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[parent] Nagell-Lutz theorem (Theorem)

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 $ \mathbb{Q}$.

Theorem 1 (Nagell-Lutz Theorem)   Let $ E/\mathbb{Q}$ be an elliptic curve with Weierstrass equation:
$\displaystyle y^2=x^3+Ax+B,\quad A,B\in \mathbb{Z}$
Then for all non-zero torsion points $ P$ we have:
  1. The coordinates of $ P$ are in $ \mathbb{Z}$, i.e.
    $\displaystyle x(P),y(P)\in \mathbb{Z}$
  2. If $ P$ is of order greater than $ 2$, then
    $\displaystyle y(P)^2\quad divides\quad 4A^3+27B^2 $
  3. If $ P$ is of order $ 2$ then
    $\displaystyle y(P)=0\quad and\quad x(P)^3+Ax(P)+B=0$

Bibliography

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.



"Nagell-Lutz theorem" is owned by alozano.
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See Also: elliptic curve, Mordell-Weil theorem, rank of an elliptic curve, the torsion subgroup of an elliptic curve injects in the reduction of the curve, the arithmetic of elliptic curves

Also defines:  Nagell-Lutz theorem
Keywords:  torsion, elliptic curve, Mazur's theorem

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Cross-references: order, coordinates, points, torsion, Weierstrass equation, elliptic curve, torsion subgroup
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This is version 1 of Nagell-Lutz theorem, born on 2003-08-18.
Object id is 4608, canonical name is NagellLutzTheorem.
Accessed 3425 times total.

Classification:
AMS MSC: 14H52 (Algebraic geometry :: Curves :: Elliptic curves)
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