×

zbMATH — the first resource for mathematics

No 17-torsion on elliptic curves over cubic number fields. (English) Zbl 1072.11037
For an integer \(d\), let \(S(d)\) denote the set of prime numbers \(p\) for which there exists an elliptic curve \(E\) over a number field \(K\) with \([K: \mathbb{Q}]= d\) and a point \(P\) in \(E(K)\) of order \(p\). In the author’s previous paper [Ann. Inst. Fourier 50, No. 3, 723–749 (2000; Zbl 0971.11030)], it is shown that \(S(3)= \{2,3,5,7,11,13\) and maybe \(17\}\).
The goal of the present article is to show that 17 does not belong to \(S(3)\). Suppose that there exists an elliptic curve over a cubic number field \(K\) endowed with a \(K\)-rational point of order 17. Then one can associate with it a point \(P= (p_1 ,p_2, p_3)\in X_1(17)^{(3)}(\mathbb{Z}[1/17])\) (symmetric power) such that \(p_i\) are generically non-cuspidal points, but \(P\) coincides in the fiber at 2 with a triplet of cusps \(P_2\in X_1(17)^{(3)}\) above the cusp \(3.\infty\in X_0(17)^{(3)}\). He focuses on a morphism \(F_{P_0}: X_1(17)^{(3)}\to J_1(17)\) defined by \(F_{P_0}(Q)= t(Q- P_0)\), where \(t\) is an element of the Hecke algebra \(\mathbb{T}_{\Gamma_1(17)}\) which kills the 2-torsion of \(J_1(17)\).
In order to derive a contradiction, he appeals to [loc. cit., 1.5] by which he is reduced to showing that 1) \(F_{P_0}\) is a formal immersion at \(P_0(\mathbb{F}_2)\) and 2) no non-cuspidal point of \(X_1(17)^{(3)}(\mathbb{Z})\) is mapped by \(F_{P_0}\) to the nontrivial section of a \(\mu_2\)-subscheme of \(J_1(17)_{/\mathbb{Z}}\). The point 1) is already shown to hold in [loc.cit., 4.3], and he finishes the proof by showing in this paper the validity of the assertion 2) using elementary theory of formal groups.

MSC:
11G05 Elliptic curves over global fields
11G18 Arithmetic aspects of modular and Shimura varieties
11F11 Holomorphic modular forms of integral weight
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] Deninger, C., Nart, E., Formal groups and L-series. Comment. Math. Helvetici65 (1990), 318-333. · Zbl 0741.14026
[2] Honda, T., On the theory of commutative formal groups. J. Math. Soc. Japan22 (1970), 213-246. · Zbl 0202.03101
[3] Ishii, N., Momose, F., Hyperelliptic modular curves. Tsukuba J. Math.15 no. 2 (1991), 413-423. · Zbl 0771.14008
[4] Kamienny, S., Torsion points on elliptic curves over all quadratic fields. Duke Math. J.53 no. 1 (1986), 157-162. · Zbl 0599.14029
[5] Kato, K., p-adic Hodge theory and values of zeta-functions of modular forms. To appear in Astérisque. · Zbl 1142.11336
[6] Mazur, B., Modular curves and the Eisenstein ideal. Publications mathématiques de l’I.H.E.S.47 (1977), 33-186. · Zbl 0394.14008
[7] Parent, P., Torsion des courbes elliptiques sur les corps cubiques. Ann. Inst. Fourier50 (2000), 723-749. · Zbl 0971.11030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.