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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.
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| ==Definition of the terms==
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| Suppose that the equation
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| :<math>y^2 = x^3 + ax^2 + bx + c \ </math>
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| 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:
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| :<math>D = -4a^3c + a^2b^2 + 18abc - 4b^3 - 27c^2.\ </math>
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| ==Statement of the theorem==
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| 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:
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| *1) ''x'' and ''y'' are integers
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| *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''.
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| ==Generalizations==
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| The Nagell–Lutz theorem generalizes to arbitrary number fields and more
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| general cubic equations.<ref name="general">See, for example,
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| [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",
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| Springer, ISBN 0-387-96203-4.</ref>
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| For curves over the rationals, the
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| generalization says that, for a nonsingular cubic curve
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| whose Weierstrass form
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| :<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
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| order must have integer coordinates, or else have order 2 and
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| coordinates of the form ''x''=''m''/4, ''y''=''n''/8, for ''m'' and ''n'' integers.
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| ==History==
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| The result is named for its two independent discoverers, the Norwegian [[Trygve Nagell]] (1895–1988) who published it in 1935, and [[Élisabeth Lutz]] (1937).
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| ==See also==
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| *[[Mordell–Weil theorem]]
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| ==References==
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| <references/>
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| * {{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 }}
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| * [[Joseph H. Silverman]], [[John Tate]] (1994), "Rational Points on Elliptic Curves", Springer, ISBN 0-387-97825-9.
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| {{DEFAULTSORT:Nagell-Lutz theorem}}
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| [[Category:Elliptic curves]] | |
| [[Category:Theorems in number theory]]
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