Editing Nagell–Lutz theorem

{{short description|Describes rational torsion points on elliptic curves over the integers}}
In [[mathematics]], the '''Nagell–Lutz theorem''' is a result in the [[diophantine equation|diophantine geometry]] of [[elliptic curve]]s, which describes [[rational number|rational]] [[Torsion (algebra)|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

:<math>y^2 = x^3 + ax^2 + bx + c </math>

defines a [[Algebraic curve#Singularities|non-singular]] [[cubic curve]] ''E'' with integer [[coefficient]]s ''a'', ''b'', ''c'', and let ''D'' be the [[discriminant]] of the cubic [[polynomial]] on the right side:

:<math>D = -4a^3c + a^2b^2 + 18abc - 4b^3 - 27c^2.</math>

==Statement of the theorem==
If <math>P = (x,y)</math> is a [[rational point]] of finite [[Group (mathematics)#order of an element|order]] on ''E'', for the [[Elliptic curve#The group law|elliptic curve group law]], then:
# ''x'' and ''y'' are integers;
# either <math>y = 0</math>, in which case ''P'' has order two, or else ''y'' divides ''D'', which immediately implies that <math>y^2</math> divides ''D''.

==Generalizations==
The Nagell–Lutz theorem generalizes to arbitrary number fields and more
general cubic equations.<ref name="general">See, for example, 
[https://books.google.com/books?id=6y_SmPc9fh4C&dq=Silverman+torsion+points&pg=PA220 Theorem VIII.7.1] of
[[Joseph H. Silverman]] (1986), "The arithmetic of elliptic curves",
Springer, {{isbn|0-387-96203-4}}.</ref>  
For curves over the rationals, the generalization says that, for a nonsingular cubic curve whose Weierstrass form
:<math>y^2 +a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6 </math>
has integer coefficients, any rational point <math>P = (x,y)</math> of finite
order must have integer coordinates, or else have order 2 and
coordinates of the form <math>x=m/4</math>, <math>y=n/8</math>, 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==
<references/>
* {{cite journal | year=1937 | pages=237–247 | volume=177 | journal=[[Crelle's Journal|J. Reine Angew. Math.]] | author-link=Élisabeth Lutz | title=Sur l'équation ''y''<sup>2</sup> = ''x''<sup>3</sup> − ''Ax'' − ''B'' dans les corps ''p''-adiques | first= Élisabeth |last=Lutz }}
* [[Joseph H. Silverman]], [[John Tate (mathematician)|John Tate]] (1994), "Rational Points on Elliptic Curves", Springer, {{isbn|0-387-97825-9}}.

{{Algebraic curves navbox}}

{{DEFAULTSORT:Nagell-Lutz theorem}}
[[Category:Elliptic curves]]
[[Category:Theorems in number theory]]