scholarly journals A Comprehensive Overview on the Formation of Homomorphic Copies in Coset Graphs for the Modular Group

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Hanan Alolaiyan ◽  
Abdul Razaq ◽  
Awais Yousaf ◽  
Rida Zahra
Keyword(s):  
Finite Field ◽  
Group Action ◽  
Modular Group ◽  
Infinite Graphs ◽  
Quadratic Fields ◽  
Comprehensive Overview ◽  
Real Quadratic Fields ◽  
Coset Graph ◽  
Real Quadratic

This work deals with the well-known group-theoretic graphs called coset graphs for the modular group G and its applications. The group action of G on real quadratic fields forms infinite coset graphs. These graphs are made up of closed paths. When M acts on the finite field Zp, the coset graph appears through the contraction of the vertices of these infinite graphs. Thus, finite coset graphs are composed of homomorphic copies of closed paths in infinite coset graphs. In this work, we have presented a comprehensive overview of the formation of homomorphic copies.

Dihedral group and classification of G-circuits of length 10

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Muhammad Nadeem Bari ◽  
Muhammad Aslam Malik ◽  
Saba Al-Kaseasbeh ◽  
Hafiz Muhammad Afzal Siddiqui ◽  
Alibek Issakhov ◽  
...  
Keyword(s):  
Cyclic Group ◽  
Group Action ◽  
Modular Group ◽  
Dihedral Group ◽  
Equivalence Classes ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Exact Number ◽  
Real Quadratic

Abstract In this paper, we classify G-circuits of length 10 with the help of the location of the reduced numbers lying on G-circuit. The reduced numbers play an important role in the study of modular group action on P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -subset of Q ( m ) \ Q $Q(\sqrt{m}){\backslash}Q$ . For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G-circuits in P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -orbits of real quadratic fields. In particular, we classify P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -orbits of Q ( m ) \ Q $Q(\sqrt{m}){\backslash}Q$ = ⋃ k ∈ N Q * k 2 m $={\bigcup }_{k\in N}{Q}^{{\ast}}\left(\sqrt{{k}^{2}m}\right)$ containing G-circuits of length 10 and determine that the number of equivalence classes of G-circuits of length 10 is 41 in number. We also use dihedral group to explore cyclically equivalence classes of circuits and use cyclic group to explore similar G-circuits of length 10 corresponding to 10 of these circuits. By using cyclically equivalent classes of circuits and similar circuits, we obtain the exact number of G-orbits and the structure of G-circuits corresponding to cyclically equivalent classes. This study also helps us in classifying the reduced numbers lying in the P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -orbits.

scholarly journals Icosahedral Group and Classification of PSL(2, Z)-Orbits of Real Quadratic Fields

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Tianlan Chen ◽  
Muhammad Nadeem Bari ◽  
Muhammad Aslam Malik ◽  
Hafiz Muhammad Afzal Siddiqui ◽  
Jia-Bao Liu
Keyword(s):  
Group Action ◽  
Modular Group ◽  
Equivalence Classes ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Icosahedral Group ◽  
Real Quadratic

Reduced numbers play an important role in the study of modular group action on the PSL2,ℤ-subset of Qm/Q. For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G-circuits in PSL2,ℤ-orbits of real quadratic fields. In particular, we classify PSL2,ℤ-orbits of Qm/Q=∪k∈NQ∗k2m containing G-circuits of length 6 and determine that the number of equivalence classes of G-circuits of length 6 is ten. We also employ the icosahedral group to explore cyclically equivalence classes of circuits and similar G-circuits of length 6 corresponding to each of these ten circuits. This study also helps us in classifying reduced numbers lying in the PSL2,ℤ-orbits.

Modular group acting on real quadratic fields

1988 ◽  
Vol 37 (2) ◽  
pp. 303-309 ◽  
Author(s):  
Q. Mushtaq
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Modular Group ◽  
Quadratic Fields ◽  
Closed Path ◽  
Irrational Numbers ◽  
Real Quadratic Fields ◽  
Quadratic Irrational ◽  
Coset Diagrams ◽  
Real Quadratic

Coset diagrams for the orbit of the modular group G = 〈x, y: x2 = y3 = 1〉 acting on real quadratic fields give some interesting information. By using these coset diagrams, we show that for a fixed value of n, a non-square positive integer, there are only a finite number of real quadratic irrational numbers of the form , where θ and its algebraic conjugate have different signs, and that part of the coset diagram containing such numbers forms a single circuit (closed path) and it is the only circuit in the orbit of θ.

scholarly journals EXTREME VALUES OF GEODESIC PERIODS ON ARITHMETIC HYPERBOLIC SURFACES

2020 ◽  
pp. 1-36
Author(s):  
Bart Michels
Keyword(s):  
Lower Bound ◽  
Closed Geodesic ◽  
Extreme Values ◽  
Theta Series ◽  
Hyperbolic Surface ◽  
Quadratic Fields ◽  
Maass Forms ◽  
Real Quadratic Fields ◽  
Central Values ◽  
Real Quadratic

Abstract Given a closed geodesic on a compact arithmetic hyperbolic surface, we show the existence of a sequence of Laplacian eigenfunctions whose integrals along the geodesic exhibit nontrivial growth. Via Waldspurger’s formula we deduce a lower bound for central values of Rankin-Selberg L-functions of Maass forms times theta series associated to real quadratic fields.

scholarly journals On the Number of Quadratic Twists with a Rational Point of Almost Minimal Height

2020 ◽  
Author(s):  
Joachim Petit
Keyword(s):  
Asymptotic Formula ◽  
Elliptic Curve ◽  
Asymptotic Estimate ◽  
Rational Point ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Quadratic Twists ◽  
The Family ◽  
Minimal Height ◽  
Real Quadratic

Abstract We investigate the number of curves having a rational point of almost minimal height in the family of quadratic twists of a given elliptic curve. This problem takes its origin in the work of Hooley, who asked this question in the setting of real quadratic fields. In particular, he showed an asymptotic estimate for the number of such fields with almost minimal fundamental unit. Our main result establishes the analogue asymptotic formula in the setting of quadratic twists of a fixed elliptic curve.

scholarly journals Prime geodesics and averages of the Zagier L-series

2021 ◽  
pp. 1-24
Author(s):  
OLGA BALKANOVA ◽  
DMITRY FROLENKOV ◽  
MORTEN S. RISAGER
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Error Term ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Prime Geodesic Theorem ◽  
Error Terms ◽  
Average Size ◽  
Real Quadratic

Abstract The Zagier L-series encode data of real quadratic fields. We study the average size of these L-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.

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