Sharp estimates of torsion of elliptic curves.

*(English. Russian original)*Zbl 0932.11036
Math. Notes 63, No. 4, 444-448 (1998); translation from Mat. Zametki 63, No. 4, 503-508 (1998).

By a theorem of Merel and Oesterlé [see L. Merel, Points rationnels et séries de Dirichlet, Doc. Math. Extra Vol. ICM 1998 II, 183-186 (1998) and Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124, 437-449 (1996)], if \(k\) is a number field of degree \(n= [k:\mathbb{Q}]\) and \(E\) is an elliptic curve over \(k\) with a \(k\)-rational torsion point of prime order \(p\), then \(p\leq (1+3^{\frac{n}{2}})^2\).

This result (which is exponential in the degree \(n\)) together with a theorem of Yu. I. Manin [The \(p\)-torsion of elliptic curves is uniformly bounded, Math. USSR, Izv. 3, 433-438 (1969); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 33, 459-465 (1969; Zbl 0191.19601)] proves the famous boundedness conjecture. (A more precise estimate was given by Parent; see the first-mentioned paper of Merel.)

The author of the paper under consideration obtains stronger bounds (which are polynomial in \(n\)) for the CM-elliptic curves over \(k\) \(F_1: y^2= x^3+s\) \((s\in k)\) and \(F_2: y^2= x^3+rz\) \((r\in k)\). He claims that if a \(k\)-rational torsion point has exact order \(m= 2^\alpha m_im_j\) on \(F_2\), then (Theorem 2), for \(\alpha\geq 2\), \[ 2^{2\alpha-4} \varphi(m_im_j) \psi(m_j)\leq 4n,\tag{1} \] while for a \(k\)-rational torsion point of exact order \(m= 2^\alpha 3^\beta m_im_j\) on \(F_1\) (Theorem 1), for \(\alpha\geq 2\), \(\beta\geq 2\), \[ 2^{2\alpha-3} 3^{2\beta-3} \varphi (m_im_j) \psi(m_j)\leq 6n. \tag{2} \] Here, \(\varphi\) is Euler’s function, \(\psi\) is defined as is \(\varphi(m)\) for \(m\in \mathbb{N}\) but with \(p-1\) replaced by \(p+1\) for the primes \(p|m\), where \[ m_i= \prod_{\kappa=1}^i p_\kappa^{\delta_\kappa}, \qquad m_j= \prod_{\lambda=1}^j p_\lambda^{\eta_\lambda}, \] and the primes are \(p_\kappa\equiv 1\pmod 4\) and \(p_\lambda\equiv 3\pmod 4\) in the first case (1) and \(p_\kappa\equiv 1\pmod 6\) and \(p_\lambda\equiv 5\pmod 6\) is the second (2). (Observe that the automorphism group of the elliptic curve over the algebraic closure of \(k\) is of order 4 in the first case and of order 6 in the second.)

In principle these results are desirable. They are, as is usual by the author, a consequence of a series of elementary, but very tedious Lemmata which, however, the reviewer was unable to verify. For the proof of Lemmata 3, 4, 6 and 7, the author refers to another paper of his.

No examples for the two theorems are given. (Reviewer’s remark: The Theorem of “Hagel-Lutz” is the Theorem of “Nagell-Lutz”, the \(x\) and \(y\) in formulas (15) and (17) are lower case letters, and the \(F_2\) in Theorem 1 should read \(F_1\). The last is a confusing error contained already in the Russian original, as is the negligible error in formulas (15) and (17)).

This result (which is exponential in the degree \(n\)) together with a theorem of Yu. I. Manin [The \(p\)-torsion of elliptic curves is uniformly bounded, Math. USSR, Izv. 3, 433-438 (1969); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 33, 459-465 (1969; Zbl 0191.19601)] proves the famous boundedness conjecture. (A more precise estimate was given by Parent; see the first-mentioned paper of Merel.)

The author of the paper under consideration obtains stronger bounds (which are polynomial in \(n\)) for the CM-elliptic curves over \(k\) \(F_1: y^2= x^3+s\) \((s\in k)\) and \(F_2: y^2= x^3+rz\) \((r\in k)\). He claims that if a \(k\)-rational torsion point has exact order \(m= 2^\alpha m_im_j\) on \(F_2\), then (Theorem 2), for \(\alpha\geq 2\), \[ 2^{2\alpha-4} \varphi(m_im_j) \psi(m_j)\leq 4n,\tag{1} \] while for a \(k\)-rational torsion point of exact order \(m= 2^\alpha 3^\beta m_im_j\) on \(F_1\) (Theorem 1), for \(\alpha\geq 2\), \(\beta\geq 2\), \[ 2^{2\alpha-3} 3^{2\beta-3} \varphi (m_im_j) \psi(m_j)\leq 6n. \tag{2} \] Here, \(\varphi\) is Euler’s function, \(\psi\) is defined as is \(\varphi(m)\) for \(m\in \mathbb{N}\) but with \(p-1\) replaced by \(p+1\) for the primes \(p|m\), where \[ m_i= \prod_{\kappa=1}^i p_\kappa^{\delta_\kappa}, \qquad m_j= \prod_{\lambda=1}^j p_\lambda^{\eta_\lambda}, \] and the primes are \(p_\kappa\equiv 1\pmod 4\) and \(p_\lambda\equiv 3\pmod 4\) in the first case (1) and \(p_\kappa\equiv 1\pmod 6\) and \(p_\lambda\equiv 5\pmod 6\) is the second (2). (Observe that the automorphism group of the elliptic curve over the algebraic closure of \(k\) is of order 4 in the first case and of order 6 in the second.)

In principle these results are desirable. They are, as is usual by the author, a consequence of a series of elementary, but very tedious Lemmata which, however, the reviewer was unable to verify. For the proof of Lemmata 3, 4, 6 and 7, the author refers to another paper of his.

No examples for the two theorems are given. (Reviewer’s remark: The Theorem of “Hagel-Lutz” is the Theorem of “Nagell-Lutz”, the \(x\) and \(y\) in formulas (15) and (17) are lower case letters, and the \(F_2\) in Theorem 1 should read \(F_1\). The last is a confusing error contained already in the Russian original, as is the negligible error in formulas (15) and (17)).

Reviewer: H.G.Zimmer (Saarbrücken)

##### MSC:

11G05 | Elliptic curves over global fields |

14H52 | Elliptic curves |

11G15 | Complex multiplication and moduli of abelian varieties |

##### Keywords:

elliptic curve; torsion group; sharp estimates; algebraic number field; complex multiplication
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\textit{V. A. Demyanenko}, Math. Notes 63, No. 4, 444--448 (1998; Zbl 0932.11036); translation from Mat. Zametki 63, No. 4, 503--508 (1998)

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##### References:

[1] | V. A. Dem’yanenko, ”On sharp estimates of the torsion of elliptic curves,” in:Third International Conference on Algebra. [in Russian], Abstracts, Krasnoyarsk (1993), p. 106. |

[2] | V. A. Dem’yanenko, ”On sharp estimates of thep-torsion of some curves of the first kind,”Mat. Zametki [Math. Notes],21, No. 1, 3–7 (1977). |

[3] | V. A. Dem’yanenko, ”On sharp estimates of the torsion of points of curves of the first kind,”Zap. Nauchn. Sem. LOMI,151, 57–65 (1986). |

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