scholarly journals On number fields with given ramification

2007 ◽  
Vol 143 (6) ◽  
pp. 1359-1373 ◽  
Author(s):  
Gaëtan Chenevier
Keyword(s):  
Prime Number ◽  
Number Field ◽  
Number Fields ◽  
Algebraic Extension ◽  
Natural Maps ◽  
Finite Set ◽  
Totally Real

AbstractLet E be a CM number field and let S be a finite set of primes of E containing the primes dividing a given prime number l and another prime u split above the maximal totally real subfield of E. If ES denotes a maximal algebraic extension of E which is unramified outside S, we show that the natural maps $\mathrm {Gal}(\overline {E_u}/E_u) \longrightarrow \mathrm {Gal}(E_S/E)$ are injective. We discuss generalizations of this result.

ON S-HARDY–LITTLEWOOD HOMOGENEOUS SPACES

1998 ◽  
Vol 09 (06) ◽  
pp. 723-757 ◽  
Author(s):  
MASANORI MORISHITA ◽  
TAKAO WATANABE
Keyword(s):  
Number Field ◽  
Homogeneous Spaces ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Positive Definite ◽  
Integral Points ◽  
Distribution Property ◽  
Totally Positive ◽  
Finite Set ◽  
Totally Real

We study the asymptotic distribution of S-integral points on affine homogeneous spaces in the light of the Hardy–Littlewood property introduced by Borovoi and Rudnick. We introduce the S-Hardy–Littlewood property for affine homogeneous spaces defined over an algebraic number field and a finite set S of places of the base field. We work with the adelic harmonic analysis on affine algebraic groups over a number field to determine the asymptotic density of S-integral points under congruence conditions. We give some new examples of strongly or relatively S-Hardy–Littlewood homogeneous spaces over number fields. As an application, we prove certain asymptotically uniform distribution property of integral points on an ellipsoid defined by a totally positive definite tenary quadratic form over a totally real number field.

scholarly journals ON NUMBER FIELDS WITH EQUIVALENT INTEGRAL TRACE FORMS

2012 ◽  
Vol 08 (07) ◽  
pp. 1569-1580 ◽  
Author(s):  
GUILLERMO MANTILLA-SOLER
Keyword(s):  
Quadratic Form ◽  
Real Number ◽  
Number Field ◽  
Number Fields ◽  
Trace Form ◽  
Integral Quadratic Form ◽  
Trace Forms ◽  
Fundamental Discriminant ◽  
Totally Real

Let K be a number field. The integral trace form is the integral quadratic form given by tr k/ℚ(x2)|OK. In this article we study the existence of non-conjugated number fields with equivalent integral trace forms. As a corollary of one of the main results of this paper, we show that any two non-totally real number fields with the same signature and same prime discriminant have equivalent integral trace forms. Additionally, based on previous results obtained by the author and the evidence presented here, we conjecture that any two totally real quartic fields of fundamental discriminant have equivalent trace zero forms if and only if they are conjugated.

On tamely ramified pro-p-extensions over -extensions of

2013 ◽  
Vol 156 (2) ◽  
pp. 281-294
Author(s):  
TSUYOSHI ITOH ◽  
YASUSHI MIZUSAWA
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Number Field ◽  
Rational Number ◽  
Prime Numbers ◽  
Finite Set

AbstractFor an odd prime number p and a finite set S of prime numbers congruent to 1 modulo p, we consider the Galois group of the maximal pro-p-extension unramified outside S over the ${\mathbb Z}_p$-extension of the rational number field. In this paper, we classify all S such that the Galois group is a metacyclic pro-p group.

scholarly journals ON MAZUR'S CONJECTURE FOR TWISTED L-FUNCTIONS OF ELLIPTIC CURVES OVER TOTALLY REAL OR CM FIELDS

2010 ◽  
Vol 53 (1) ◽  
pp. 207-210
Author(s):  
CRISTIAN VIRDOL
Keyword(s):  
Functional Equation ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Order ◽  
Number Field ◽  
Meromorphic Continuation ◽  
Minimal Order ◽  
Finite Set ◽  
Entire Complex ◽  
Totally Real

Let E be an elliptic curve defined over a number field F, and let Σ be a finite set of finite places of F. Let L(s, E, ψ) be the L-function of E twisted by a finite-order Hecke character ψ of F. It is conjectured that L(s, E, ψ) has a meromorphic continuation to the entire complex plane and satisfies a functional equation s ↔ 2 − s. Then one can define the so called minimal order of vanishing ats = 1 of L(s, E, ψ), denoted by m(E, ψ) (see Section 2 for the definition).

scholarly journals Weak Arithmetic Equivalence

2015 ◽  
Vol 58 (1) ◽  
pp. 115-127 ◽  
Author(s):  
Guillermo Mantilla-Soler
Keyword(s):  
Zeta Function ◽  
Real Number ◽  
Number Field ◽  
Number Fields ◽  
Trace Form ◽  
Root Numbers ◽  
Local Root ◽  
Arithmetic Equivalence ◽  
Totally Real ◽  
Local Root Numbers

AbstractInspired by the invariant of a number field given by its zeta function, we define the notion of weak arithmetic equivalence and show that under certain ramification hypotheses this equivalence determines the local root numbers of the number field. This is analogous to a result of Rohrlich on the local root numbers of a rational elliptic curve. Additionally, we prove that for tame non-totally real number fields, the integral trace form is invariant under arithmetic equivalence

scholarly journals On Modified Circular Units and Annihilation of real Classes

2005 ◽  
Vol 177 ◽  
pp. 77-115 ◽  
Author(s):  
Jean-Robert Belliard ◽  
Thống Nguyễn-Quang-Ðỗ
Keyword(s):  
Real Number ◽  
Prime Number ◽  
Number Field ◽  
Class Group ◽  
Euler Systems ◽  
Totally Real ◽  
Circular Units ◽  
Functorial Approach ◽  
Real Number Field

For an abelian totally real number field F and an odd prime number p which splits totally in F, we present a functorial approach to special “p-units” previously built by D. Solomon using “wild” Euler systems. This allows us to prove a conjecture of Solomon on the annihilation of the p-class group of F (in the particular context here), as well as related annihilation results and index formulae.

