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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.

11G05 Elliptic curves over global fields
11G18 Arithmetic aspects of modular and Shimura varieties
11F11 Holomorphic modular forms of integral weight
Full Text: DOI Numdam EuDML
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