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.

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 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 Fitting Ideals in Number Theory and Arithmetic

2021 ◽  
Author(s):  
Cornelius Greither
Keyword(s):  
Number Theory ◽  
Galois Group ◽  
Classical Result ◽  
Number Fields ◽  
Class Groups ◽  
Stark Conjecture

AbstractWe describe classical and recent results concerning the structure of class groups of number fields as modules over the Galois group. When presenting more modern developments, we can only hint at the much broader context and the very powerful general techniques that are involved, but we endeavour to give complete statements or at least examples where feasible. The timeline goes from a classical result proved in 1890 (Stickelberger’s Theorem) to a recent (2020) breakthrough: the proof of the Brumer-Stark conjecture by Dasgupta and Kakde.

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 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.

On non-abelian Brumer and Brumer–Stark conjecture for monomial CM-extensions

2014 ◽  
Vol 10 (04) ◽  
pp. 817-848 ◽  
Author(s):  
Jiro Nomura
Keyword(s):  
Dihedral Group ◽  
Number Fields ◽  
Alternating Group ◽  
Quaternion Group ◽  
Finite Set ◽  
Generalized Quaternion Group ◽  
Monomial Group ◽  
Quaternion Case ◽  
Stark Conjecture ◽  
Generalized Quaternion

Let K/k be a finite Galois CM-extension of number fields whose Galois group G is monomial and S a finite set of places of k. Then the "Stickelberger element" θK/k,S is defined. Concerning this element, Andreas Nickel formulated the non-abelian Brumer and Brumer–Stark conjectures and their "weak" versions. In this paper, when G is a monomial group, we prove that the weak non-abelian conjectures are reduced to the weak conjectures for abelian subextensions. We write D4p, Q2n+2 and A4 for the dihedral group of order 4p for any odd prime p, the generalized quaternion group of order 2n+2 for any natural number n and the alternating group on four letters respectively. Suppose that G is isomorphic to D4p, Q2n+2 or A4 × ℤ/2ℤ. Then we prove the l-parts of the weak non-abelian conjectures, where l = 2 in the quaternion case, and l is an arbitrary prime which does not split in ℚ(ζp) in the dihedral case and in ℚ(ζ3) in the alternating case. In particular, we do not exclude the 2-part of the conjectures and do not assume that S contains all finite places which ramify in K/k in contrast with Nickel's formulation.

scholarly journals The size function for cyclic cubic fields

2018 ◽  
Vol 14 (02) ◽  
pp. 399-415
Author(s):  
Ha Thanh Nguyen Tran ◽  
Peng Tian
Keyword(s):  
Number Field ◽  
Algebraic Curve ◽  
Number Fields ◽  
Size Function ◽  
Rank One ◽  
Cubic Fields ◽  
Cyclic Cubic Fields

The size function for a number field is an analogue of the dimension of the Riemann–Roch spaces of divisors on an algebraic curve. It was conjectured to attain its maximum at the trivial class of Arakelov divisors. This conjecture was proved for many number fields with unit groups of rank one. Our research confirms that the conjecture also holds for cyclic cubic fields, which have unit groups of rank two.

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