 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.
Contents
Definition of the terms
Suppose that the equation
defines a nonsingular cubic curve with integer coefficients a, b, c, and let D be the discriminant of the cubic polynomial on the right side:
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 y^{2} 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
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).
See also
References
 ^ See, for example, Theorem VIII.7.1 of Joseph H. Silverman (1986), "The arithmetic of elliptic curves", Springer, ISBN 0387962034.
 Élisabeth Lutz (1937). "Sur l’équation y^{2} = x^{3} − Ax − B dans les corps padiques". J. Reine Angew. Math. 177: 237–247.
 Joseph H. Silverman, John Tate (1994), "Rational Points on Elliptic Curves", Springer, ISBN 0387978259.
Categories: Elliptic curves
 Theorems in number theory
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