Icosahedral Galois extensions and elliptic curves (Q1360923)

This paper is devoted to the last unsolved case of the Artin conjecture in two dimensions. Given an irreducible two-dimensional complex representation of the absolute Galois group of a number field \(F\), the Artin conjecture states that the associated \(L\)-function is entire. The conjecture has been proven in all cases except the icosahedral one. The way one proves it is to use lifting techniques to construct a cuspidal modular form whose \(L\)-function coincides with the given one. In the paper, icosahedral representations \(\rho\) for \(F=\mathbb{Q}(\sqrt 5)\) are constructed and their \(L\)-series are computed explicitly. Then it is shown that the existence of a modular form \(f\) whose \(L\)-function coincides with \(L(s,\rho)\) only up to a certain ideal will already imply the Artin conjecture for this representation \(\rho\).