Harman Patil (Editor)

Nagell–Lutz theorem

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

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.

Contents

Definition of the terms

Suppose that the equation

y 2 = x 3 + a x 2 + b x + 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 = − 4 a 3 c + a 2 b 2 + 18 a b c − 4 b 3 − 27 c 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. 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

    Nagell–Lutz theorem Wikipedia
    (Text) CC BY-SA