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Torsion des courbes elliptiques sur les corps cubiques. (Torsion of elliptic curves over cubic fields). (French) Zbl 0971.11030
Given a number field $$K$$ and an elliptic curve $$E$$ defined over $$K$$, there exist only finitely many torsion points in $$E(K)$$; this is the theorem of L. Mérel in [Invent. Math. 124, 437-449 (1996; Zbl 0936.11037)]. How is the torsion subgroup of $$E(K)$$ made? Let $$d$$ be a positive integer, let $$S(d)$$ be the set of all the prime numbers $$p$$ such that there exists a number field $$K$$ of degree $$d$$ over $${\mathbb Q}$$ and an elliptic curve $$E$$ defined over $$K$$ with a rational point of order $$p$$. The set $$S(d)$$ is finite, and $$S(1)=\{2,3,5,7\}$$ [B. Mazur, Publ. Math., Inst. Hautes Étud. Sci. 47, 33-186 (1977; Zbl 0394.14008)], $$S(2)=\{2,3,5,7,11,13\}$$ [S. Kamienny, Commun. Algebra 23, 2167-2169 (1995; Zbl 0832.14023)]. More generally if $$p\in S(d)$$ then $$p\leq(1+3^{d/2})^2$$ if $$d\not=3$$ and $$p\leq 37<(1+3^{3/2})^2$$ or $$p=43$$ if $$d=3$$ (J. Oesterlé).
In this article the author explores in more detail the case $$d=3$$. He can indeed prove the following: If for all prime numbers $$17\leq p\leq 43$$ the winding quotient of $$J_1(p)$$ has rank $$0$$ over $${\mathbb Q}$$ (*), then $$\{2,3,5,7,11,13\}\subset S(3)\subset\{2,3,5,7,11,13,17\}$$.
This result answers a question of S. Kamienny and B. Mazur in [Astérisque 228, 81-98; appendix 99-100 (1995; Zbl 0846.14012)]. Note that the condition (*) is certainly satisfied if the Birch – Swinnerton-Dyer conjecture is assumed to be true.
In his proof the author deals with new difficulties arising essentially from the fact that to get such a sharp result, one has to “reduce at the prime number $$\ell=2$$” instead of $$\ell=3$$: It turns out that $$X_0(p)$$ has its genus too small for all the interesting prime numbers $$p$$, and Kamienny’s formal immersion technique is no longer useful for $$\ell=2$$. The author must work with a suitable model of $$X_1(p)$$ instead of $$X_0(p)$$, so that he can appeal to Mazur’s formal immersion technique at $$\ell=2$$, and this explains why a conjecture occurs, on the rank of $$J_1^e({\mathbb Q})$$.

##### MSC:
 11G05 Elliptic curves over global fields 14G05 Rational points 14H52 Elliptic curves
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##### References:
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