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On the structure of elliptic fields. I. (English. Russian original) Zbl 0595.14022
Math. USSR, Izv. 27, 39-51 (1986); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 49, No. 4, 719-730 (1985).
Let \({\mathcal T}\) be the elliptic curve \(y^ 2=x^ 3+rx+s\) over a field k. For any natural number \(m\in {\mathbb{N}}\), \(m\geq 2\), denote by k(m) (resp. \(K(m^{\nu})\), where \(\nu\in {\mathbb{N}})\) the field obtained from k by adjoining the coordinates of a primitive torsion point on \({\mathcal T}\) of order m (resp. of all torsion points on \({\mathcal T}\) of order \(m^{\nu})\). Thus \(K(m^{\nu})\) is the field of \(m^{\nu}\)-division points of \({\mathcal T}\) over k. Note that \(K(m^{\nu})\) is a normal extension of k which is Galois if the characteristic p of k does not divide m. In particular, K(m) is the normal closure of k(m)/k and is the Galois closure of k(m)/k provided that \(p\nmid m.\)
Choosing the ground field \(k:={\mathbb{Q}}(r,s)\), the author gives an explicit description of the structure of the field extensions \(K(m^{\nu})/k(m)\). In fact he shows by explicit calculation that, for \(m>2\) and \(\nu\in {\mathbb{N}}\), \(K(m)=k(m)(\beta_ 1,\beta_ 2)\) and \(K(m^{\nu +1})=K(m^{\nu})(\alpha_{1\nu},\alpha_{2\nu},\alpha_{3\nu},\alpha_{4n})\), where \(\beta_ 1:=e^{2\pi i/m}\), \(\alpha_{1\nu}:=^{m^{\nu}}\sqrt{\beta_ 1}\) and \(\beta_ 2\) or \(\alpha_{2\nu},\alpha_{3\nu},\alpha_{4\nu}\) are essentially an m-th root of certain products of differences of x-coordinates of m-division points or \(m^{\nu}\)-division points on \({\mathcal T}\), respectively. A similar theorem holds also for \(m=2\).
The explicit and involved calculations rely partly on earlier work of the author and are hard to follow.
Reviewer: H.G.Zimmer

MSC:
14H45 Special algebraic curves and curves of low genus
14H52 Elliptic curves
11R58 Arithmetic theory of algebraic function fields
14H05 Algebraic functions and function fields in algebraic geometry
11R32 Galois theory
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