A symplectic case of Artin’s Conjecture

Abstract

An equivalent form of this conjecture is the following. If ρ is irreducible and non-trivial, then L(s, ρ) is entire. Artin proved this conjecture in the case where ρ is 1-dimensional using his reciprocity law together with a result of Hecke. As L-functions are inductive, this proved the conjecture if ρ is monomial, i.e. induced from a degree one representation of some subgroup. Now consider the case where ρ is 2-dimensional. Let ρ : G → PGL2(C) be the composition of ρ : G → GL2(C) with the natural projection of GL2(C) to PGL2(C). Let Ḡ denote the image of G under ρ. So Ḡ is a finite subgroup of PGL2(C), which is isomorphic to SO3(C). The only finite subgroups of SO3(C) are cyclic, dihedral, tetrahedral (isomorphic to A4), octahedral (isomorphic to S4) and icosahedral (isomorphic to A5). We thus classify ρ according the isomorphism type of Ḡ. If Ḡ is cyclic or dihedral, then ρ is reducible or monomial and L(s, ρ) is entire by Artin’s result. Much later, Langlands applied to this problem the theory of automorphic representations, which also have associated L-functions. For cuspidal automorphic representations of GLn the associated L-function is known to be entire [Ja]. Both automorphic and Artin L-functions can be written as Euler products of local factors L(s, π) = ∏ v L(s, πv) and L(s, ρ) = ∏ v L(s, ρv). Langlands formulated the following amazing conjecture [La1].