rank of an elliptic curve

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 coordinatesin $K$.

Theorem 1 (Mordell-Weil).

$E(K)$ is a finitely generated abeliangroup.

Proof.

The proof of this theorem is fairly involved. Themain two ingredients are the so called “weak Mordell-Weil theorem”(see below), the concept of height function for abelian groups andthe “descent” theorem.
See [2], Chapter VIII, page189.∎

Theorem 2 (Weak Mordell-Weil).

$E(K)/mE(K)$ isfinite for all $m\mathrm{\ge }\mathrm{2}$.

The Mordell-Weil theorem implies that for any elliptic curve $E/K$the group of points has the following structure:

$$E(K)\simeq E_{\mathrm{torsion}}(K)\oplus {\mathbb{Z}}^R$$

where $E_{\mathrm{torsion}}(K)$ denotes the set of points of finite order (or torsion group),and $R$ is a non-negative integer which is called the $rank$ of theelliptic curve. It is not known how big this number $R$ can getfor elliptic curves over $\mathbb{Q}$. The largest rank known foran elliptic curve over $\mathbb{Q}$ is 28 http://www.math.hr/ duje/tors/tors.htmlElkies (2006).

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

Examples:

1. The elliptic curve $E_1/\mathbb{Q}:y^2=x^3+6$ has rank 0and $E_1(\mathbb{Q})\simeq 0$.

2. Let $E_2/\mathbb{Q}:y^2=x^3+1$, then$E_2(\mathbb{Q})\simeq \mathbb{Z}/6\mathbb{Z}$. The torsion groupis generated by the point $(2,3)$.

3. Let $E_3/\mathbb{Q}:y^2=x^3+109858299531561$, then$E_3(\mathbb{Q})\simeq \mathbb{Z}/3\mathbb{Z}\oplus {\mathbb{Z}}^5$. Seehttp://math.bu.edu/people/alozano/Torsion.htmlgeneratorshere.

4. Let $E_4/\mathbb{Q}:y^2+1951/164xy-3222367/40344y=x^3+3537/164x^2-40302641/121032x$, then$E_4(\mathbb{Q})\simeq {\mathbb{Z}}^{10}$. Seehttp://math.bu.edu/people/alozano/Examples.htmlgeneratorshere.