The trace form for tame, abelian extensions of number fields

1996 ◽  
Vol 119 (2) ◽  
pp. 209-230
Author(s):  
M. J. Taylor
Keyword(s):  
Galois Group ◽  
Grothendieck Group ◽  
Number Fields ◽  
Abelian Extension ◽  
Base Field ◽  
Trace Form ◽  
Abelian Extensions ◽  
Ring Of Integers ◽  
Object Of Study ◽  
Image Position

Let N/K be a tame, abelian extension of number fields, whose Galois group is denoted by Γ. The basic object of study in this paper is the ring of integers of N, endowed with the trace form TN/K; the pair is then a Hermitian module (where we abbreviate ), and it restricts to a -Hermitian module (The reader is referred to Section 2 for the basics on Hermitian modules.) Ideally one would like to determine completely the class of this Hermitian module in K0H(ℤΓ), the Grothendieck group of ℤΓ-Hermitian modules modulo orthogonal sums; however, in general when Γ is even, one knows that even the ℚΓ-Hermitian module given by restricting (N, TN/K) is difficult to classify. (See for instance [S] and [F2].) To circumvent this difficulty we proceed in the following fashion, as suggested by the recent work of P. Lawrence (see [L]): let D = D(Γ) denote the anti-diagonal of Γ in Γ × Γ, that is to sayand let N(2) = (N⊗KN)D so that N(2) is a Galois algebra over K with Galois group Γ × Γ/D(Γ) ≅ Γ. Write for the trace form of N(2)/K, and define the -order it is then easy to see that is isomorphic as an -module to (see (3·1·6)). To be precise we ought really to write etc.; however, the base field will always be clear from the context.

Ramification Groups of Abelian Local Field Extensions

1971 ◽  
Vol 23 (2) ◽  
pp. 271-281 ◽  
Author(s):  
Murray A. Marshall
Keyword(s):  
Prime Ideal ◽  
Local Field ◽  
Galois Extension ◽  
Residue Class ◽  
Abelian Extensions ◽  
Ring Of Integers ◽  
Field Extensions ◽  
Valued Field ◽  
Residue Class Field ◽  
Image Position

Let k be a local field; that is, a complete discrete-valued field having a perfect residue class field. If L is a finite Galois extension of k then L is also a local field. Let G denote the Galois group GL|k. Then the nth ramification group Gn is defined bywhere OL, denotes the ring of integers of L, and PL is the prime ideal of OL. The ramification groups form a descending chain of invariant subgroups of G:1In this paper, an attempt is made to characterize (in terms of the arithmetic of k) the ramification filters (1) obtained from abelian extensions L\k.

Trace form associated to cyclic number fields of ramified odd prime degree

2019 ◽  
Vol 19 (04) ◽  
pp. 2050080
Author(s):  
Robson R. Araujo ◽  
Ana C. M. M. Chagas ◽  
Antonio A. Andrade ◽  
Trajano P. Nóbrega Neto
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Trace Form ◽  
Prime Degree ◽  
Ring Of Integers ◽  
Algebraic Lattices ◽  
Center Density

In this work, we computate the trace form [Formula: see text] associated to a cyclic number field [Formula: see text] of odd prime degree [Formula: see text], where [Formula: see text] ramified in [Formula: see text] and [Formula: see text] belongs to the ring of integers of [Formula: see text]. Furthermore, we use this trace form to calculate the expression of the center density of algebraic lattices constructed via the Minkowski embedding from some ideals in the ring of integers of [Formula: see text].

Estimates of lattice points in the discriminant aspect over abelian extension fields

2018 ◽  
Vol 30 (3) ◽  
pp. 767-773 ◽  
Author(s):  
Wataru Takeda ◽  
Shin-ya Koyama
Keyword(s):  
Number Field ◽  
Zeta Functions ◽  
Number Fields ◽  
Upper Bounds ◽  
Abelian Extension ◽  
Lattice Points ◽  
Abelian Extensions ◽  
Lindelöf Hypothesis ◽  
Error Terms

AbstractWe estimate the number of relatively r-prime lattice points in {K^{m}} with their components having a norm less than x, where K is a number field. The error terms are estimated in terms of x and the discriminant D of the field K, as both x and D grow. The proof uses the bounds of Dedekind zeta functions. We obtain uniform upper bounds as K runs through number fields of any degree under assuming the Lindelöf hypothesis. We also show unconditional results for abelian extensions with a degree less than or equal to 6.

