# Nagell–Lutz theorem

﻿
Nagell–Lutz theorem

In mathematics, the Nagell–Lutz theorem is a result in the diophantine geometry of elliptic curves, which describes rational torsion points on elliptic curves over the integers.

## Definition of the terms

Suppose that the equation

$y^2 = x^3 + ax^2 + bx + c \$

defines a non-singular cubic curve with integer coefficients a, b, c, and let D be the discriminant of the cubic polynomial on the right side:

$D = -4a^3c + a^2b^2 + 18abc - 4b^3 - 27c^2.\$

## Statement of the theorem

If P = (x,y) is a rational point of finite order on C, for the elliptic curve group law, then:

• 1) x and y are integers
• 2) either y = 0, in which case P has order two, or else y divides D, which immediately implies that y2 divides D.

## Generalizations

The Nagell–Lutz theorem generalizes to arbitrary number fields and more general cubic equations.[1] For curves over the rationals, the generalization says that, for a nonsingular cubic curve whose Weierstrass form

$y^2 +a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6 \$

has integer coefficients, any rational point P=(x,y) of finite order must have integer coordinates, or else have order 2 and coordinates of the form x=m/4, y=n/8, for m and n integers.

## History

The result is named for its two independent discoverers, the Norwegian Trygve Nagell (1895–1988) who published it in 1935, and Élisabeth Lutz (1937).

## References

1. ^ See, for example, Theorem VIII.7.1 of Joseph H. Silverman (1986), "The arithmetic of elliptic curves", Springer, ISBN 0-387-96203-4.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Théorème de Nagell-Lutz —  Ne pas confondre avec le théorème de Nagel. En mathématiques, le théorème de Nagell Lutz est un résultat de sur la géométrie diophantienne des courbes elliptiques. Supposons que C, définie par soit une courbe cubique non singulière avec les …   Wikipédia en Français

• Lutz — may refer to:People with the surname Lutz*Adolfo Lutz, Brazilian physician*Alois Lutz, Austrian figure skater, for whom the Lutz jump is named *Al Lutz, American internet journalist *Berta Lutz, Brazilian scientist and feminist *Bob Lutz,… …   Wikipedia

• Nagell — may refer to: Dominicus Nagell Lemvig Brun (1790–1874), a Norwegian military officer and politician Richard Case Nagell, a former military officer who claimed to have had foreknowledge of the John F. Kennedy assassination Tinius Nagell Erichsen… …   Wikipedia

• De Franchis theorem — In mathematics, the de Franchis theorem is one of a number of closely related statements applying to compact Riemann surfaces, or, more generally, algebraic curves, X and Y, in the case of genus g > 1. The simplest is that the automorphism… …   Wikipedia

• Élisabeth Lutz — was a 20th century French mathematician.She was a student of André Weil at the Université de Strasbourg , from 1934 to 1938. In 1935 she began working on aspects of elliptic curves over p adic fields. An elliptic curve over Q can be put in the… …   Wikipedia

• List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… …   Wikipedia

• Riemann surface — For the Riemann surface of a subring of a field, see Zariski–Riemann space. Riemann surface for the function ƒ(z) = √z. The two horizontal axes represent the real and imaginary parts of z, while the vertical axis represents the real… …   Wikipedia

• List of number theory topics — This is a list of number theory topics, by Wikipedia page. See also List of recreational number theory topics Topics in cryptography Contents 1 Factors 2 Fractions 3 Modular arithmetic …   Wikipedia

• Plane curve — In mathematics, a plane curve is a curve in a Euclidean plane (cf. space curve). The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. A smooth plane curve is a curve in a …   Wikipedia

• List of mathematics articles (N) — NOTOC N N body problem N category N category number N connected space N dimensional sequential move puzzles N dimensional space N huge cardinal N jet N Mahlo cardinal N monoid N player game N set N skeleton N sphere N! conjecture Nabla symbol… …   Wikipedia