Nonmonomial characters and Artin’s conjecture

Abstract

If E / F E/F is a Galois extension of number fields with solvable Galois group G G , the main result of this paper proves that if the Dedekind zeta-function of E E has a zero of order less than M G {\mathcal {M}_G} at the complex point s 0 ≠ 1 {s_0} \ne 1 , then all Artin L L -series for G G are holomorphic at s 0 {s_0} — here M G {\mathcal {M}_G} is the smallest degree of a nonmonomial character of any subgroup of G G . The proof relies only on certain properties of L L -functions which are axiomatized to give a purely character-theoretic statement of this result.