classically describes the time evolution of the angle of the pendulum is best rewritten in terms of the altitude of the pendulum: the law of energy conservation is then written as
$p^2 = q(q-q_0)(q-2l) = P(q)$
where 0 and 2l are the extremal values of the altitude $q$, $q_0$ is the highest altitude which can be reached with a given energy (even if $q_0 > 2l$, which corresponds to the pendulum make full rotations around its axis), and $p$ is the vertical momentum of the pendulum.

In this setting, there are classical Hamilton relations $dq = p dt$, $dp = P'(q) dt$, so the differential form $dt = dq/p$ turns out to be the canonical non-vanishing abelian differential on the elliptic curve. This explains why the period of the pendulum is an elliptic integral, which can be calculed by an arithmetic-geometric mean, and why the position of the pendulum at $t = t_1 + t_2$ can be deduced from its position at times $t_1$ and $t_2$ by the classical secant-tangent law.

The notes for the talk (in French) are available here.