Elliptic Curves Over the Rational Numbers

Figure 6.5: Louis J. Mordell

Let $E$ be an elliptic curve defined over $\mathbb {Q}$ . The following is a deep theorem about the group $E(\mathbb{Q})$ .

Theorem 6.5 (Mordell)   The group $E(\mathbb{Q})$ is finitely generated. That is, 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 integers $n_1, \ldots n_s\in\mathbb{Z}$ .

Mordell's theorem implies that it makes sense to ask whether or not we can compute $E(\mathbb{Q})$ , where by ``compute'' we mean find a finite set $P_1,\ldots, P_s$ of points on $E$ that generate $E(\mathbb{Q})$ as an abelian group. There is a systematic approach to computing $E(\mathbb{Q})$ called ``descent'' (see e.g., [#!cremona:algs!#,#!mwrank!#,#!silverman:aec!#]). It is widely believed that the method of descent will always succeed, but nobody has yet proved that it will. Proving that descent works for all curves is one of the central open problem in number theory, and is closely related to the Birch and Swinnerton-Dyer conjecture (one of the Clay Math Institute's million dollar prize problems). The crucial difficulty amounts to deciding whether or not certain explicitly given curves have any rational points on them or not (these are curves that have points over  $\mathbb {R}$ and modulo $n$ for all $n$ ).

The details of using descent to computing $E(\mathbb{Q})$ are beyond the scope of this book. In several places below we will simply assert that $E(\mathbb{Q})$ has a certain structure or is generated by certain elements. In each case, we computed $E(\mathbb{Q})$ using a computer implementation of this method.