Free quotients of subgroups of the Bianchi groups whose kernels contain many elementary matrices

1994 ◽  
Vol 116 (2) ◽  
pp. 253-273
Author(s):  
A. W. Mason ◽  
R. W. K. Odoni
Keyword(s):  
Normal Subgroup ◽  
Positive Integer ◽  
Commutator Subgroup ◽  
Quadratic Number ◽  
Congruence Subgroups ◽  
Quadratic Number Field ◽  
Ring Of Integers ◽  
Imaginary Quadratic Number ◽  
Low Dimensional ◽  
Fundamental Paper

AbstractLet d be a square-free positive integer and let be the ring of integers of the imaginary quadratic number field ℚ(√ − d) The Bianchi groups are the groups SL2() (or PSL2(). Let m be the order of index m in . In this paper we prove that for each d there exist infinitely many m for which SL2(m)/NE2(m) has a free, non-cyclic quotient, where NE2(m) is the normal subgroup of SL2(m) generated by the elementary matrices. When d is not a prime congruent to 3 (mod 4) this result is true for all but finitely many m. The proofs are based on the fundamental paper of Zimmert and its generalization due to Grunewald and Schwermer.The results are used to extend earlier work of Lubotzky on non-congruence subgroups of SL2(), which involves the concept of the ‘non-congruence crack’. In addition the results highlight a number of low-dimensional anomalies. For example, it is known that [SLn(m), SLnm)] = En(m), when n ≥ 3, where [SLn(m), SLn(m)] is the commutator subgroup of SL(m) and En(m) is the subgroup of SLn(m) generated by the elementary matrices. Our results show that this is not always true when n = 2.

scholarly journals Higher torsion in the Abelianization of the full Bianchi groups

2013 ◽  
Vol 16 ◽  
pp. 344-365 ◽  
Author(s):  
Alexander D. Rahm
Keyword(s):  
Positive Integer ◽  
Finite Order ◽  
Quotient Space ◽  
Prime Numbers ◽  
Integral Homology ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Ring Of Integers ◽  
Imaginary Quadratic Number ◽  
Order Of Elements

AbstractDenote by $ \mathbb{Q} ( \sqrt{- m} )$, with $m$ a square-free positive integer, an imaginary quadratic number field, and by ${ \mathcal{O} }_{- m} $ its ring of integers. The Bianchi groups are the groups ${\mathrm{SL} }_{2} ({ \mathcal{O} }_{- m} )$. In the literature, so far there have been no examples of $p$-torsion in the integral homology of the full Bianchi groups, for $p$ a prime greater than the order of elements of finite order in the Bianchi group, which is at most 6. However, extending the scope of the computations, we can observe examples of torsion in the integral homology of the quotient space, at prime numbers as high as for instance $p= 80\hspace{0.167em} 737$ at the discriminant $- 1747$.Supplementary materials are available with this article.

Trace formulae and applications to class numbers

2014 ◽  
Vol 12 (6) ◽  
Author(s):  
Nicole Raulf
Keyword(s):  
Quadratic Forms ◽  
Hecke Operators ◽  
Class Numbers ◽  
Asymptotic Result ◽  
Quadratic Number ◽  
Binary Quadratic Forms ◽  
Quadratic Number Field ◽  
Ring Of Integers ◽  
Imaginary Quadratic Number ◽  
Trace Formulae

AbstractIn this paper we compute the trace formula for Hecke operators acting on automorphic forms on the hyperbolic 3-space for the group PSL2($\mathcal{O}_K $) with $\mathcal{O}_K $ being the ring of integers of an imaginary quadratic number field K of class number H K > 1. Furthermore, as a corollary we obtain an asymptotic result for class numbers of binary quadratic forms.

scholarly journals Geometric-progression-free sets over quadratic number fields

2017 ◽  
Vol 147 (2) ◽  
pp. 245-262
Author(s):  
Andrew Best ◽  
Karen Huan ◽  
Nathan McNew ◽  
Steven J. Miller ◽  
Jasmine Powell ◽  
...  
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Geometric Progression ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Ring Of Integers ◽  
Quadratic Number Fields ◽  
Geometric Progressions ◽  
Upper Density ◽  
Free Sets

In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a particular substructure. A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding three-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid three-term geometric progressions. When unique factorization fails, or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets ‘greedily’, a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometric-progression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.

