Special values of Artin L-series
Let k be a number field, and \(\rho\) a complex representation of Gal\((\bar k/k)\) with character \(\chi\). Let \(L(\rho,s)=L(\chi,s)\) be the Artin L- series of \(\rho\). Assume that the values of \(L(\rho,s)\) at the negative integers are not all zero. As is well known, this implies that k is totally real and that \(\rho\) is totally-even or totally-odd (see chapter 1). The purpose of this paper is to prove two results concerning \(\{L(\rho,-h),\quad h=0,1,2,...\}.\) Our first result asserts that this sequence determines \(L(\rho,s)\) as a Dirichlet series with a functional equation of Artin type. - Our second result concerns a certain power series in (T-1), \(f(\rho,T)\), which is canonically associated to \(\{L(\rho,-h),\quad h=0,1,2,...\}\) and which has coefficients in the field \({\mathbb{Q}}(\chi)\) obtained by adjoining the values of \(\chi\) to \({\mathbb{Q}}\). We show that, under the assumption of the p-adic Artin conjecture and the vanishing of Iwasawa's \(\mu\)-invariant, all the coefficients of \(f(\rho,T)\), except perhaps the first, are integral at all primes \({\mathfrak p}\) of \({\mathbb{Q}}(\chi)\) that lie over p. Conversely, the integrality of this power series at \({\mathfrak p}\) will imply the p-adic Artin conjecture for \(\rho\) and its twists by the Teichmüller character.
scientific article; zbMATH DE number 4095621