To Infinity!

At first glance, the above construction doesn't work if $x_1=x_2$. [draw picture]. Fortunately, there is a natural sense in which the graph of $E$ is missing one point, and when $x_1=x_2$ this one missing point is the third point of intersection.

The graph of $E$ that we drew above is a graph in the plan $\mathbb{R}^2$. The plane is a subset of the projective plane $\P ^2$, which I will define in just a moment. The closure of the graph of $y^2 = x^3 + ax+b$ in $\P ^2$ has exactly one extra point, which has rational coordinates, and which we denote by $\infty$. Formally, $\P ^2$ can be viewed as the set of triples $(a,b,c)$ with $a,b,c$ not all 0 modulo the equivalence relation

$$(a,b,c)\sim (\lambda a ,\lambda b , \lambda c)$$

for any nonzero $\lambda$. Denote by $(a:b:c)$ the equivalence class of $(a,b,c)$. The closure of the graph of $y^2 = x^3 + ax+b$ is the graph of $y^2 z = x^3 + axz^2 + bz^3$ and the extra point $\infty$ is $(0:1:0)$.

Venerable Problem: Find an algorithm that, given an elliptic curve $E$ over  $\mathbb{Q}$, outputs a complete description of the set of rational points $(x_0, y_0)$ on $E$.

This problem is difficult. In fact, so far it has stumped everyone! There is a conjectural algorithm, but nobody has succeeded in proving that it is really an algorithm, in the sense that it terminates for any input curve $E$. Several of your profs at Harvard, including Barry Mazur, myself, and Christophe Cornut (who will teach Math 129 next semester) have spent, or will probably spend, a huge chunk of their life thinking about this problem. (Am I being overly pessimistic?)

How could one possible ``describe'' the set of rational points on $E$ in the first place? In 1923, Louis Mordell proved an amazing theorem, which implies that there is a reasonable way to describe the rational points on $E$. To state his theorem, we introduce the ``group law'' on $E$.