Number fields generated by the 3-torsion points of an elliptic curve
Identifieur interne : 000160 ( Main/Exploration ); précédent : 000159; suivant : 000161Number fields generated by the 3-torsion points of an elliptic curve
Auteurs : Andrea Bandini [Italie] ; Laura Paladino [Italie]Source :
- Monatshefte für Mathematik [ 0026-9255 ] ; 2012-11-01.
English descriptors
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Abstract
Abstract: Let $${{\mathcal{E}}}$$ be an elliptic curve, m a positive number and $${\mathcal{E}[m]}$$ the m-torsion subgroup of $${\mathcal{E}}$$ . Let P 1 = (x 1, y 1), P 2 = (x 2, y 2) form a basis of $${\mathcal{E}[m]}$$ . We prove that $${\mathbb Q(\mathcal{E}[m]) = \mathbb Q(x_1, x_2, \zeta_m, y_1)}$$ in general. For the case m = 3 we provide a description of all the possible extensions $${\mathbb Q(\mathcal{E}[3])}$$ in terms of generators, degree and Galois groups.
Url:
DOI: 10.1007/s00605-012-0377-x
Affiliations:
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<front><div type="abstract" xml:lang="en">Abstract: Let $${{\mathcal{E}}}$$ be an elliptic curve, m a positive number and $${\mathcal{E}[m]}$$ the m-torsion subgroup of $${\mathcal{E}}$$ . Let P 1 = (x 1, y 1), P 2 = (x 2, y 2) form a basis of $${\mathcal{E}[m]}$$ . We prove that $${\mathbb Q(\mathcal{E}[m]) = \mathbb Q(x_1, x_2, \zeta_m, y_1)}$$ in general. For the case m = 3 we provide a description of all the possible extensions $${\mathbb Q(\mathcal{E}[3])}$$ in terms of generators, degree and Galois groups.</div>
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