# 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

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:

## 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 0-387-96203-4.

- Élisabeth Lutz (1937). "Sur l'équation
*y*^{2}=*x*^{3}−*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.