Sylow 2-subgroups of Galois Groups Arising as Minimal Counterexamples to Artin’s Conjecture

Let E/F be a Galois extension of number fields with Galois group G. The purpose of this paper is to place limitations on the structure of a Sylow 2-subgroup of G in the case when the extension E/F is a minimal counterexample to Artin’s Conjecture on the holomorphy of L-series. More specifically, assume for some s0 ∈ C − {1} and some irreducible character χ of G that the Artin L-series L(s, χ,E/F ) has a pole at s0. Assume further that this is a minimal configuration in the sense that: for no intermediate Galois extension E1/F1 of smaller degree, where F ⊆ F1 ⊆ E1 ⊆ E, and no irreducible character ψ of Gal(E1/F1) does L(s, ψ,E1/F1) have a pole at s0. Then we show that a Sylow 2-subgroup of G has a faithful complex representation of degree r, where r is the order of zero of the zeta function of E at s0. In particular, this implies that the 2-rank of G is at most r.