Modularity of Solvable Artin Representations of Go(4)-type

Let F be a number field, and (ρ, V ) a continuous, n-dimensional representation of the absolute Galois group Gal(F/F ) on a finite-dimensional C-vector space V . Denote by L(s, ρ) the associated L-function, which is known to be meromorphic with a functional equation. Artin’s conjecture predicts that L(s, ρ) is holomorphic everywhere except possibly at s = 1, where its order of pole is the multiplicity of the trivial representation in V . The modularity conjecture of Langlands for such representations ([La3]), often called the strong Artin conjecture, asserts that there should be an associated (isobaric) automorphic form π = π∞⊗πf on GL(n)/F such that L(s, ρ) = L(s, πf ). Since L(s, πf ) possesses the requisite properties ([JS]), the modularity conjecture implies the Artin conjecture.