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 \(J_1^e({\mathbb Q})\).
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 \(J_1^e({\mathbb Q})\).
Reviewer: Federico Pellarin (Caen)
Keywords:
elliptic curves over number fields with small degree; torsion points; modular symbols; winding quotient; Birch Swinnerton-Dyer conjectureReferences:
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