Nagell-Lutz theorem

The following theorem, proved independently by E. Lutz and T. Nagell, gives a very efficient method to compute the torsion subgroup of an elliptic curve defined over $\mathbb{Q}$ .

Theorem 1 (Nagell-Lutz Theorem).

Let $E/\mathbb{Q}$ be an elliptic curve with Weierstrass equation:

$y^{2}=x^{3}+Ax+B,\quad A,B\in\mathbb{Z}$

Then for all non-zero torsion points $P$ we have:

The coordinates of $P$ are in $\mathbb{Z}$ , i.e.

$x(P),y(P)\in\mathbb{Z}$

If $P$ is of order greater than $2$ , then

$y(P)^{2}\quad divides\quad 4A^{3}+27B^{2}$

If $P$ is of order $2$ then

$y(P)=0\quad and\quad x(P)^{3}+Ax(P)+B=0$

References

1 E. Lutz, Sur l’equation $y^{2}=x^{3}-Ax-B$ dans les corps p-adic, J. Reine Angew. Math. 177 (1937), 431-466.
2 T. Nagell, Solution de quelque problemes dans la theorie arithmetique des cubiques planes du premier genre, Wid. Akad. Skrifter Oslo I, 1935, Nr. 1.
3 James Milne, Elliptic Curves, http://www.jmilne.org/math/CourseNotes/math679.htmlonline course notes.
4 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.

Title	Nagell-Lutz theorem
Canonical name	NagellLutzTheorem
Date of creation	2013-03-22 13:52:02
Last modified on	2013-03-22 13:52:02
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	4
Author	alozano (2414)
Entry type	Theorem
Classification	msc 14H52
Related topic	EllipticCurve
Related topic	MordellWeilTheorem
Related topic	RankOfAnEllipticCurve
Related topic	TorsionSubgroupOfAnEllipticCurveInjectsInTheReductionOfTheCurve
Related topic	ArithmeticOfEllipticCurves
Defines	Nagell-Lutz theorem