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Zbl 1072.11037
Parent, Pierre
No 17-torsion on elliptic curves over cubic number fields.
(English)
[J] J. ThÃ©or. Nombres Bordx. 15, No. 3, 831-838 (2003). ISSN 1246-7405

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 $[K: \bbfQ]= d$ 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 $17\}$.\par 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 $P= (p_1 ,p_2, p_3)\in X_1(17)^{(3)}(\bbfZ[1/17])$ (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 $\bbfT_{\Gamma_1(17)}$ which kills the 2-torsion of $J_1(17)$.\par 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 $P_0(\bbfF_2)$ and 2) no non-cuspidal point of $X_1(17)^{(3)}(\bbfZ)$ is mapped by $F_{P_0}$ to the nontrivial section of a $\mu_2$-subscheme of $J_1(17)_{/\bbfZ}$. 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.
[Fumio Hazama (Hatoyama)]
MSC 2000:
*11G05 Elliptic curves over global fields
11G18 Arithmetic aspects of modular and Shimura varieties
11F11 Modular forms, one variable

Keywords: elliptic curve; rational point; modular curve

Citations: Zbl 0971.11030

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