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. It is named for Trygve Nagell and Élisabeth Lutz.

Definition of the terms Suppose that the equation

y 2 = x 3 + a x 2 + b x + c {\displaystyle y^{2}=x^{3}+ax^{2}+bx+c}

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

D = − 4 a 3 c + a 2 b 2 + 18 a b c − 4 b 3 − 27 c 2 . {\displaystyle D=-4a^{3}c+a^{2}b^{2}+18abc-4b^{3}-27c^{2}.}

Statement of the theorem If P = ( x , y ) {\displaystyle P=(x,y)} is a rational point of finite order on E, for the elliptic curve group law, then:

x and y are integers; either y = 0 {\displaystyle y=0} , in which case P has order two, or else y divides D, which immediately implies that y 2 {\displaystyle y^{2}} divides D.

Generalizations The Nagell–Lutz theorem generalizes to arbitrary number fields and more general cubic equations. 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 {\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}}

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

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