scholarly journals Hyperbolic tessellations and generators of for imaginary quadratic fields

2021 ◽  
Vol 9 ◽  
Author(s):  
David Burns ◽  
Rob de Jeu ◽  
Herbert Gangl ◽  
Alexander D. Rahm ◽  
Dan Yasaki
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Quadratic Fields ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Quadratic Number Fields ◽  
Formula Group ◽  
Imaginary Quadratic Number ◽  
Direct Calculations

Abstract We develop methods for constructing explicit generators, modulo torsion, of the $K_3$ -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic $3$ -space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite $K_3$ -group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for $ K_3 $ of any field, predict the precise power of $2$ that should occur in the Lichtenbaum conjecture at $ -1 $ and prove that this prediction is valid for all abelian number fields.

scholarly journals Finiteness of lattice points on varieties F(y) = F(g(𝕏)) + r(𝕏) over imaginary quadratic fields

2021 ◽  
Vol 27 (1) ◽  
pp. 76-90
Author(s):  
Lukasz Nizio ◽  
Keyword(s):  
Finite Number ◽  
Cyclotomic Field ◽  
Number Fields ◽  
Lattice Points ◽  
Quadratic Fields ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Integral Points ◽  
Imaginary Quadratic Number ◽  
Ring Of Algebraic Integers

We construct affine varieties over \mathbb{Q} and imaginary quadratic number fields \mathbb{K} with a finite number of \alpha-lattice points for a fixed \alpha\in \mathcal{O}_\mathbb{K}, where \mathcal{O}_\mathbb{K} denotes the ring of algebraic integers of \mathbb{K}. These varieties arise from equations of the form F(y) = F(g(x_1,x_2,\ldots, x_k))+r(x_1,x_2\ldots, x_k), where F is a rational function, g and r are polynomials over \mathbb{K}, and the degree of r is relatively small. We also give an example of an affine variety of dimension two, with a finite number of algebraic integral points. This variety is defined over the cyclotomic field \mathbb{Q}(\xi_3)=\mathbb{Q}(\sqrt{-3}).

scholarly journals The Hecke Group H λ 4 Acting on Imaginary Quadratic Number Fields

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Abdulaziz Deajim
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Relevant Literature ◽  
Number Fields ◽  
Quadratic Number ◽  
Hecke Group ◽  
Quadratic Number Fields ◽  
Imaginary Quadratic Number ◽  
Imaginary Number

Let H λ 4 be the Hecke group x , y : x 2 = y 4 = 1 and, for a square-free positive integer n , consider the subset ℚ ∗ − n = a + − n / c | a , b = a 2 + n / c ∈ ℤ ,   c ∈ 2 ℤ of the quadratic imaginary number field ℚ − n . Following a line of research in the relevant literature, we study the properties of the action of H λ 4 on ℚ ∗ − n . In particular, we calculate the number of orbits arising from this action for every such n . Some illustrative examples are also given.

Nonadjacent Radix-τ Expansions of Integers in Euclidean Imaginary Quadratic Number Fields

2008 ◽  
Vol 60 (6) ◽  
pp. 1267-1282 ◽  
Author(s):  
Ian F. Blake ◽  
V. Kumar Murty ◽  
Guangwu Xu
Keyword(s):  
Number Fields ◽  
Quadratic Number ◽  
Efficient Point ◽  
Koblitz Curves ◽  
Quadratic Number Fields ◽  
Imaginary Quadratic Number

AbstractIn his seminal papers, Koblitz proposed curves for cryptographic use. For fast operations on these curves, these papers also initiated a study of the radix-τ expansion of integers in the number fields and . The (window) nonadjacent form of τ -expansion of integers in was first investigated by Solinas. For integers in , the nonadjacent form and the window nonadjacent form of the τ -expansion were studied. These are used for efficient point multiplications on Koblitz curves. In this paper, we complete the picture by producing the (window) nonadjacent radix-τ expansions for integers in all Euclidean imaginary quadratic number fields.

ON THE CLASSIFICATION OF EVEN UNIMODULAR LATTICES WITH A COMPLEX STRUCTURE

2012 ◽  
Vol 08 (04) ◽  
pp. 983-992 ◽  
Author(s):  
MICHAEL HENTSCHEL ◽  
ALOYS KRIEG ◽  
GABRIELE NEBE
Keyword(s):  
Modular Group ◽  
Complex Structure ◽  
Theta Series ◽  
Number Fields ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Rings Of Integers ◽  
Hermitian Modular Form ◽  
Imaginary Quadratic Number

This paper classifies the even unimodular lattices that have a structure as a Hermitian [Formula: see text]-lattice of rank r ≤ 12 for rings of integers in imaginary quadratic number fields K of class number 1. The Hermitian theta series of such a lattice is a Hermitian modular form of weight r for the full modular group, therefore we call them theta lattices. For arbitrary imaginary quadratic fields we derive a mass formula for the principal genus of theta lattices which is applied to show completeness of the classifications.

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