Icosahedral galois extensions and elliptic curves

Summary This paper is devoted to the last unsolved case of the Artin Conjecture in two dimensions. Given an irreducible 2-dimensional complex representation of the absolute Galois group of a number fieldF, the Artin Conjecture states that the associatedL-series is entire. The conjecture has been proved for all cases except the icosahedral one. In this paper we construct icosahedral representations of the absolute Galois group of ℚ(√5) by means of 5-torsion points of an elliptic curve defined 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 representation. 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 from the representation. In this paper we obtain the following result: letρ be an elliptic Galois representation over ℚ(√5) of the type above, and letL(s, ρ) be the corresponding L-series. If there exists a Hilbert modular formf of weight one such thatL(s, f) ≡L(s, ρ) modulo a certain ideal above (√5), then the Artin conjecture is true forρ. This article was processed by the author using the LATEX style filecljour1m from Springer-Verlag.