HERMITIAN POINTS IN MARKOV SPECTRA

2010 ◽  
Vol 06 (04) ◽  
pp. 713-730 ◽  
Author(s):  
L. YA VULAKH
Keyword(s):  
Discrete Group ◽  
Class Group ◽  
Quadratic Field ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Space Model ◽  
Continuous Parameter ◽  
Markov Spectrum ◽  
Group B ◽  
Group Of Isometries

Let Hn be the upper half-space model of the n-dimensional hyperbolic space. For n=3, Hermitian points in the Markov spectrum of the extended Bianchi group Bd are introduced for any d. If ν is a Hermitian point in the spectrum, then there is a set of extremal geodesics in H3 with diameter 1/ν, which depends on one continuous parameter. It is shown that ν2 ≤ |D|/24 for any imaginary quadratic field with discriminant D, whose ideal-class group contains no cyclic subgroup of order 4, and in many other cases. Similarly, in the case of n = 4, if ν is a Hermitian point in the Markov spectrum for SV(Z4), some discrete group of isometries of H4, then the corresponding set of extremal geodesics depends on two continuous parameters.

scholarly journals Annihilators of the Ideal Class Group of a Cyclic Extension of an Imaginary Quadratic Field

2019 ◽  
Vol 71 (6) ◽  
pp. 1395-1419
Author(s):  
Hugo Chapdelaine ◽  
Radan Kučera
Keyword(s):  
Sylow Subgroup ◽  
Class Group ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Cyclic Extension ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Elliptic Units ◽  
The Ideal

AbstractThe aim of this paper is to study the group of elliptic units of a cyclic extension $L$ of an imaginary quadratic field $K$ such that the degree $[L:K]$ is a power of an odd prime $p$. We construct an explicit root of the usual top generator of this group, and we use it to obtain an annihilation result of the $p$-Sylow subgroup of the ideal class group of $L$.

DIOPHANTINE APPROXIMATION IN $\mathbf{Q}(\sqrt{-5})$ AND $\mathbf{Q}(\sqrt{-6})$

2006 ◽  
Vol 02 (01) ◽  
pp. 25-48 ◽  
Author(s):  
L. YA. VULAKH
Keyword(s):  
Diophantine Approximation ◽  
Hyperbolic Plane ◽  
Three Dimensional ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Space Model ◽  
Continuous Parameter ◽  
Markov Spectrum ◽  
Group B ◽  
Discrete Part

The complete description of the discrete part of the Lagrange and Markov spectra of the imaginary quadratic fields with discriminants -20 and -24 are given. Farey polygons associated with the extended Bianchi groups Bd, d = 5, 6, are used to reduce the problem of finding the discrete part of the Markov spectrum for the group Bd to the corresponding problem for one of its maximal Fuchsian subgroup. Hermitian points in the Markov spectrum of Bd are introduced for any d. Let H3 be the upper half-space model of the three-dimensional hyperbolic space. If ν is a hermitian point in the spectrum, then there is a set of extremal geodesics in H3 with diameter 1/ν, which depends on one continuous parameter. This phenomenon does not take place in the hyperbolic plane.

scholarly journals ON THE IDEAL CLASS GROUP OF CERTAIN QUADRATIC FIELDS

2010 ◽  
Vol 52 (3) ◽  
pp. 575-581 ◽  
Author(s):  
YASUHIRO KISHI
Keyword(s):  
Class Number ◽  
Class Group ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Ideal Class ◽  
Real Quadratic Field ◽  
Ideal Class Group ◽  
The Real ◽  
Real Quadratic ◽  
The Ideal

AbstractLet n(≥ 3) be an odd integer. Let k:= $\Q(\sqrt{4-3^n})\)$ be the imaginary quadratic field and k′:= $\Q(\sqrt{-3(4-3^n)})\)$ the real quadratic field. In this paper, we prove that the class number of k is divisible by 3 unconditionally, and the class number of k′ is divisible by 3 if n(≥ 9) is divisible by 3. Moreover, we prove that the 3-rank of the ideal class group of k is at least 2 if n(≥ 9) is divisible by 3.

MINIMA OF BINARY INDEFINITE HERMITIAN FORMS

2010 ◽  
Vol 06 (02) ◽  
pp. 411-435 ◽  
Author(s):  
L. YA. VULAKH
Keyword(s):  
Modular Group ◽  
Quadratic Field ◽  
Three Dimensional ◽  
Hermitian Forms ◽  
Space Model ◽  
Continuous Parameter ◽  
Hilbert Modular Group ◽  
Hilbert Modular ◽  
Group B

Classification of binary indefinite primitive Hermitian forms modulo the action of the extended Bianchi group (or Hilbert modular group) Bd is given. When the discriminant of the quadratic field [Formula: see text] (and d) is negative, the results obtained can be applied to classify the maximal non-elementary Fuchsian subgroups of Bd, and to find the Hermitian points in the Markov spectrum of Bd. If ν is a Hermitian point in the spectrum, then there is a set of extremal geodesics in H3, the upper half-space model of the three-dimensional hyperbolic space, with diameter 1/ν, which depends on one continuous parameter.

scholarly journals Factorizations in Bounded Hereditary Noetherian Prime Rings

2018 ◽  
Vol 62 (2) ◽  
pp. 395-442 ◽  
Author(s):  
Daniel Smertnig
Keyword(s):  
Class Group ◽  
Structure Theory ◽  
Projective Modules ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Prime Rings ◽  
Sets Of Lengths ◽  
Irreducible Elements ◽  
Transfer Homomorphism ◽  
Zero Sum

AbstractIf H is a monoid and a = u1 ··· uk ∈ H with atoms (irreducible elements) u1, … , uk, then k is a length of a, the set of lengths of a is denoted by Ⅼ(a), and ℒ(H) = {Ⅼ(a) | a ∈ H} is the system of sets of lengths of H. Let R be a hereditary Noetherian prime (HNP) ring. Then every element of the monoid of non-zero-divisors R• can be written as a product of atoms. We show that if R is bounded and every stably free right R-ideal is free, then there exists a transfer homomorphism from R• to the monoid B of zero-sum sequences over a subset Gmax(R) of the ideal class group G(R). This implies that the systems of sets of lengths, together with further arithmetical invariants, of the monoids R• and B coincide. It is well known that commutative Dedekind domains allow transfer homomorphisms to monoids of zero-sum sequences, and the arithmetic of the latter has been the object of much research. Our approach is based on the structure theory of finitely generated projective modules over HNP rings, as established in the recent monograph by Levy and Robson. We complement our results by giving an example of a non-bounded HNP ring in which every stably free right R-ideal is free but which does not allow a transfer homomorphism to a monoid of zero-sum sequences over any subset of its ideal class group.

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