The rational characterization of certain sets of relatively Abelian extensions

1959 ◽  
Vol 251 (998) ◽  
pp. 385-425 ◽  
Keyword(s):  
Field Theory ◽  
Galois Group ◽  
Class Group ◽  
Class Field Theory ◽  
Abelian Extension ◽  
Galois Groups ◽  
Class Field ◽  
Abelian Extensions ◽  
Rational Field

Let H be a class group— in the sense of class-field theory— in the rational field P, whose order is some power of a prime l . With H there is associated an Abelian extension K of P. The purpose of this paper is to determine in rational terms and for all fields K given in the described manner, the set T(K/P) of cyclic extensions A of K of relative degree l , which are absolutely normal. In particular we shall find the ramification laws for these fields A, and the possible extension types of a group of order l by the Galois group of K, which are realized in Galois groups of fields in T(K/P). It is fundamental to the programme outlined, that we aim at obtaining purely rational criteria of determination.

scholarly journals Leading terms of Artin L-series at negative integers and annihilation of higher K-groups

2011 ◽  
Vol 151 (1) ◽  
pp. 1-22 ◽  
Author(s):  
ANDREAS NICKEL
Keyword(s):  
Galois Group ◽  
Galois Extension ◽  
Number Fields ◽  
Ring Of Integers ◽  
Annihilator Ideal ◽  
Galois Action ◽  
Higher Dimensional ◽  
Totally Real ◽  
Tamagawa Number Conjecture ◽  
Special Case

AbstractLet L/K be a finite Galois extension of number fields with Galois group G. We use leading terms of Artin L-series at strictly negative integers to construct elements which we conjecture to lie in the annihilator ideal associated to the Galois action on the higher dimensional algebraic K-groups of the ring of integers in L. For abelian G our conjecture coincides with a conjecture of Snaith and thus generalizes also the well-known Coates–Sinnott conjecture. We show that our conjecture is implied by the appropriate special case of the equivariant Tamagawa number conjecture (ETNC) provided that the Quillen–Lichtenbaum conjecture holds. Moreover, we prove induction results for the ETNC in the case of Tate motives h0(Spec(L))(r), where r is a strictly negative integer. In particular, this implies the ETNC for the pair (h0(Spec(L))(r), ), where L is totally real, r < 0 is odd and is a maximal order containing ℤ[]G, and will also provide some evidence for our conjecture.

scholarly journals PRINCIPALIZATION OF IDEALS IN ABELIAN EXTENSIONS OF NUMBER FIELDS

2009 ◽  
Vol 05 (03) ◽  
pp. 527-539
Author(s):  
SÉBASTIEN BOSCA
Keyword(s):  
Number Fields ◽  
Abelian Extension ◽  
Ideal Class ◽  
Abelian Extensions ◽  
Rational Field ◽  
General Conjecture ◽  
Field L ◽  
Class C ◽  
Totally Real

We give a self-contained proof of a general conjecture of Gras on principalization of ideals in Abelian extensions of a given field L, which was solved by Kurihara in the case of totally real extensions L of the rational field ℚ. More precisely, for any given extension L/K of number fields, in which at least one infinite place of K totally splits, and for any ideal class cL of L, we construct a finite Abelian extension F/K, in which all infinite places totally split, such that cL become principal in the compositum M = LF.

Variation of the algebraic λ-invariant over a solvable extension

2019 ◽  
pp. 1-24
Author(s):  
DANIEL DELBOURGO
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Galois Representation ◽  
Exact Formula ◽  
Abelian Extension ◽  
Hida Family ◽  
Solvable Extension ◽  
Image Position