A SYSTEM OF RELATIVE PELLIAN EQUATIONS AND A RELATED FAMILY OF RELATIVE THUE EQUATIONS

2006 ◽  
Vol 02 (04) ◽  
pp. 569-590 ◽  
Author(s):  
BORKA JADRIJEVIĆ ◽  
VOLKER ZIEGLER
Keyword(s):  
Number Field ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Root Of Unity ◽  
Related Family ◽  
The Family ◽  
Thue Equation ◽  
Imaginary Quadratic Number ◽  
Pellian Equations

In this paper we consider the family of systems (2c + 1)U2 - 2cV2 = μ and (c - 2)U2 - cZ2 = -2μ of relative Pellian equations, where the parameter c and the root of unity μ are integers in the same imaginary quadratic number field [Formula: see text]. We show that for |c| ≥ 3 only certain values of μ yield solutions of this system, and solve the system completely for |c| ≥ 1544686. Furthermore we will consider the related relative Thue equation [Formula: see text] and solve it by the method of Tzanakis under the same assumptions.

scholarly journals Picard Groups and Refined Discrete Logarithms

2005 ◽  
Vol 8 ◽  
pp. 1-16 ◽  
Author(s):  
W. Bley ◽  
M. Endres
Keyword(s):  
Abelian Group ◽  
Number Field ◽  
Finite Abelian Group ◽  
Discrete Logarithm ◽  
Quadratic Number ◽  
Prime Degree ◽  
Quadratic Number Field ◽  
Picard Groups ◽  
Imaginary Quadratic Number ◽  
Ring Of Algebraic Integers

AbstractLet K denote a number field, and G a finite abelian group. The ring of algebraic integers in K is denoted in this paper by $/cal{O}_K$, and $/cal{A}$ denotes any $/cal{O}_K$-order in K[G]. The paper describes an algorithm that explicitly computes the Picard group Pic($/cal{A}$), and solves the corresponding (refined) discrete logarithm problem. A tamely ramified extension L/K of prime degree l of an imaginary quadratic number field K is used as an example; the class of $/cal{O}_L$ in Pic($/cal{O}_K[G]$) can be numerically determined.

scholarly journals On a criterion for the class number of a quadratic number field to be one

1980 ◽  
Vol 79 ◽  
pp. 123-129 ◽  
Author(s):  
Masakazu Kutsuna
Keyword(s):  
Number Field ◽  
Class Number ◽  
Euclidean Algorithm ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Imaginary Quadratic Number ◽  
Necessary And Sufficient

G. Rabinowitsch [3] generalized the concept of the Euclidean algorithm and proved a theorem on a criterion in order that the class number of an imaginary quadratic number field is equal to one:Theorem.It is necessary and sufficient for the class number of an imaginary quadratic number fieldD= 1 — 4m, m> 0,to be one that x2—x+m is prime for any integer x such that1 ≤x≤m— 2.

scholarly journals GLOBAL UNITS MODULO ELLIPTIC UNITS AND IDEAL CLASS GROUPS

2012 ◽  
Vol 08 (03) ◽  
pp. 569-588
Author(s):  
STÉPHANE VIGUIÉ
Keyword(s):  
Class Group ◽  
Finite Extension ◽  
Projective Limit ◽  
Ideal Class ◽  
Quadratic Number ◽  
Class Groups ◽  
Quadratic Number Field ◽  
Ideal Class Groups ◽  
Imaginary Quadratic Number ◽  
Elliptic Units

Let p be a prime number, and let k be an imaginary quadratic number field in which p decomposes into two distinct primes 𝔭 and [Formula: see text]. Let k∞ be the unique ℤp-extension of k which is unramified outside of 𝔭, and let K∞ be a finite extension of k∞, abelian over k. Following closely the ideas of Belliard in [1], we prove that in K∞, the projective limit of the p-class group and the projective limit of units modulo elliptic units share the same μ-invariant and the same λ-invariant. We deduce that a version of the classical main conjecture, which is known to be true for p ∉ {2, 3}, holds also for p ∈ {2, 3} once we neglect the μ-invariants.

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