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 Singular values of multiple eta-quotients for ramified primes

2013 ◽  
Vol 16 ◽  
pp. 407-418 ◽  
Author(s):  
Andreas Enge ◽  
Reinhard Schertz
Keyword(s):  
Complex Multiplication ◽  
Singular Values ◽  
Number Fields ◽  
Algebraic Numbers ◽  
Modular Functions ◽  
Class Invariants ◽  
Free Level ◽  
Yield Class ◽  
Imaginary Quadratic Number ◽  
Class Fields

AbstractWe determine the conditions under which singular values of multiple $\eta $-quotients of square-free level, not necessarily prime to six, yield class invariants; that is, algebraic numbers in ring class fields of imaginary-quadratic number fields. We show that the singular values lie in subfields of the ring class fields of index ${2}^{{k}^{\prime } - 1} $ when ${k}^{\prime } \geq 2$ primes dividing the level are ramified in the imaginary-quadratic field, which leads to faster computations of elliptic curves with prescribed complex multiplication. The result is generalised to singular values of modular functions on ${ X}_{0}^{+ } (p)$ for $p$ prime and ramified.

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.

Fermat curves and a refinement of the reciprocity law on cyclotomic units

2018 ◽  
Vol 2018 (741) ◽  
pp. 255-273 ◽  
Author(s):  
Tomokazu Kashio
Keyword(s):  
Beta Function ◽  
Zeta Functions ◽  
Alternative Proof ◽  
Base Field ◽  
Algebraic Numbers ◽  
Special Values ◽  
Reciprocity Law ◽  
Stark's Conjecture ◽  
Fermat Curves ◽  
Cyclotomic Units

Abstract We define a “period-ring-valued beta function” and give a reciprocity law on its special values. The proof is based on some results of Rohrlich and Coleman concerning Fermat curves. We also have the following application. Stark’s conjecture implies that the exponentials of the derivatives at s=0 of partial zeta functions are algebraic numbers which satisfy a reciprocity law under certain conditions. It follows from Euler’s formulas and properties of cyclotomic units when the base field is the rational number field. In this paper, we provide an alternative proof of a weaker result by using the reciprocity law on the period-ring-valued beta function. In other words, the reciprocity law given in this paper is a refinement of the reciprocity law on cyclotomic units.

RAMANUJAN INVARIANTS FOR DISCRIMINANTS CONGRUENT TO 5 (mod 24)

2012 ◽  
Vol 08 (01) ◽  
pp. 265-287 ◽  
Author(s):  
ELISAVET KONSTANTINOU ◽  
ARISTIDES KONTOGEORGIS
Keyword(s):  
Number Theory ◽  
Computer Science ◽  
Modular Functions ◽  
Minimal Polynomials ◽  
Reciprocity Law ◽  
Class Invariants ◽  
Lecture Notes ◽  
Yield Class ◽  
Class Fields

In this paper we compute the minimal polynomials of Ramanujan values [Formula: see text] for discriminants D ≡ 5 ( mod 24). Our method is based on Shimura Reciprocity Law as which was made computationally explicit by Gee and Stevenhagen in [Generating class fields using Shimura reciprocity, in Algorithmic Number Theory, Lecture Notes in Computer Science, Vol. 1423 (Springer, Berlin, 1998), pp. 441–453; MR MR1726092 (2000m:11112)]. However, since these Ramanujan values are not class invariants, we present a modification of the method used in [Generating class fields using Shimura reciprocity, in Algorithmic Number Theory, Lecture Notes in Computer Science, Vol. 1423 (Springer, Berlin, 1998), pp. 441–453; MR MR1726092 (2000m:11112)] which can be applied on modular functions that do not necessarily yield class invariants.

scholarly journals Algebraic Properties of Chromatic Roots

10.37236/6578 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Peter J. Cameron ◽  
Kerri Morgan
Keyword(s):  
Algebraic Integer ◽  
Tutte Polynomial ◽  
Chromatic Polynomial ◽  
Galois Groups ◽  
Algebraic Numbers ◽  
Isaac Newton ◽  
Isaac Newton Institute ◽  
Chromatic Root ◽  
The Subject ◽  
Chromatic Roots

A chromatic root is a root of the chromatic polynomial of a graph.  Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008.  The purpose of this paper is to report on the seminar and subsequent developments.We conjecture that, for every algebraic integer $\alpha$, there is a natural number n such that $\alpha+n$ is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

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.

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