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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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.
KeywordsExact Sequence Elliptic Curve Elliptic Curf Galois Group Abelian Variety
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