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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)).
##### MSC:
 11G05 Elliptic curves over global fields 14H52 Elliptic curves 11G15 Complex multiplication and moduli of abelian varieties
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##### References:
  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.  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).  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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