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**On icosahedral Artin representations. II.**
*(English)*
Zbl 1031.11031

Let \(\rho: G_{\mathbb{Q}}\to GL_2(\mathbb{C})\) be an irreducible continuous representation with \(\text{det }\rho(c)= -1\) (here \(c\) denotes a complex conjugation). Suppose that if the projective image of \(\rho\) is isomorphic to \(A_5\) then the projective image of the inertia group at \(3\) has odd order and the projective image of the decomposition group at \(5\) is unramified of order \(2\). The author proves the Artin conjecture in this case: \(\rho\) is modular and its Artin \(L\)-function \(L(\rho,s)\) is entire (Theorem A). He gives some concrete examples where all the conditions are satisfied.

This paper is a sequel to [K. Buzzard, M. Dickinson, N. Shepherd-Barron and R. Taylor, Duke Math. J. 109, 283-318 (2001; Zbl 1015.11021)], where they proved the Artin conjecture for certain odd icosahedral representations of \(G_{\mathbb{Q}}\) by proving that they were modular. The key innovation in this paper is to work with the prime \(5\) rather than the prime \(2\). The key ingredients in the proof of the main result are: base change arguments combined with a method of Ramakrishna [Ramakrishna, preprint], and an extension to totally real fields of results of Wiles and Taylor [A. Wiles, Ann. Math. (2) 141, 443-551 (1995; Zbl 0823.11029); R. Taylor and A. Wiles, Ann. Math. (2) 141, 553-572 (1995; Zbl 0823.11030)].

This paper is a sequel to [K. Buzzard, M. Dickinson, N. Shepherd-Barron and R. Taylor, Duke Math. J. 109, 283-318 (2001; Zbl 1015.11021)], where they proved the Artin conjecture for certain odd icosahedral representations of \(G_{\mathbb{Q}}\) by proving that they were modular. The key innovation in this paper is to work with the prime \(5\) rather than the prime \(2\). The key ingredients in the proof of the main result are: base change arguments combined with a method of Ramakrishna [Ramakrishna, preprint], and an extension to totally real fields of results of Wiles and Taylor [A. Wiles, Ann. Math. (2) 141, 443-551 (1995; Zbl 0823.11029); R. Taylor and A. Wiles, Ann. Math. (2) 141, 553-572 (1995; Zbl 0823.11030)].

Reviewer: A.Dąbrowski (Szczecin)

### MSC:

11F80 | Galois representations |

11F33 | Congruences for modular and \(p\)-adic modular forms |

11G18 | Arithmetic aspects of modular and Shimura varieties |