Zeros of order 2 of Dedekind zeta functions and Artin's conjecture
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 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 \(s_ 0\in {\mathbb{C}}\setminus \{1\}\). If G is solvable and if the Dedekid \(\zeta\)-function of E has a zero of order \(\leq 2\) at \(s_ 0\), then all Artin L-series are analytic at \(s_ 0.\) Finally it is shown that the groups \(SL_ 2(p)\), p prime, \(p\geq 5\), and \(_ 2(3)^{{\hat{\;}}}\) cannot be eliminated as minimal obstructions by arguments which solely rely on the properties of L-functions used in the paper. Theorem III. Suppose G is a finite group with a virtual character \(\theta\) satisfying: (1) \(\theta\) (1)\(\leq 2\), (2) \(\theta\) is not a character of G but its restriction to every proper subgroup is a character, and (3) if \(\chi\) is any irreducible constituent of \(\theta\) such that \(<\theta,\chi ><0\) then \(\chi\) is faithful, nonlinear, and not induced from any proper subgroup of G. Then \(\theta (1)=2\), \(G\cong SL_ 2(p)\) or \(_ 2(3)^{{\hat{\;}}}\) for some prime \(p\geq 5\), and \(\theta\) is one of the virtual characters described in Propositions I, II, and III in Section 3 of this paper. The proofs of the theorems are given by means of a massive excursion through the character theory of finite groups, Goldschmidt's transfer theorem, the classification of the finite simple groups. To close with, let me say that I was once again impressed by the force of group theory in dealing with Artin's conjecture on L-functions.
scientific article; zbMATH DE number 4152510
scientific article