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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.

##### MSC:
 11G05 Elliptic curves over global fields 11G18 Arithmetic aspects of modular and Shimura varieties 11F11 Holomorphic modular forms of integral weight
##### Keywords:
elliptic curve; rational point; modular curve
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##### References:
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