Algebraic extensions in nonstandard models and Hilbert's irreducibility theorem

1988 ◽  
Vol 53 (2) ◽  
pp. 470-480 ◽  
Author(s):  
Masahiro Yasumoto
Keyword(s):  
Natural Number ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Algebraic Extension ◽  
Algebraic Integers ◽  
Nonstandard Models ◽  
Ring Of Algebraic Integers

LetKbe an algebraic number field andIKthe ring of algebraic integers inK. *Kand *IKdenote enlargements ofKandIKrespectively. LetxЄ *K–K. In this paper, we are concerned with algebraic extensions ofK(x)within *K. For eachxЄ *K–Kand each natural numberd, YK(x,d)is defined to be the number of algebraic extensions ofK(x)of degreedwithin *K.xЄ *K–Kis called a Hilbertian element ifYK(x,d)= 0 for alldЄ N,d> 1; in other words,K(x)has no algebraic extension within *K. In their paper [2], P. C. Gilmore and A. Robinson proved that the existence of a Hilbertian element is equivalent to Hilbert's irreducibility theorem. In a previous paper [9], we gave many Hilbertian elements of nonstandard integers explicitly, for example, for any nonstandard natural numberω, 2ωPωand 2ω(ω3+ 1) are Hilbertian elements in*Q, where pωis theωth prime number.

scholarly journals VALUES AT s = -1 OF L-FUNCTIONS FOR MULTI-QUADRATIC EXTENSIONS OF NUMBER FIELDS, AND THE FITTING IDEAL OF THE TAME KERNEL

2009 ◽  
Vol 05 (03) ◽  
pp. 383-405
Author(s):  
JONATHAN W. SANDS
Keyword(s):  
Galois Extension ◽  
Number Fields ◽  
Sufficient Condition ◽  
Tate Conjecture ◽  
Quadratic Extensions ◽  
Fitting Ideal ◽  
Tame Kernel ◽  
Finite Set ◽  
Stickelberger Ideal ◽  
Totally Real

Fix a Galois extension [Formula: see text] of totally real number fields such that the Galois group G has exponent 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in [Formula: see text], let [Formula: see text] denote the primes of [Formula: see text] lying above those in S, and let [Formula: see text] denote the ring of [Formula: see text]-integers of [Formula: see text]. We then compare the Fitting ideal of [Formula: see text] as a ℤ[G]-module with a higher Stickelberger ideal. The two extend to the same ideal in the maximal order of ℚ[G], and hence in ℤ[1/2][G]. Results in ℤ[G] are obtained under the assumption of the Birch–Tate conjecture, especially for biquadratic extensions, where we compute the index of the higher Stickelberger ideal. We find a sufficient condition for the Fitting ideal to contain the higher Stickelberger ideal in the case where [Formula: see text] is a biquadratic extension of F containing the first layer of the cyclotomic ℤ2-extension of F, and describe a class of biquadratic extensions of F = ℚ that satisfy this condition.

scholarly journals The lattice of primary ideals of orders in quadratic number fields

2016 ◽  
Vol 12 (07) ◽  
pp. 2025-2040 ◽  
Author(s):  
Giulio Peruginelli ◽  
Paolo Zanardo
Keyword(s):  
Prime Ideal ◽  
Prime Number ◽  
Number Field ◽  
Number Fields ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Ring Of Integers ◽  
The Core ◽  
Number Formula ◽  
Primary Ideals

Let [Formula: see text] be an order in a quadratic number field [Formula: see text] with ring of integers [Formula: see text], such that the conductor [Formula: see text] is a prime ideal of [Formula: see text], where [Formula: see text] is a prime. We give a complete description of the [Formula: see text]-primary ideals of [Formula: see text]. They form a lattice with a particular structure by layers; the first layer, which is the core of the lattice, consists of those [Formula: see text]-primary ideals not contained in [Formula: see text]. We get three different cases, according to whether the prime number [Formula: see text] is split, inert or ramified in [Formula: see text].

scholarly journals Computation of L(0, χ) and of relative class numbers of CM-fields

2001 ◽  
Vol 161 ◽  
pp. 171-191 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Real Number ◽  
Number Field ◽  
Class Group ◽  
Number Fields ◽  
Class Numbers ◽  
Finite Part ◽  
Relative Class ◽  
Relative Class Number ◽  
Totally Real ◽  
Real Number Field

Let χ be a nontrivial Hecke character on a (strict) ray class group of a totally real number field L of discriminant dL. Then, L(0, χ) is an algebraic number of some cyclotomic number field. We develop an efficient technique for computing the exact values at s = 0 of such abelian Hecke L-functions over totally real number fields L. Let fχ denote the norm of the finite part of the conductor of χ. Then, roughly speaking, we can compute L(0, χ) in O((dLfx)0.5+∊) elementary operations. We then explain how the computation of relative class numbers of CM-fields boils down to the computation of exact values at s = 0 of such abelian Hecke L-functions over totally real number fields L. Finally, we give examples of relative class number computations for CM-fields of large degrees based on computations of L(0, χ) over totally real number fields of degree 2 and 6.

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