Definition of the terms
Suppose that the equation
Statement of the theorem
- 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.
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
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.
- See, for example, Theorem VIII.7.1 of Joseph H. Silverman (1986), "The arithmetic of elliptic curves", Springer, ISBN 0-387-96203-4.
- Élisabeth Lutz (1937). "Sur l'équation y2 = x3 − Ax − B dans les corps p-adiques". J. Reine Angew. Math. 177: 237–247.
- Joseph H. Silverman, John Tate (1994), "Rational Points on Elliptic Curves", Springer, ISBN 0-387-97825-9.