Mordell's Theorem

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 might spend, a huge chunk of their life thinking about this problem.

How could one possible ``describe'' the group $E(\mathbb{Q})$, since it can be infinite? In 1923, Mordell proved that there is always a reasonable way to describe $E(\mathbb{Q})$.

Theorem 5.1 (Mordell)   The group $E(\mathbb{Q})$ is finitely generated.

This means that there are points $P_1,\ldots, P_s \in E(\mathbb{Q})$ such that every element of $E(\mathbb{Q})$ is of the form $n_1 P_1 + \cdots + n_s P_s$ for some $n_1, \ldots n_s\in\mathbb{Z}$. I will not prove Mordell's theorem in this course, but see §1.3 of [Kato et al.].

Example 5.2   Consider the elliptic curve $E$ given by $y^2 = x^3 -6x - 4$. Then $E(\mathbb{Q})\approx (\mathbb{Z}/2\mathbb{Z})\times \mathbb{Z}$ with generators $(-2,0)$ and $(-1,1)$. We have

$$5(-1,1) = \left(-\frac{131432401}{121462441} , -\frac{1481891884199}{1338637562261}\right).$$

Trying finding that point without knowing about the group law!