Zeros of order 2 of Dedekind zeta functions and Artin's conjecture
type: Item
instance of: scholarly article
title: Zeros of order 2 of Dedekind zeta functions and Artin's conjecture
zbMATH Open document ID: 0703.11063
DOI: 10.1016/0021-8693(90)90173-L
author: Elsie Sterbin Gottlieb
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: In this paper the following beautiful theorems are proved, (after introducing the notation \(_ 2(3)^{{\hat{\;}}}\), which stands for any non-trivial semi-direct product of the quaternion group of order eight by a cyclic 3-group): Theorem I. Let E/F be a Galois extension of number fields with Galois group G and let \(s_ 0\in {\mathbb{C}}\setminus \{1\}\). Assume the Dedekind \(\zeta\)-function of E has a zero of order \(\leq 2\) at \(s_ 0\). Assume further that if \(F\subseteq F_ 1\subseteq E_ 1\subseteq E\), with \(E_ 1/F_ 1\) a Galois extension with Galois group isomorphic to \(SL_ 2(p)\), for some odd prime p, or isomorphic to \(_ 2(3)^{{\hat{\;}}}\), then all Artin L-series \(L(s,\psi,E_ 1/F_ 1)\) are analytic at \(s_ 0\) for all characters \(\psi\) of \(Gal(E_ 1/F_ 1)\). Then L(s,\(\chi\),E/F) is analytic at \(s_ 0\), for all characters \(\chi\) of G. Corollary. Let E/F be a Galois extension of number fields with Galois group G and let \(s_ 0\in {\mathbb{C}}\setminus \{1\}\). Assume the Dedekind \(\zeta\)-function of E has a zero of order \(\leq 2\) at \(s_ 0\). Any of the following conditions is sufficient to ensure that all Artin L-series are analytic at \(s_ 0:\) (i) the order of G is prime to 3, (ii) G has abelian Sylow 2-subgroups, or (iii) G has dihedral Sylow 2-subgroups. By quoting the resuls of R. Langlands on base change, it is even true that Artin's conjecture on the integrality of L-functions for any Galois extension of a number field whose Galois group modulo its center is isomorphic to \(A_ 4\). Therefore we can reformulate Theorem I into Theorem II: Theorem II. Let E/F be a Galois extension of number fields with Galois group G and let \(s_ 0\in {\mathbb{C}}\setminus \{1\}\). Assume the Dedekind \(\zeta\)-function of E has a zero of order \(\leq 2\) at \(s_ 0\). Assume further that if \(F\subseteq F_ 1\subseteq E_ 1\subseteq E\) with \(E_ 1/F_ 1\) a Galois extension with Galois group isomorphic to \(SL_ 2(p)\) for some odd prime \(p\geq 5\), then a…
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 20C10
Mathematics Subject Classification ID: 20D05
zbMATH DE Number: 4152510
zbMATH Keywords: Artin conjecture on L-functions
zbMATH Keywords: Artin L-functions
zbMATH Keywords: Dedekind \(\zeta \) - function
zbMATH Keywords: character theory of finite groups
zbMATH Keywords: Goldschmidt's transfer theorem
zbMATH Keywords: finite simple groups
reviewed by: Robert W. van der Waall
MaRDI profile type: Publication
Wikidata QID: Q123111004
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description: scientific article; zbMATH DE number 4152510