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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)).
11G05 Elliptic curves over global fields
14H52 Elliptic curves
11G15 Complex multiplication and moduli of abelian varieties
Full Text: DOI
[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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