ABSTRACT. This note gives a simple proof that certain values of Artin's L-function, for a representation ρ with character χ ρ , are stable under twisting by an even Dirichlet character χ, up to the dim(ρ)th power of the Gauss sum τ(χ) and an element generated over Q by the values of χ and χ ρ . This extends a result due to J. Coates and S. Lichtenbaum.
INTRODUCTIONWe let K|F denote a finite Galois extension of algebraic number fields, and χ will denote the character associated with a representation of Gal(K|F). Via C.L. Siegel and H. Klingen [4,8], if χ denotes a Dirichlet character and K is a totally real number field, then the Dirichlet L-function L(χ, s) lies in Q(χ) when s takes negative integer values. For a representation ρ, the associated character will simply be written as χ ρ . For general representations ρ when K is a finite Galois extension of Q, J. Coates and S. Lichtenbaum [1] decomposed the factors at infinity in the functional equation [5, VII.12.6to show that, at a negative integer s = m that is a critical point for the Artin Lfunction L(K|Q, χ ρ , s), either (I) m is odd, and the fixed field K ρ of the kernel of ρ is totally real; or (II) K ρ is totally imaginary, conjugation is central in Gal(K ρ |Q), and χ ρ (σ ) = − dim(ρ). They then employed Brauer's theorem on induced characters and functorial properties of Artin's L-function to prove that L(K|Q, χ ρ , m) ∈ Q(χ ρ ) when m is a negative integer and critical point of L(K|Q, χ ρ , s) (for all other negative integers m, this L-function trivially takes the value zero).We consider a finite Galois extension K|Q, and a representation χ ρ of Gal(K|Q). We let χ denote an even Dirichlet character, which is viewed as either a onedimensional character of Gal(K|Q) acting trivially on the conjugation automorphism, or any one-dimensional character of Gal(K|Q) if K is totally real. We also let Q(χ ρ , χ) denote the field generated over Q by the values of χ ρ and χ. The 1991 Mathematics Subject Classification. 11F67, 11F80, 11L05, 11M06.