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).

$E(K)$ is a finitely generated abelian group.

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 $r a n k$ 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.

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

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