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
\( {\displaystyle y^{2}=x^{3}+ax^{2}+bx+c} \)
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:
\( {\displaystyle D=-4a^{3}c+a^{2}b^{2}+18abc-4b^{3}-27c^{2}.} \)
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.[1] For curves over the rationals, the generalization says that, for a nonsingular cubic curve whose Weierstrass form
\( {\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}} \)
has integer coefficients, any rational point P=(x,y) 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
Mordell–Weil theorem
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 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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
