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Torsion of rational elliptic curves over cubic fields. (English) Zbl 1358.11068

The aim of this article is to determine the torsion groups over cubic fields of an elliptic curve \(E\) defined over \(\mathbb Q\) with the torsion group over \(\mathbb Q\) isomorphic to a given group \(G\). The set \(\Phi_{\mathbb Q}(d)\) of possible structures of torsion groups over number fields of degree \(1,2\) and \(3\) of elliptic curves rational over \(\mathbb Q\) are determined by B. Mazur [Publ. Math., Inst. Hautes Étud. Sci. 47, 33–186 (1977; Zbl 0394.14008); Invent. Math. 44, 129–162 (1978; Zbl 0386.14009)] in the case \(d=1\) and the second author [Math. Res. Lett. 23, No. 1, 245–272 (2016; Zbl 1416.11084)] in other cases. The authors determine the set \(\Phi_{\mathbb Q}(3,G)\) of possible structures of torsion groups over a cubic number field of an elliptic curve \(E\) defined over \(\mathbb Q\) such that \(E(\mathbb Q)_{\text{tors}}\simeq G\in \Phi_{\mathbb Q}(1)\). Further, they determine explicitly all elliptic curves \(E\) and all cubic fields \(K\) up to isomorphisms such that \(E(\mathbb Q)_{\text{tors}}\neq E(K)_{\text{tors}}\). In particular, for an elliptic curve \(E\), they show that there is at most one cubic field \(K\), up to isomorphisms, such that \(E(\mathbb Q)_{\text{tors}}\neq E(K)_{\text{tors}}\simeq H\) for a fixed \(H\in\Phi_{\mathbb Q}(3)\) and that there are at most three non-isomorphic cubic fields \(K\) such that \(E(\mathbb Q)_{\text{tors}}\neq E(K)_{\text{tors}}\).

MSC:

11G05 Elliptic curves over global fields
11R16 Cubic and quartic extensions
14G05 Rational points
14H52 Elliptic curves

Software:

Magma
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References:

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