Let $K$ be a number field and let $E$ be an elliptic curve over $K$ . By $E(K)$ we denote the set of points in $E$ with coordinates in $K$ .

Proof. The proof of this theorem is fairly involved. The main two ingredients are the so called ``weak Mordell-Weil theorem'' (see below), the concept of height function for abelian groups and the ``descent'' theorem.
See $\cite{silverman}$ , Chapter VIII, page 189.

The Mordell-Weil theorem implies that for any elliptic curve $E/K$ the group of points has the following structure: $$E(K)\simeq E_{\operatorname{torsion}}(K)\bigoplus {\mathbb{Z}}^R$$ where $E_{\operatorname{torsion}}(K)$ denotes the set of points of finite order (or torsion group), and $R$ is a non-negative integer which is called the $rank$ of the elliptic curve. It is not known how big this number $R$ can get for elliptic curves over $\mathbb{Q}$ . The largest rank known for an elliptic curve over $\mathbb{Q}$ is 28 Elkies (2006).

Note: see Mazur's theorem for an account of the possible torsion subgroups over $\mathbb{Q}$ .

Examples:

1. The elliptic curve $E_1/\mathbb{Q}\colon y^2=x^3+6$ has rank 0 and $E_1(\mathbb{Q})\simeq {0}$ .
2. Let $E_2/\mathbb{Q}\colon y^2=x^3+1$ , then $E_2(\mathbb{Q})\simeq \mathbb{Z}/6\mathbb{Z}$ . The torsion group is generated by the point $(2,3)$ .
3. Let $E_3/\mathbb{Q}\colon y^2=x^3+109858299531561$ , then $E_3(\mathbb{Q})\simeq \mathbb{Z}/3\mathbb{Z}\bigoplus {\mathbb{Z}}^5$ . See generators here.
4. Let $E_4/\mathbb{Q}\colon y^2 +1951/164xy -3222367/40344y=x^3+3537/164x^2-40302641/121032x$ , then $E_4(\mathbb{Q})\simeq {\mathbb{Z}}^{10}$ . See generators here.

Bibliography

1

James Milne, Elliptic Curves, online course notes. http://www.jmilne.org/math/CourseNotes/math679.html

2

Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.

3

Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.

4

Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.