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.
It is named for Trygve Nagell and Élisabeth Lutz.

Definition of the terms

Suppose that the equation

:$y^2 = x^3 + ax^2 + bx + c$

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:

:$D = -4a^3c + a^2b^2 + 18abc - 4b^3 - 27c^2.$

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 ''y''² divides ''D''.

Generalizations

The Nagell–Lutz theorem generalizes to arbitrary number fields and more
general cubic equations.See, for example,
Theorem VIII.7.1
of
Joseph H. Silverman (1986), "The arithmetic of elliptic curves",
Springer, .
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).

See also

* Mordell–Weil theorem

References

*
* Joseph H. Silverman, John Tate (1994), "Rational Points on Elliptic Curves", Springer, .

{{DEFAULTSORT:Nagell-Lutz theorem
Elliptic curves
Theorems in number theory