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Nagell-Lutz theorem
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(Theorem)
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The following theorem, proved independently by E. Lutz and T. Nagell, gives a very efficient method to compute the torsion subgroup of an elliptic curve defined over $\Rats$ .
Theorem 1 (Nagell-Lutz Theorem) Let $E/\Rats$ be an elliptic curve with Weierstrass equation: $$y^2=x^3+Ax+B,\quad A,B\in \Ints$$ Then for all non-zero torsion points $P$ we have:
- The coordinates of $P$ are in $\Ints$ , i.e. $$x(P),y(P)\in \Ints$$
- If $P$ is of order greater than $2$ , then $$y(P)^2\quad divides\quad 4A^3+27B^2 $$
- If $P$ is of order $2$ then $$y(P)=0\quad and\quad x(P)^3+Ax(P)+B=0$$
- 1
- E. Lutz, Sur l'equation $y^2=x^3-Ax-B$ dans les corps p-adic, J. Reine Angew. Math. 177 (1937), 431-466.
- 2
- T. Nagell, Solution de quelque problemes dans la theorie arithmetique des cubiques planes du premier genre, Wid. Akad. Skrifter Oslo I, 1935, Nr. 1.
- 3
- James Milne, Elliptic Curves, online course notes.
- 4
- Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
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"Nagell-Lutz theorem" is owned by alozano.
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Cross-references: order, coordinates, points, torsion, Weierstrass equation, elliptic curve, torsion subgroup, theorem
There are 3 references to this entry.
This is version 1 of Nagell-Lutz theorem, born on 2003-08-18.
Object id is 4608, canonical name is NagellLutzTheorem.
Accessed 4848 times total.
Classification:
| AMS MSC: | 14H52 (Algebraic geometry :: Curves :: Elliptic curves) |
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Pending Errata and Addenda
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