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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

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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 A 5-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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Literature

  • [Buh] Buhler, J.P., Icosahedral Galois Representations, L.N.M. 654, Springer, Berlin, 1978.

    MATH  Google Scholar 

  • [F-K] Frey, G., Kani, E., Curves of Genus 2 covering elliptic curves and an arithmetical application, in: v.d. Geer, G., Oort, F., Steenbrink, J., Arithmetic Algebraic Geometry, Progress in Math. 89, Birkhäuser, Boston, 1991.

    Google Scholar 

  • [Fri] Fricke, R., Lehrbuch der Algebra, Bd. 2, Vieweg, Braunschweig, 1926.

    MATH  Google Scholar 

  • [Kan] Kani, E., Curves of Genus 2 with Elliptic Differentials and the Height Conjecture for Elliptic Curves, in: Frey, G. (Ed.), Proceedings of the Conference on Number Theory and Arithmetical Geometry, Inst. f. Exp. Math., Essen, 1991.

    Google Scholar 

  • [Kin] Kinzelbach, M., Konstruktion von zweidimensionalen Galoisdarstel-lungen mit Hilfe von Kurven vom Geschlecht≤2, Dissertation, Univ. Essen, 1993.

    Google Scholar 

  • [Kle] Klein, F., Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, Teubner, Leipzig, 1884.

    MATH  Google Scholar 

  • [La 1] Lang, S., Abelian Varieties, Springer, New York, 1983.

    CrossRef  MATH  Google Scholar 

  • [La 2] Lang, S., Elliptic Functions, G.T.M. 112, New York, 1987.

    Google Scholar 

  • [Sch] Scharlau, W., Quadratic and Hermitian Forms, Grundl. d. math. Wiss. 270, Springer, Berlin, 1985.

    MATH  Google Scholar 

  • [Se 1] Serre, J.-P., Extension icosaédriques, Séminaire de Théorie des Nombres de Bordeaux 1979/80, Exp. 19 or Ęvres III, 123, pp 550–554.

    Google Scholar 

  • [Se 2] Serre, J.-P., L'invariant de Witt de la forme Tr(x2), Comm. Math. Hel. 59 (1984), pp 651–676 or Ęvres III,131, pp 675–700.

    Google Scholar 

  • [Shi] Shih, K., p-Division Points on Certain Elliptic Curves, Composito Mathematica 36/2, (1978), pp. 113–129.

    MathSciNet  MATH  Google Scholar 

  • [Sil] Silverman, J.H., The Arithmetic of Elliptic Curves, G.T.M. 106, Springer, New York, 1986.

    MATH  Google Scholar 

  • [Slo] Slodowy, P., Das Ikosaeder und die Gleichungen fünften Grades, in: Mathematische Miniaturen 3, Birkhäuser, Basel, 1986.

    Google Scholar 

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© 1994 Springer-Verlag

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Kinzelbach, M. (1994). 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. In: Frey, G. (eds) On Artin's Conjecture for Odd 2-dimensional Representations. Lecture Notes in Mathematics, vol 1585. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0074109

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  • DOI: https://doi.org/10.1007/BFb0074109

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58387-5

  • Online ISBN: 978-3-540-48681-7

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