Construction of ray class fields by elliptic units
type: Item
instance of: scholarly article
title: Construction of ray class fields by elliptic units
zbMATH Open document ID: 0902.11047
author: Reinhard Schertz
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1998-09-21T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/227251
full work available at URL: jtnb9-2.html
review text: Let \(K\) be an imaginary quadratic field, and let \(K(\mathfrak f)\) be a ray class field over \(K\), where the conductor \(\mathfrak f\) is an integral ideal of the ring of integers of \(K\). Let \(\omega_1,\omega_2\) be a \(\mathbb Z\)-basis of \(\mathfrak f\) with \(\text{Im} (\omega_1/\omega_2)>0\). The normalized Klein form is defined by \[ \varphi(z)=2\pi i\exp(-zz^\ast/2)\sigma(z\mid\mathfrak f) \eta(\omega_1/\omega_2)^2\omega_2^{-1}, \] where \(\sigma\) denotes the \(\sigma\)-function of \(\mathfrak f\), \(\eta\) is the Dedekind \(\eta\)-function, and \(z^\ast\) is defined by \(z^\ast=z_1\eta_1+ z_2\eta_2\) with the real coordinates \(z_1,z_2\) of \(z=z_1\omega_1+z_2\omega_2\) and the quasi-periods \(\eta_1,\eta_2\) of the elliptic Weierstrass \(\zeta\)-function of \(\mathfrak f\) belonging to \(\omega_1,\omega_2.\) \textit{K. Ramachandra} [Ann. Math. (2) 80, 104-148 (1964; Zbl 0142.29804)] has proved that \(K(\mathfrak f)\) can be generated by a complicated product involving high powers of singular values of \(\varphi\) and singular values of the discriminant. The author's aim is to show that in many cases it is sufficient to take a power of one singular value of \(\varphi\) or a quotient of two such values. Numerical examples indicate that the coefficients of the minimal polynomials of the constructed numbers are rather small.
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11G16
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 1164094
zbMATH Keywords: elliptic unit
zbMATH Keywords: imaginary quadratic field
zbMATH Keywords: ray class field
zbMATH Keywords: normalized Klein form
reviewed by: Veikko Ennola
MaRDI profile type: Publication
OpenAlex ID: W2042333717
cites work: Q3953865
cites work: Q3246482
cites work: Some applications of Kronecker's limit formulas
cites work: Galoismodulstruktur und elliptische Funktionen. (Galois module structure and elliptic functions)
cites work: Problèmes de construction en multiplication complexe
cites work: Q913858
cites work: \(L\)-functions at \(s=1\). IV: First derivatives at \(s=0\)
DOI: 10.5802/JTNB.209
Recommended article: Lower powers of elliptic units
Recommended article: Singular values of some modular functions and their applications to class fields
Recommended article: Normal bases of ray class fields over imaginary quadratic fields
Recommended article: Ray class invariants over imaginary quadratic fields
Recommended article: GENERATION OF CLASS FIELDS BY SIEGEL-RAMACHANDRA INVARIANTS
Recommended article: On a problem of Hasse and Ramachandra
Recommended article: Ray class fields generated by torsion points of certain elliptic curves
Recommended article: Generation of ray class fields by elliptic units
Recommended article: Q4312957
Recommended article: Construction of ray-class fields by smaller generators and their applications
description: scientific article; zbMATH DE number 1164094