scholarly journals THE EQUIVALENCE OF RUBIN'S CONJECTURE AND THE ETNC/LRNC FOR CERTAIN BIQUADRATIC EXTENSIONS

2013 ◽  
Vol 56 (2) ◽  
pp. 335-353 ◽  
Author(s):  
PAUL BUCKINGHAM
Keyword(s):  
Number Fields ◽  
Abelian Extension ◽  
Root Number ◽  
Tamagawa Number Conjecture

AbstractFor an abelian extension L/K of number fields, the Equivariant Tamagawa Number Conjecture (ETNC) at s = 0, which is equivalent to the Lifted Root Number Conjecture (LRNC), implies Rubin's Conjecture by work of Burns [3]. We show that, for relative biquadratic extensions L/K satisfying a certain condition on the splitting of places, Rubin's Conjecture in turn implies the ETNC/LRNC. We conclude with some examples.

scholarly journals Determining Fitting ideals of minus class groups via the equivariant Tamagawa number conjecture

2007 ◽  
Vol 143 (6) ◽  
pp. 1399-1426 ◽  
Author(s):  
Cornelius Greither
Keyword(s):  
Number Fields ◽  
Rational Numbers ◽  
Abelian Extension ◽  
Galois Module ◽  
Class Groups ◽  
Fitting Ideal ◽  
Tamagawa Number Conjecture

AbstractWe assume the validity of the equivariant Tamagawa number conjecture for a certain motive attached to an abelian extension K/k of number fields, and we calculate the Fitting ideal of the dual of clK− as a Galois module, under mild extra hypotheses on K/k. This builds on concepts and results of Tate, Burns, Ritter and Weiss. If k is the field of rational numbers, our results are unconditional.

scholarly journals GEOMETRIC GALOIS THEORY, NONLINEAR NUMBER FIELDS AND A GALOIS GROUP INTERPRETATION OF THE IDELE CLASS GROUP

2005 ◽  
Vol 16 (06) ◽  
pp. 567-593
Author(s):  
T. M. GENDRON ◽  
A. VERJOVSKY
Keyword(s):  
Galois Theory ◽  
Class Group ◽  
Number Fields ◽  
Field Extension ◽  
Abelian Extension ◽  
Galois Groups ◽  
Algebraic Number Fields ◽  
Infinite Degree ◽  
Groups Of Automorphisms ◽  
Holomorphic Extensions

This paper concerns the description of holomorphic extensions of algebraic number fields. After expanding the notion of adele class group to number fields of infinite degree over ℚ, a hyperbolized adele class group [Formula: see text] is assigned to every number field K/ℚ. The projectivization of the Hardy space ℙ𝖧•[K] of graded-holomorphic functions on [Formula: see text] possesses two operations ⊕ and ⊗ giving it the structure of a nonlinear field extension of K. We show that the Galois theory of these nonlinear number fields coincides with their discrete counterparts in that 𝖦𝖺𝗅(ℙ𝖧•[K]/K) = 1 and 𝖦𝖺𝗅(ℙ𝖧•[L]/ℙ𝖧•[K]) ≅ 𝖦𝖺𝗅(L/K) if L/K is Galois. If K ab denotes the maximal abelian extension of K and 𝖢K is the idele class group, it is shown that there are embeddings of 𝖢K into 𝖦𝖺𝗅⊕(ℙ𝖧•[K ab ]/K) and 𝖦𝖺𝗅⊗(ℙ𝖧•[K ab ]/K), the "Galois groups" of automorphisms preserving ⊕ (respectively, ⊗) only.

scholarly journals THE FRACTIONAL GALOIS IDEAL FOR ARBITRARY ORDER OF VANISHING

2011 ◽  
Vol 07 (01) ◽  
pp. 87-99 ◽  
Author(s):  
PAUL BUCKINGHAM
Keyword(s):  
Class Group ◽  
Arbitrary Order ◽  
Number Fields ◽  
Simple Zero ◽  
Abelian Extension ◽  
Class Groups ◽  
Original Proof ◽  
Galois Modules ◽  
Fitting Ideal ◽  
Stickelberger Ideal

We propose a candidate, which we call the fractional Galois ideal after Snaith's fractional ideal, for replacing the classical Stickelberger ideal associated to an abelian extension of number fields. The Stickelberger ideal can be seen as gathering information about those L-functions of the extension which are non-zero at the special point s = 0, and was conjectured by Brumer to give annihilators of class-groups viewed as Galois modules. An earlier version of the fractional Galois ideal extended the Stickelberger ideal to include L-functions with a simple zero at s = 0, and was shown by the present author to provide class-group annihilators not existing in the Stickelberger ideal. The version presented in this paper deals with L-functions of arbitrary order of vanishing at s = 0, and we give evidence using results of Popescu and Rubin that it is closely related to the Fitting ideal of the class-group, a canonical ideal of annihilators. Finally, we prove an equality involving Stark elements and class-groups originally due to Büyükboduk, but under a slightly different assumption, the advantage being that we need none of the Kolyvagin system machinery used in the original proof.

scholarly journals p-Adic measures associated with zeta values and p-adic log multiple gamma functions

2019 ◽  
Vol 15 (05) ◽  
pp. 991-1007
Author(s):  
Tomokazu Kashio
Keyword(s):  
Number Fields ◽  
Abelian Extension ◽  
Gamma Functions ◽  
Rank One ◽  
Zeta Values ◽  
Stark Conjecture ◽  
Explicit Relation ◽  
Text Unit

We study a relation between two refinements of the rank one abelian Gross–Stark conjecture. For a suitable abelian extension [Formula: see text] of number fields, a Gross–Stark unit is defined as a [Formula: see text]-unit of [Formula: see text] satisfying certain properties. Let [Formula: see text]. Yoshida and the author constructed the symbol [Formula: see text] by using [Formula: see text]-adic [Formula: see text] multiple gamma functions, and conjectured that the [Formula: see text] of a Gross–Stark unit can be expressed by [Formula: see text]. Dasgupta constructed the symbol [Formula: see text] by using the [Formula: see text]-adic multiplicative integration, and conjectured that a Gross–Stark unit can be expressed by [Formula: see text]. In this paper, we give an explicit relation between [Formula: see text] and [Formula: see text] and prove that two refinements are consistent.

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.

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.

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.

scholarly journals The Genus Field and Genus Number in Algebraic Number Fields

1967 ◽  
Vol 29 ◽  
pp. 281-285 ◽  
Author(s):  
Yoshiomi Furuta
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Abelian Extension ◽  
Normal Extension ◽  
Finite Degree ◽  
Algebraic Number Fields ◽  
Genus Field ◽  
Genus Number

Let k be an algebraic number field and K be its normal extension of finite degree. Then the genus field K* of K over k is defined as the maximal unramified extension of K which is obtained from K by composing an abelian extension over k2). We call the degree (K*: K) the genus number of K over k.

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.

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