scholarly journals Cohomology groups of Fermat curves via ray class fields of cyclotomic fields

2020 ◽  
Vol 554 ◽  
pp. 78-105
Author(s):  
Rachel Davis ◽  
Rachel Pries
Keyword(s):  
Cyclotomic Fields ◽  
Cohomology Groups ◽  
Fermat Curves ◽  
Class Fields ◽  
Ray Class Fields

Completely normal elements in some finite abelian extensions

2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Ja Koo ◽  
Dong Shin
Keyword(s):  
Function Fields ◽  
Modular Function ◽  
Rational Numbers ◽  
Quadratic Fields ◽  
Cyclotomic Fields ◽  
Imaginary Quadratic Fields ◽  
Abelian Extensions ◽  
Class Fields ◽  
Normal Element ◽  
Ray Class Fields

AbstractWe present some completely normal elements in the maximal real subfields of cyclotomic fields over the field of rational numbers, relying on the criterion for normal element developed in [Jung H.Y., Koo J.K., Shin D.H., Normal bases of ray class fields over imaginary quadratic fields, Math. Z., 2012, 271(1–2), 109–116]. And, we further find completely normal elements in certain abelian extensions of modular function fields in terms of Siegel functions.

Constructing ray class fields of a real quadratic field using elliptic curves

2019 ◽  
Vol 187 (3) ◽  
pp. 219-232
Author(s):  
Takashi Fukuda ◽  
Keiichi Komatsu ◽  
Kiichiro Hashimoto
Keyword(s):  
Elliptic Curves ◽  
Quadratic Field ◽  
Real Quadratic Field ◽  
Class Fields ◽  
Ray Class Fields ◽  
Real Quadratic

scholarly journals Ray class fields generated by torsion points of certain elliptic curves

2012 ◽  
Vol 28 (3) ◽  
pp. 341-360
Author(s):  
Ja Kyung Koo ◽  
Dong Hwa Shin ◽  
Dong Sung Yoon
Keyword(s):  
Elliptic Curves ◽  
Torsion Points ◽  
Class Fields ◽  
Ray Class Fields

Construction of ray-class fields by smaller generators and their applications

2017 ◽  
Vol 147 (4) ◽  
pp. 781-812 ◽  
Author(s):  
Ja Kyung Koo ◽  
Dong Sung Yoon
Keyword(s):  
Diophantine Equations ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Minimal Polynomials ◽  
Class Invariants ◽  
Class Fields ◽  
Ray Class Fields

We generate ray-class fields over imaginary quadratic fields in terms of Siegel–Ramachandra invariants, which are an extension of a result of Schertz. By making use of quotients of Siegel–Ramachandra invariants we also construct ray-class invariants over imaginary quadratic fields whose minimal polynomials have relatively small coefficients, from which we are able to solve certain quadratic Diophantine equations.

scholarly journals Siegel families with application to class fields

2018 ◽  
Vol 148 (4) ◽  
pp. 751-771 ◽  
Author(s):  
Ja Kyung Koo ◽  
Dong Hwa Shin ◽  
Dong Sung Yoon
Keyword(s):  
Galois Groups ◽  
Algebraic Numbers ◽  
Modular Functions ◽  
Special Values ◽  
Reciprocity Law ◽  
Class Fields ◽  
Ray Class Fields ◽  
Natural Way

We investigate certain families of meromorphic Siegel modular functions on which Galois groups act in a natural way. By using Shimura's reciprocity law we construct some algebraic numbers in the ray class fields of CM-fields in terms of special values of functions in these Siegel families.

scholarly journals Generating ray class fields of real quadratic fields via complex equiangular lines

2020 ◽  
Vol 192 (3) ◽  
pp. 211-233 ◽  
Author(s):  
Marcus Appleby ◽  
Steven Flammia ◽  
Gary McConnell ◽  
Jon Yard
Keyword(s):  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Class Fields ◽  
Ray Class Fields ◽  
Equiangular Lines ◽  
Real Quadratic

scholarly journals On some Fricke families and application to the Lang–Schertz conjecture

2016 ◽  
Vol 146 (4) ◽  
pp. 723-740 ◽  
Author(s):  
Ho Yun Jung ◽  
Ja Kyung Koo ◽  
Dong Hwa Shin
Keyword(s):  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Special Values ◽  
Class Fields ◽  
Ray Class Fields

We investigate two kinds of Fricke families, those consisting of Fricke functions and those consisting of Siegel functions. In terms of their special values we then generate ray class fields of imaginary quadratic fields over the Hilbert class fields, which are related to the Lang–Schertz conjecture.

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