On icosahedral Artin representations
The authors prove the (strong) Artin conjecture in the icosahedral case for an infinite class of representations. Let \(\rho: \text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})\to \text{GL}_2 (\mathbb{C})\) be an odd, continuous, irreducible representation. They prove that for \(\rho\) icosahedral (under some restrictions) the Artin \(L\)-series for \(\rho\) is the Mellin transform of a weight 1 newform and is an entire function (Theorem A). The proof follows a strategy outlined in [\textit{R. Taylor}, Pac. J. Math., Spec. Iss., 337-347 (1998; Zbl 0942.11031)]. The main result follows from the special case (Theorem 4.1) and slight extensions of results in [\textit{N. Shepherd-Barron} and \textit{R. Taylor}, J. Am. Math. Soc. 10, 283-298 (1997; Zbl 1015.11019), \textit{M. Dickinson}, Duke Math. J. 109, 319-382 (2001; Zbl 1015.11020) and \textit{K. Buzzard} and \textit{R. Taylor}, Ann. Math. (2) 149, 905-919 (1999; Zbl 0965.11019)].
scientific article; zbMATH DE number 1820840