Kullback–Leibler divergence: Difference between revisions
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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. | |||
==Definition of the terms== | |||
Suppose that the equation | |||
:<math>y^2 = x^3 + ax^2 + bx + c \ </math> | |||
defines a [[non-singular]] [[cubic curve]] 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 ''P'' = (''x'',''y'') is a [[rational point]] of finite [[Group (mathematics)#order_of_an_element|order]] on ''C'', for the [[Elliptic curve#The group law|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''<sup>2</sup> divides ''D''. | |||
==Generalizations== | |||
The Nagell–Lutz theorem generalizes to arbitrary number fields and more | |||
general cubic equations.<ref name="general">See, for example, | |||
[http://books.google.com/books?id=6y_SmPc9fh4C&pg=PA220&dq=Silverman+torsion+points#v=onepage&q=&f=false 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 ''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== | |||
<references/> | |||
* {{cite journal | year=1937 | pages=237–247 | volume=177 | journal=[[J. Reine Angew. Math.]] | authorlink=Élisabeth Lutz | title=Sur l’équation ''y''<sup>2</sup> = ''x''<sup>3</sup> − ''Ax'' − ''B'' dans les corps ''p''-adiques | author= Élisabeth Lutz }} | |||
* [[Joseph H. Silverman]], [[John Tate]] (1994), "Rational Points on Elliptic Curves", Springer, ISBN 0-387-97825-9. | |||
{{DEFAULTSORT:Nagell-Lutz theorem}} | |||
[[Category:Elliptic curves]] | |||
[[Category:Theorems in number theory]] | |||
Revision as of 12:26, 30 December 2013
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 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
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.
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