Abstract Fix an odd prime p. Let $\mathcal{D}_n$ denote a non-abelian extension of a number field K such that $K\cap\mathbb{Q}(\mu_{p^{\infty}})=\mathbb{Q}, $ and whose Galois group has the form $ \text{Gal}\big(\mathcal{D}_n/K\big)\cong \big(\mathbb{Z}/p^{n'}\mathbb{Z}\big)^{\oplus g}\rtimes \big(\mathbb{Z}/p^n\mathbb{Z}\big)^{\times}\ $ where g > 0 and $0 \lt n'\leq n$ . Given a modular Galois representation $\overline{\rho}:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F})$ which is p-ordinary and also p-distinguished, we shall write $\mathcal{H}(\overline{\rho})$ for the associated Hida family. Using Greenberg’s notion of Selmer atoms, we prove an exact formula for the algebraic λ-invariant \begin{equation} \lambda^{\text{alg}}_{\mathcal{D}_n}(f) \;=\; \text{the number of zeroes of } \text{char}_{\Lambda}\big(\text{Sel}_{\mathcal{D}_n^{\text{cy}}}\big(f\big)^{\wedge}\big) \end{equation} at all $f\in\mathcal{H}(\overline{\rho})$ , under the assumption $\mu^{\text{alg}}_{K(\mu_p)}(f_0)=0$ for at least one form f0. We can then easily deduce that $\lambda^{\text{alg}}_{\mathcal{D}_n}(f)$ is constant along branches of $\mathcal{H}(\overline{\rho})$ , generalising a theorem of Emerton, Pollack and Weston for $\lambda^{\text{alg}}_{\mathbb{Q}(\mu_{p})}(f)$ . For example, if $\mathcal{D}_{\infty}=\bigcup_{n\geq 1}\mathcal{D}_n$ has the structure of a p-adic Lie extension then our formulae include the cases where: either (i) $\mathcal{D}_{\infty}/K$ is a g-fold false Tate tower, or (ii) $\text{Gal}\big(\mathcal{D}_{\infty}/K(\mu_p)\big)$ has dimension ≤ 3 and is a pro-p-group.

Some self-dual local rings of integers not free over their associated orders

1991 ◽  
Vol 110 (1) ◽  
pp. 5-10 ◽  
Author(s):  
N. P. Byott
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Group Ring ◽  
Finite Extension ◽  
Free Module ◽  
Abelian Extension ◽  
Local Rings ◽  
Valuation Rings ◽  
Rings Of Integers ◽  
Image Position

Let p be a prime number, and let K be a finite extension of the rational p-adic field ℚp. Let L/K be a finite abelian extension with Galois group G, and let L, K denote the valuation rings of L, K respectively. Then L is a free module of rank 1 over the group algebra KG. Defining the associated order of the extension L/K byL can be viewed as a module over the ring , and a fortiori over the group ring KG.

scholarly journals On nilpotent extensions of algebraic number fields I

1992 ◽  
Vol 125 ◽  
pp. 1-14
Author(s):  
Katsuya Miyake ◽  
Hans Opolka
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Algebraic Number ◽  
Number Fields ◽  
Abelian Extension ◽  
Central Series ◽  
Schur Multiplier ◽  
Absolute Galois Group ◽  
Algebraic Number Fields ◽  
The Absolute

The lower central series of the absolute Galois group of a field is obtained by iterating the process of forming the maximal central extension of the maximal nilpotent extension of a given class, starting with the maximal abelian extension. The purpose of this paper is to give a cohomological description of this central series in case of an algebraic number field. This description is based on a result of Tate which states that the Schur multiplier of the absolute Galois group of a number field is trivial. We are in a profinite situation throughout which requires some functorial background especially for treating the dual of the Schur multiplier of a profinite group. In a future paper we plan to apply our results to construct a nilpotent reciprocity map.

scholarly journals On separable abelian extensions of rings

1982 ◽  
Vol 5 (4) ◽  
pp. 779-784
Author(s):  
George Szeto
Keyword(s):  
Automorphism Group ◽  
Galois Group ◽  
Abelian Extension ◽  
Azumaya Algebra ◽  
Abelian Extensions

LetRbe a ring with1,G(=〈ρ1〉×…×〈ρm〉)a finite abelian automorphism group ofRof ordernwhere〈ρi〉is cyclic of orderni. for some integersn,ni, andm, andCthe center ofRwhose automorphism group induced byGis isomorphic withG. Then an abelian extensionR[x1,…,xm]is defined as a generalization of cyclic extensions of rings, andR[x1,…,xm]is an Azumaya algebra overK(=CG={c   in   C/(c)ρi=c   for each   ρi   in   G})such thatR[x1,…,xm]≅RG⊗KC[x1,…,xm]if and only ifCis Galois overKwith Galois groupG(the Kanzaki hypothesis).

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