Class numbers of maximal binary quadratic lattices over totally real number fields

1978 ◽  
Vol 30 (1) ◽  
pp. 398-399 ◽  
Author(s):  
Meinhard Peters
Keyword(s):  
Real Number ◽  
Number Fields ◽  
Class Numbers ◽  
Quadratic Lattices ◽  
Totally Real

Universal positive quaternary quadratic lattices over totally real number fields

Mathematika ◽  
1997 ◽  
Vol 44 (2) ◽  
pp. 342-347 ◽  
Author(s):  
A. G. Earnest ◽  
Azar Khosravani
Keyword(s):  
Real Number ◽  
Number Fields ◽  
Quadratic Lattices ◽  
Totally Real

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.

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.

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.

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