ICOSAHEDRAL GALOIS EXTENSIONS AND ELLIPTIC-CURVES

This paper is devoted to the last unsolved case of the Artin Conjectur e in two dimensions. Given an irreducible 2-dimensional complex repres entation of the absolute Galois group of a number field F, the Artin C onjecture states that the associated L-series is entire. The conjectur e has been proved for all cases except the icosahedral one. In this pa per we construct icosahedral representations of the absolute Galois gr oup of Q(root 5) by means of 5-torsion points of an elliptic curve def ined over Q. We compute the L-series explicitely as an Euler product, giving algorithms for determining the factors at the difficult primes. We also prove a formula for the conductor of the elliptic representat ion. A feasible way of proving the Artin Conjecture in a given case is to construct a modular form whose L-series matches the one obtained f rom the representation. In this paper we obtain the following result: let rho be an elliptic Galois representation over Q(root 5) of the typ e above, and let L(s, rho) be the corresponding L-series. If there exi sts a Hilbert modular form f of weight one such that L(s, f) = L(s, rh o) module a certain ideal above (root 5), then the Artin conjecture is true for rho.