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 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.

scholarly journals Constructions of Dense Lattices of Full Diversity

2020 ◽  
Vol 21 (2) ◽  
pp. 299
Author(s):  
A. A. Andrade ◽  
A. J. Ferrari ◽  
J. C. Interlando ◽  
R. R. Araujo
Keyword(s):  
Real Number ◽  
Fading Channels ◽  
Packing Density ◽  
Signal Transmission ◽  
Cyclotomic Field ◽  
Sphere Packing ◽  
Number Fields ◽  
Trace Form ◽  
Full Diversity ◽  
Totally Real

A lattice construction using Z-submodules of rings of integers of number fields is presented. The construction yields rotated versions of the laminated lattices A_n for n = 2,3,4,5,6, which are the densest lattices in their respective dimensions. The sphere packing density of a lattice is a function of its packing radius, which in turn can be directly calculated from the minimum squared Euclidean norm of the lattice. Norms in a lattice that is realized by a totally real number field can be calculated by the trace form of the field restricted to its ring of integers. Thus, in the present work, we also present the trace form of the maximal real subfield of a cyclotomic field. Our focus is on totally real number fields since their associated lattices have full diversity. Along with high packing density, the full diversity feature is desirable in lattices that are used for signal transmission over both Gaussian and Rayleigh fading channels.

On the Preservation of Root Numbers and the Behavior of Weil Characters under Reciprocity Equivalence

1997 ◽  
Vol 40 (4) ◽  
pp. 402-415
Author(s):  
Jenna P. Carpenter
Keyword(s):  
Number Field ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Witt Ring ◽  
Root Numbers ◽  
Local Root ◽  
Necessary And Sufficient ◽  
Selection Of ◽  
Local Root Numbers

AbstractThis paper studies how the local root numbers and the Weil additive characters of the Witt ring of a number field behave under reciprocity equivalence. Given a reciprocity equivalence between two fields, at each place we define a local square class which vanishes if and only if the local root numbers are preserved. Thus this local square class serves as a local obstruction to the preservation of local root numbers. We establish a set of necessary and sufficient conditions for a selection of local square classes (one at each place) to represent a global square class. Then, given a reciprocity equivalence that has a finite wild set, we use these conditions to show that the local square classes combine to give a global square class which serves as a global obstruction to the preservation of all root numbers. Lastly, we use these results to study the behavior of Weil characters under reciprocity equivalence.

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.

A note on values of the Dedekind zeta-function at odd positive integers

2021 ◽  
pp. 1-12
Author(s):  
M. Ram Murty ◽  
Siddhi S. Pathak
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Imaginary Quadratic Fields ◽  
Fixed Integer ◽  
Dedekind Zeta Function ◽  
Totally Real ◽  
Positive Integers ◽  
Real Number Field

For an algebraic number field [Formula: see text], let [Formula: see text] be the associated Dedekind zeta-function. It is conjectured that [Formula: see text] is transcendental for any positive integer [Formula: see text]. The only known case of this conjecture was proved independently by Siegel and Klingen, namely that, when [Formula: see text] is a totally real number field, [Formula: see text] is an algebraic multiple of [Formula: see text] and hence, is transcendental. If [Formula: see text] is not totally real, the question of whether [Formula: see text] is irrational or not remains open. In this paper, we prove that for a fixed integer [Formula: see text], at most one of [Formula: see text] is rational, as [Formula: see text] varies over all imaginary quadratic fields. We also discuss a generalization of this theorem to CM-extensions of number fields.

Estimates for Kloosterman sums for totally real number fields

2001 ◽  
Vol 2001 (535) ◽  
Author(s):  
R. W. Bruggeman ◽  
R. J. Miatello ◽  
I. Pacharoni
Keyword(s):  
Real Number ◽  
Number Fields ◽  
Kloosterman Sums ◽  
Totally Real

On linear relations for special L-values over certain totally real number fields

2021 ◽  
pp. 1-18
Author(s):  
Seiji Kuga
Keyword(s):  
Real Number ◽  
Fourier Coefficients ◽  
Number Fields ◽  
Arithmetic Functions ◽  
Hilbert Modular Form ◽  
Linear Relations ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Hilbert Modular ◽  
Totally Real

In this paper, we give linear relations between the Fourier coefficients of a special Hilbert modular form of half integral weight and some arithmetic functions. As a result, we have linear relations for the special [Formula: see text]-values over certain totally real number fields.

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