An Explicit Construction of Initial Perfect Quadratic Forms over Some Families of Totally Real Number Fields

2008 ◽  
pp. 240-252
Author(s):  
Alar Leibak
Keyword(s):  
Real Number ◽  
Quadratic Forms ◽  
Number Fields ◽  
Explicit Construction ◽  
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.

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.

scholarly journals There are no universal ternary quadratic forms over biquadratic fields

2020 ◽  
Vol 63 (3) ◽  
pp. 861-912 ◽  
Author(s):  
Jakub Krásenský ◽  
Magdaléna Tinková ◽  
Kristýna Zemková
Keyword(s):  
Quadratic Forms ◽  
Number Fields ◽  
Ring Of Integers ◽  
Ternary Quadratic Forms ◽  
Mild Conditions ◽  
Positive Unit ◽  
Positive Elements ◽  
Totally Positive ◽  
Biquadratic Fields ◽  
Totally Real

AbstractWe study totally positive definite quadratic forms over the ring of integers $\mathcal {O}_K$ of a totally real biquadratic field $K=\mathbb {Q}(\sqrt {m}, \sqrt {s})$. We restrict our attention to classic forms (i.e. those with all non-diagonal coefficients in $2\mathcal {O}_K$) and prove that no such forms in three variables are universal (i.e. represent all totally positive elements of $\mathcal {O}_K$). Moreover, we show the same result for totally real number fields containing at least one non-square totally positive unit and satisfying some other mild conditions. These results provide further evidence towards Kitaoka's conjecture that there are only finitely many number fields over which such forms exist. One of our main tools are additively indecomposable elements of $\mathcal {O}_K$; we prove several new results about their properties.

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.

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