|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| 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.
| | Aleta is [http://pinterest.com/search/pins/?q=what%27s+written what's written] relating to her birth certificate life style she doesn't really for example , being called like that. Managing people happens to be what she does so she plans on updating it. She's always loved living South Carolina. To dr is something her dad doesn't really like having said that she does. She is running and maintaining a blog here: http://prometeu.net<br><br>Here is my web blog clash of clans hack no survey - [http://prometeu.net click here.], |
| | |
| ==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]]
| |
Aleta is what's written relating to her birth certificate life style she doesn't really for example , being called like that. Managing people happens to be what she does so she plans on updating it. She's always loved living South Carolina. To dr is something her dad doesn't really like having said that she does. She is running and maintaining a blog here: http://prometeu.net
Here is my web blog clash of clans hack no survey - click here.,