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A. Geometrical construction of 2-dimensional galois representations of A5-type. B. On the realisation of the groups PSL2(1) as galois groups over number fields by means of l-torsion points of elliptic curves

  • Martin Kinzelbach
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Part of the Lecture Notes in Mathematics book series (LNM, volume 1585)

Abstract

Let k be a number field which contains the fifth roots of unity and K/k an A5-extension. According to Klein and Serre K/k can be described by adjunction of the 5-torsion points of a suitable elliptic curve E. It is well known that there exists such a curve E defined over k if and only if there exists a quadratic overfield M of K such that M is Galois over k with group <5. This corresponds to a 2-dimensional Galois representation of the Galois group G(k) of k with trivial determinant.

In general there exists a curve E defined over a quadratic extension of k. We show that if there exists an overfield N of K such that N is Galois over k with group <2 (corresponding to a 2-dimensional Galois representation of G(k) with determinant of order 2), and if N is not of a special exceptional type, then there exists a curve D of genus 2 defined over k such that K/k can be described by means of coordinates of 5-torsion points of D.

Keywords

Exact Sequence Elliptic Curve Elliptic Curf Galois Group Abelian Variety 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Martin Kinzelbach
    • 1
  1. 1.Institut für Experimentelle MathematikUniversität GH EssenEssenGermany

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