On The Determination Of Number Fields By Artin L-Functions (original) (raw)

Are number fields determined by Artin L-functions?

Journal of Number Theory, 2016

Let k be a number field, K/k a finite Galois extension with Galois group G, χ a faithful character of G. We prove that the Artin L-function L(s, χ, K/k) determines the Galois closure of K over Q. In the special case k = Q it also determines the character χ.

Artin's L-functions and one-dimensional characters

Journal of Number Theory, 2007

Let K/Q be a finite Galois extension with the Galois group G, let χ 1 , . . . , χ r be the irreducible non-trivial characters of G, and let A be the C-algebra generated by the Artin L-functions L(s, χ 1 ), . . . , L(s, χ r ). Let B be the subalgebra of A generated by the L-functions corresponding to induced characters of non-trivial one-dimensional characters of subgroups of G. We prove: (1) B is of Krull dimension r and has the same quotient field as A;

A Note on the Artin Conjecture

2006

Let K/Q be a Galois extension of Q and ρ : Gal(K/Q) −→ GL(n,C) a nontrivial irre-ducible representation of its Galois group. E. Artin [1] associated to this data an L-function L(s,ρ), defined for Res > 1, which he conjectured to continue analytically to an entire func-tion on the whole ...

On p-adic L-functions and cyclotomic fields

Nagoya Mathematical Journal, 1975

Let p be a prime. If one adjoins to Q all pn -th roots of unity for n = 1, 2, 3, …, then the resulting field will contain a unique subfield Q ∞ such that Q ∞ is a Galois extension of Q with Gal the additive group of p-adic integers. We will denote Gal(Q ∞/Q ) by Γ.

On l-adic Galois L-functions

2014

Introduction 1 1. Action of the absolut Galois group on fundamental groups 9 2. Measures associated to towers of projective lines 10 3. Inclusions 16 4. Inversion 18 5. Measures K 1 (z) 21 6. Congruences between coefficients 25 7. ℓ-adic poly-multi-zeta functions? 28 8. ℓ-adic L-functions of Kubota-Leopoldt 28 9. ℓ-adic functions associated to measure K 1 (−1) 31 10. Hurwitz zeta functions and Dirichlet L-series 33 11. ℓ-adic L-functions of Z[1/m] 39 References 41

3 on the Structure of the Galois Group of the Abelian Closure of a Number Field

2016

Following a paper by Athanasios Angelakis and Peter Stevenhagen on the determination of imaginary quadratic fields having the same absolute Abelian Galois group A, we study this property for arbitrary number fields. We show that such a property is probably not easily generalizable, apart from imaginary quadratic fields, because of some p-adic obstructions coming from the global units. By restriction to the p-Sylow subgroups of A, we show that the corresponding study is related to a generalization of the classical notion of p-rational fields. However, we obtain some non-trivial information about the structure of the profinite group A, for every number field, by application of results published in our book on class field theory. Résumé.-A partir d'un article de Athanasios Angelakis et Peter Stevenhagen sur la détermination de corps quadratiques imaginaires ayant le même groupe de Galois Abélien absolu A, nous étudions cette propriété pour les corps de nombres quelconques. Nous montrons qu'une telle propriété n'est probablement pas facilement généralisable, en dehors des corps quadratiques imaginaires, en raison d'obstructions p-adiques provenant des unités globales. En se restreignant aux p-sous-groupes de Sylow de A, nous montrons que l'étude correspondante est liée à une généralisation de la notion classique de corps p-rationnels. Cependant, nous obtenons des informations non triviales sur la structure du groupe profini A, pour tout corps de nombres, par application de résultats publiés dans notre livre sur la théorie du corps de classes.

On p-adic Artin L-functions

Nagoya Mathematical Journal, 1983

In this paper we will discussp-adic ArtinL-functions. The existence of these functions is a simple consequence of a theorem of Deligne and Ribet [4]. One can formulate a “p-adic Artin conjecture” for these functions. Our primary purpose here is to relate this conjecture to the “main conjecture” discussed by Coates in [3]. We will describe the precise formulations of these conjectures that we will use later. Our main result will be that in fact the main conjecture implies thep-adic Artin conjecture.