scholarly journals Infrastructure: Structure Inside the Class Group of a Real Quadratic Field

2014 ◽  
Vol 61 (01) ◽  
pp. 36 ◽  
Author(s):  
Michael J. Jacobson ◽  
R. Scheidler
Keyword(s):  
Class Group ◽  
Quadratic Field ◽  
Real Quadratic Field ◽  
Real Quadratic

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.

The group Gal(k3(2)|k) for k = ℚ(−3,d) of type (3,3)

2016 ◽  
Vol 12 (07) ◽  
pp. 1951-1986 ◽  
Author(s):  
Abdelmalek Azizi ◽  
Mohamed Talbi ◽  
Mohammed Talbi ◽  
Aïssa Derhem ◽  
Daniel C. Mayer
Keyword(s):  
Galois Group ◽  
Class Group ◽  
Quadratic Field ◽  
Galois Groups ◽  
Numerical Verification ◽  
Real Quadratic Field ◽  
Class Field ◽  
Type Formula ◽  
Biquadratic Fields ◽  
Real Quadratic

Let [Formula: see text] denote the discriminant of a real quadratic field. For all bicyclic biquadratic fields [Formula: see text], having a [Formula: see text]-class group of type [Formula: see text], the possibilities for the isomorphism type of the Galois group [Formula: see text] of the second Hilbert [Formula: see text]-class field [Formula: see text] of [Formula: see text] are determined. For each coclass graph [Formula: see text], [Formula: see text], in the sense of Eick, Leedham-Green, Newman and O’Brien, the roots [Formula: see text] of even branches of exactly one coclass tree and, in the case of even coclass [Formula: see text], additionally their siblings of depth [Formula: see text] and defect [Formula: see text], turn out to be admissible. The principalization type [Formula: see text] of [Formula: see text]-classes of [Formula: see text] in its four unramified cyclic cubic extensions [Formula: see text] is given by [Formula: see text] for [Formula: see text], and by [Formula: see text] for [Formula: see text]. The theory is underpinned by an extensive numerical verification for all [Formula: see text] fields [Formula: see text] with values of [Formula: see text] in the range [Formula: see text], which supports the assumption that all admissible vertices [Formula: see text] will actually be realized as Galois groups [Formula: see text] for certain fields [Formula: see text], asymptotically.

scholarly journals Use of the rabinowitsch polynomial to determine the class groups of a real quadratic field

1994 ◽  
Vol 50 (3) ◽  
pp. 435-443
Author(s):  
R.A. Mollin
Keyword(s):  
Class Group ◽  
Quadratic Field ◽  
Sufficient Condition ◽  
Real Quadratic Field ◽  
Necessary And Sufficient Condition ◽  
Class Groups ◽  
Necessary And Sufficient ◽  
Real Quadratic

The main result is a necessary and sufficient condition for the class group of a real quadratic field to be determined by primality properties of the well-known Rabinowitsch polynomial.

HEURISTICS FOR -CLASS TOWERS OF REAL QUADRATIC FIELDS

2019 ◽  
pp. 1-24
Author(s):  
Nigel Boston ◽  
Michael R. Bush ◽  
Farshid Hajir
Keyword(s):  
Infinite Order ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Ideal Class ◽  
Real Quadratic Field ◽  
Ground Field ◽  
Real Quadratic Fields ◽  
Ideal Class Groups ◽  
Formula Group ◽  
Real Quadratic

Let $p$ be an odd prime. For a number field $K$ , we let $K_{\infty }$ be the maximal unramified pro- $p$ extension of  $K$ ; we call the group $\text{Gal}(K_{\infty }/K)$ the $p$ -class tower group of  $K$ . In a previous work, as a non-abelian generalization of the work of Cohen and Lenstra on ideal class groups, we studied how likely it is that a given finite $p$ -group occurs as the $p$ -class tower group of an imaginary quadratic field. Here we do the same for an arbitrary real quadratic field $K$ as base. As before, the action of $\text{Gal}(K/\mathbb{Q})$ on the $p$ -class tower group of $K$ plays a crucial role; however, the presence of units of infinite order in the ground field significantly complicates the possibilities for the groups that can occur. We also sharpen our results in the imaginary quadratic field case by removing a certain hypothesis, using ideas of Boston and Wood. In the appendix, we show how the probabilities introduced for finite $p$ -groups can be extended in a consistent way to the infinite pro- $p$ groups which can arise in both the real and imaginary quadratic settings.

