scholarly journals Wave Transport and Localization in Prime Number Landscapes

2021 ◽  
Vol 9 ◽  
Author(s):  
Luca Dal Negro ◽  
David Taylor Henderson ◽  
Fabrizio Sgrignuoli
Keyword(s):  
Prime Number ◽  
Scattering Problem ◽  
Prime Numbers ◽  
Graph Connectivity ◽  
Quadratic Fields ◽  
Strong Coupling Regime ◽  
Fluctuation Analysis ◽  
Imaginary Quadratic Fields ◽  
Wave Transport ◽  
Green’S Matrix

In this paper, we study the wave transport and localization properties of novel aperiodic structures that manifest the intrinsic complexity of prime number distributions in imaginary quadratic fields. In particular, we address structure-property relationships and wave scattering through the prime elements of the nine imaginary quadratic fields (i.e., of their associated rings of integers) with class number one, which are unique factorization domains (UFDs). Our theoretical analysis combines the rigorous Green’s matrix solution of the multiple scattering problem with the interdisciplinary methods of spatial statistics and graph theory analysis of point patterns to unveil the relevant structural properties that produce wave localization effects. The onset of a Delocalization-Localization Transition (DLT) is demonstrated by a comprehensive study of the spectral properties of the Green’s matrix and the Thouless number as a function of their optical density. Furthermore, we employ Multifractal Detrended Fluctuation Analysis (MDFA) to establish the multifractal scaling of the local density of states in these complex structures and we discover a direct connection between localization, multifractality, and graph connectivity properties. Finally, we use a semi-classical approach to demonstrate and characterize the strong coupling regime of quantum emitters embedded in these novel aperiodic environments. Our study provides access to engineering design rules for the fabrication of novel and more efficient classical and quantum sources as well as photonic devices with enhanced light-matter interaction based on the intrinsic structural complexity of prime numbers in algebraic fields.

scholarly journals On Greenberg’s generalized conjecture for imaginary quartic fields

2020 ◽  
pp. 1-11
Author(s):  
Naoya Takahashi
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Number Field ◽  
Group Ring ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Prime Numbers ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Number Formula

For an algebraic number field [Formula: see text] and a prime number [Formula: see text], let [Formula: see text] be the maximal multiple [Formula: see text]-extension. Greenberg’s generalized conjecture (GGC) predicts that the Galois group of the maximal unramified abelian pro-[Formula: see text] extension of [Formula: see text] is pseudo-null over the completed group ring [Formula: see text]. We show that GGC holds for some imaginary quartic fields containing imaginary quadratic fields and some prime numbers.

scholarly journals REMARKS ON THE DIVISIBILITY OF THE CLASS NUMBERS OF IMAGINARY QUADRATIC FIELDS

2011 ◽  
Vol 53 (2) ◽  
pp. 379-389 ◽  
Author(s):  
AKIKO ITO
Keyword(s):  
Prime Number ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Imaginary Quadratic Fields ◽  
Positive Integers

AbstractWe consider the divisibility of the class numbers of imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - q^n})$, where q is an odd prime number, k and n are positive integers. Suppose that k ≡ 1 mod 2 or n ≢ 3 mod 6. We show that the class numbers of imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - q^n})$ ≠ $\mathbb{Q}(\sqrt{-3})$ are divisible by n for q ≡ 3 mod 8. This is a generalization of the result of Kishi for imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - 3^n})$ when k ≡ 1 mod 2 or n ≢ 3 mod 6. We also show that the class numbers of imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - q^n})$ ≠ $\mathbb{Q}(\sqrt{-1})$ are divisible by n for q ≡ 1 mod 4 and the class numbers of imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - q^n})$ ≠ $\mathbb{Q}(\sqrt{-3})$ are divisible by n for q ≡ 7 mod 8.

scholarly journals On the Gras Conjecture for Imaginary Quadratic Fields

2013 ◽  
Vol 56 (1) ◽  
pp. 148-160
Author(s):  
Hassan Oukhaba ◽  
Stéphane Viguié
Keyword(s):  
Quadratic Field ◽  
Prime Numbers ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Roots Of Unity ◽  
Imaginary Quadratic Fields ◽  
Abelian Extensions

AbstractIn this paper we extend K. Rubin's methods to prove the Gras conjecture for abelian extensions of a given imaginary quadratic field k and prime numbers p that divide the number of roots of unity in k.

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.

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