Poles of Artin L - functions and the strong Artin

We show that if the L-function of an irreducible 2-dimensional complex Galois representation over Q is not automorphic then it has infinitely many poles. In particular, the Artin conjecture for a single representation implies the corresponding strong Artin conjecture. Introduction Let ρ : Gal(Q/Q) → GLn(C) be an irreducible continuous representation of the absolute Galois group of Q. Brauer [2] proved that the Artin L-function L(s, ρ) associated to ρ has meromorphic continuation to the complex plane and satisfies a functional equation of the form (1) γ(s)L(s, ρ) = εN1/2−sγ(1 − s)L(1 − s, ρ̄), where ρ̄ is the conjugate representation, |ε| = 1, N is a positive integer, and γ(s) is a certain product of Γ functions canonically associated to ρ. The famous Artin conjecture [1] asserts that L(s, ρ) is entire, with the exception of a pole at s = 1 if ρ is trivial. Moreover, Langlands’ modularity conjecture, also called the strong Artin conjecture, predicts that L(s, ρ) is automorphic, i.e. equal to L(s, π) for some cuspidal automorphic representation π of GLn(AQ). Taken as global statements, these conjectures, which are not settled in any dimension n ≥ 2, are equivalent in dimensions 2 and 3; this follows from the converse theorems of Weil [20] and Jacquet, Piatetski-Shapiro and Shalika [11], [12], together with a calculation of the ε-factors carried out by Langlands [15] for GL(2) and later simplified by Deligne [8]. However, for a single representation, the strong Artin conjecture appears strictly stronger in general, given our current state of knowledge. ∗Partially supported by the Summer Program in Japan of the NSF and Monbukagakusho. 1090 ANDREW R. BOOKER We are interested in 2-dimensional representations ρ, in which case γ(s) = π−sΓ ( s+a 2 )2 with a = 0 or 1 if ρ is even (meaning det(ρ)(c) = 1, where c denotes complex conjugation), and γ(s) = (2π)−sΓ(s) if ρ is odd (det(ρ)(c) = −1). We may further assume that ρ is icosahedral, i.e. its image in PGL2(C) is isomorphic to A5, as all other 2-dimensional cases have been shown by Langlands [16] and Tunnell [19] to be automorphic. In this article we investigate what happens should L(s, ρ) or one of its twists by a Dirichlet character have a pole. Our main result is: Theorem. If some twist L(s, ρ ⊗ χ) of L(s, ρ) by a Dirichlet character χ has a pole then L(s, ρ) has infinitely many poles. Combining this with the GL(2) converse theorem [20], we have Corollary. If L(s, ρ) is not automorphic then it has infinitely many poles. In particular, the Artin conjecture for ρ implies the strong Artin conjecture for ρ. For brevity we will describe in detail the argument for even representations and give a summary of the differences in the odd case. There are two reasons for focusing on even representations. First, there are a few technical difficulties in the even case that do not arise in the odd case. Second, there is already much in the way of evidence, both theoretical and numerical, in support of the strong Artin conjecture for odd representations (see [4], [3], [9], [14], [13], [5]); there are so far no known examples in the even case. Proceeding, consider the sum