scholarly journals On the Doi-Naganuma lifting associated with imaginary quadratic fields

1978 ◽  
Vol 71 ◽  
pp. 149-167 ◽  
Author(s):  
Tetsuya Asai
Keyword(s):  
Theta Function ◽  
Function Method ◽  
Quadratic Field ◽  
Discrete Subgroup ◽  
Real Quadratic Field ◽  
Field Case ◽  
The Real ◽  
Concrete Form ◽  
Elliptic Modular Form ◽  
Real Quadratic

Similarly to the real quadratic field case by Doi and Naganuma ([3], [9]) there is a lifting from an elliptic modular form to an automorphic form on SL2(C) with respect to an arithmetic discrete subgroup relative to an imaginary quadratic field. This fact is contained in his general theory of Jacquet ([6]) as a special case. In this paper, we try to reproduce this lifting in its concrete form by using the theta function method developed first by Niwa ([10]); also Kudla ([7]) has treated the real quadratic field case on the same line. The theta function method will naturally lead to a theory of lifting to an orthogonal group of general signature (cf. Oda [11]), and the present note will give a prototype of non-holomorphic case.

scholarly journals On Petersson’s partition limit formula

2020 ◽  
pp. 1-14
Author(s):  
Carlos Castaño-Bernard ◽  
Florian Luca
Keyword(s):  
Quadratic Field ◽  
Character Formula ◽  
Real Quadratic Field ◽  
The Real ◽  
Limit Formula ◽  
Real Quadratic

For each prime [Formula: see text] consider the Legendre character [Formula: see text]. Let [Formula: see text] be the number of partitions of [Formula: see text] into parts [Formula: see text] such that [Formula: see text]. Petersson proved a beautiful limit formula for the ratio of [Formula: see text] to [Formula: see text] as [Formula: see text] expressed in terms of important invariants of the real quadratic field [Formula: see text]. But his proof is not illuminating and Grosswald conjectured a more natural proof using a Tauberian converse of the Stolz–Cesàro theorem. In this paper, we suggest an approach to address Grosswald’s conjecture. We discuss a monotonicity conjecture which looks quite natural in the context of the monotonicity theorems of Bateman–Erdős.

scholarly journals Solvability of the diophantine equation x2 − Dy2 = ± 2 and new invariants for real quadratic fields

1994 ◽  
Vol 134 ◽  
pp. 137-149 ◽  
Author(s):  
Hideo Yokoi
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Real Quadratic Field ◽  
Real Quadratic Fields ◽  
Real Quadratic

In our recent papers [3, 4, 5], we defined some new D-invariants for any square-free positive integer D and considered their properties and interrelations among them. Especially, as an application of it, we discussed in [5] the characterization of real quadratic field Q() of so-called Richaud-Degert type in terms of these new D-invariants.

scholarly journals Volumes of Hyperbolic Three-Manifolds Associated with Modular Links

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1206 ◽  
Author(s):  
Alex Brandts ◽  
Tali Pinsky ◽  
Lior Silberman
Keyword(s):  
Class Group ◽  
Quadratic Field ◽  
Finite Collection ◽  
Ideal Class Group ◽  
Modular Surface ◽  
Closed Curves ◽  
Hyperbolic Three Manifolds ◽  
Unit Tangent Bundle ◽  
Real Quadratic ◽  
The Ideal

Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle PSL 2 ( Z ) ∖ PSL 2 ( R ) . A finite collection of such orbits is a collection of disjoint closed curves in a 3-manifold, in other words a link. The complement of those links is always a hyperbolic 3-manifold, and hence has a well-defined volume. We present strong numerical evidence that, in the case of the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesics. This is not the case for general sets of geodesics.

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