# 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 coordinates in $K$.

###### Theorem 1 (Mordell-Weil).

$E(K)$

###### 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 [2], Chapter VIII, page 189. ∎

###### Theorem 2 (Weak Mordell-Weil).

$E(K)/mE(K)$ is finite for all $m\geq 2$.

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

 $E(K)\simeq E_{\operatorname{torsion}}(K)\bigoplus{\mathbb{Z}}^{R}$

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 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. 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. 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. 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 http://math.bu.edu/people/alozano/Torsion.htmlgenerators here.

4. 4.

Let $E_{4}/\mathbb{Q}\colon y^{2}+1951/164xy-3222367/40344y=x^{3}+3537/164x^{2}-403% 02641/121032x$, then $E_{4}(\mathbb{Q})\simeq{\mathbb{Z}}^{10}$. See http://math.bu.edu/people/alozano/Examples.htmlgenerators here.

## References

• 1 James Milne, Elliptic Curves, online course notes. http://www.jmilne.org/math/CourseNotes/math679.htmlhttp://www.jmilne.org/math/CourseNotes/math679.html
• 2 Joseph H. Silverman, . 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.
 Title rank of an elliptic curve Canonical name RankOfAnEllipticCurve Date of creation 2013-03-22 13:49:12 Last modified on 2013-03-22 13:49:12 Owner alozano (2414) Last modified by alozano (2414) Numerical id 14 Author alozano (2414) Entry type Definition Classification msc 14H52 Synonym rank Related topic EllipticCurve Related topic HeightFunction Related topic MordellWeilTheorem Related topic SelmerGroup Related topic MazursTheoremOnTorsionOfEllipticCurves Related topic NagellLutzTheorem Related topic ArithmeticOfEllipticCurves Defines weak Mordell-Weil theorem Defines rank of an elliptic curve