scholarly journals On Greenberg's conjecture on a certain real quadratic field

2000 ◽  
Vol 76 (10) ◽  
pp. 163-164 ◽  
Author(s):  
Yasushi Mizusawa
Keyword(s):  
Quadratic Field ◽  
Real Quadratic Field ◽  
Greenberg’S Conjecture ◽  
Real Quadratic

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.

Properties of real quadratic irrational numbers under the action of the group H = áx, y: x2 = y4 = 1ñ

2005 ◽  
Vol 42 (4) ◽  
pp. 371-386
Author(s):  
M. Aslam Malik ◽  
S. M. Husnine ◽  
Abdul Majeed
Keyword(s):  
Quadratic Field ◽  
Infinite Group ◽  
Algebraic Structures ◽  
Real Quadratic Field ◽  
Irrational Numbers ◽  
The Real ◽  
Basic Properties ◽  
Quadratic Irrational ◽  
Real Quadratic

Studying groups through their actions on different sets and algebraic structures has become a useful technique to know about the structure of the groups. The main object of this work is to examine the action of the infinite group \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $H = \langle x,y : x^{2} = y^{4} = 1\rangle$ \end{document} where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $x (z) = \frac{-1}{2z}$ \end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $y (z) = \frac{-1}{2(z+1)}$ \end{document} on the real quadratic field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Q}\left(\sqrt{n}\,\right)$ \end{document} and find invariant subsets of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Q}\left(\sqrt{n}\,\right)$ \end{document} under the action of the group \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $H$ \end{document}. We also discuss some basic properties of elements of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Q}\left(\sqrt{n}\,\right)$ \end{document} under the action of the group H.

scholarly journals SYMMETRIES FOR BORCHERDS LIFTS ON HILBERT MODULAR GROUPS AND HIRZEBRUCH–ZAGIER DIVISORS

2013 ◽  
Vol 24 (08) ◽  
pp. 1350065 ◽  
Author(s):  
BERNHARD HEIM ◽  
ATSUSHI MURASE
Keyword(s):  
Modular Group ◽  
Quadratic Field ◽  
Hecke Operators ◽  
The Other ◽  
Real Quadratic Field ◽  
Hilbert Modular Group ◽  
Modular Groups ◽  
Hilbert Modular ◽  
The One ◽  
Real Quadratic

We show certain symmetries for Borcherds lifts on the Hilbert modular group over a real quadratic field. We give two different proofs, the one analytic and the other arithmetic. The latter proof yields an explicit description of the action of Hecke operators on Borcherds lifts.

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