scholarly journals RELATIVE INTEGRAL BASES OVER A RAY CLASS FIELD

2006 ◽  
Vol 43 (1) ◽  
pp. 77-80
Author(s):  
So-Young Choi
Keyword(s):  
Class Field ◽  
Integral Bases ◽  
Ray Class Field

scholarly journals On a problem of Hasse and Ramachandra

2019 ◽  
Vol 17 (1) ◽  
pp. 131-140
Author(s):  
Ja Kyung Koo ◽  
Dong Hwa Shin ◽  
Dong Sung Yoon
Keyword(s):  
Positive Integer ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Integral Ideal ◽  
Weber Function ◽  
Class Field ◽  
Ray Class Field

Abstract Let K be an imaginary quadratic field, and let 𝔣 be a nontrivial integral ideal of K. Hasse and Ramachandra asked whether the ray class field of K modulo 𝔣 can be generated by a single value of the Weber function. We completely resolve this question when 𝔣 = (N) for any positive integer N excluding 2, 3, 4 and 6.

scholarly journals Central extensions and rational quadratic forms

1993 ◽  
Vol 130 ◽  
pp. 177-182 ◽  
Author(s):  
Yoshiomi Furuta ◽  
Tomio Kubota
Keyword(s):  
Central Extension ◽  
Quadratic Forms ◽  
Class Group ◽  
Quadratic Field ◽  
Abelian Extension ◽  
Narrow Sense ◽  
Maximal Extension ◽  
Central Extensions ◽  
Class Field ◽  
Ray Class Field

The purpose of this paper is to characterize by means of simple quadratic forms the set of rational primes that are decomposed completely in a non-abelian central extension which is abelian over a quadratic field. More precisely, let L = Q be a bicyclic biquadratic field, and let K = Q. Denote by the ray class field mod m of K in narrow sense for a large rational integer m. Let be the maximal abelian extension over Q contained in and be the maximal extension contained in such that Gal(/L) is contained in the center of Gal(/Q). Then we shall show in Theorem 2.1 that any rational prime p not dividing d1d2m is decomposed completely in /Q if and only if p is representable by rational integers x and y such that x ≡ 1 and y ≡ 0 mod m as followswhere a, b, c are rational integers such that b2 − 4ac is equal to the discriminant of K and (a) is a norm of a representative of the ray class group of K mod m.Moreover is decomposed completely in if and only if .

scholarly journals RAY CLASS GROUPS OF QUADRATIC AND CYCLOTOMIC FIELDS

2010 ◽  
Vol 06 (05) ◽  
pp. 1169-1182
Author(s):  
JING LONG HOELSCHER
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Quadratic Number ◽  
Class Groups ◽  
Quadratic Number Field ◽  
Cyclotomic Number ◽  
Class Field ◽  
Ray Class Field ◽  
Real Quadratic ◽  
Mod P

This paper studies Galois extensions over real quadratic number fields or cyclotomic number fields ramified only at one prime. In both cases, the ray class groups are computed, and they give restrictions on the finite groups that can occur as such Galois groups. Let [Formula: see text] be a real quadratic number field with a prime P lying above p in ℚ. If p splits in K/ℚ and p does not divide the big class number of K, then any pro-p extension of K ramified only at P is finite cyclic. If p is inert in K/ℚ, then there exist infinite extensions of K ramified only at P. Furthermore, for big enough integer k, the ray class field (mod Pk+1) is obtained from the ray class field (mod Pk) by adjoining ζpk+1. In the case of a regular cyclotomic number field K = ℚ(ζp), the explicit structure of ray class groups (mod Pk) is given for any positive integer k, where P is the unique prime in K above p.

scholarly journals Unramified Subextensions of Ray Class Field Towers

2002 ◽  
Vol 249 (2) ◽  
pp. 528-543 ◽  
Author(s):  
Farshid Hajir ◽  
Christian Maire
Keyword(s):  
Class Field ◽  
Class Field Towers ◽  
Ray Class Field

scholarly journals The Lambda Invariants at CM Points

2019 ◽  
Author(s):  
Tonghai Yang ◽  
Hongbo Yin ◽  
Peng Yu
Keyword(s):  
Algebraic Integer ◽  
Independent Interest ◽  
Whittaker Functions ◽  
Explicit Formulas ◽  
Class Field ◽  
Cm Points ◽  
Ray Class Field

Abstract In this paper, we show that $\lambda (z_1) -\lambda (z_2)$, $\lambda (z_1)$, and $1-\lambda (z_1)$ are all Borcherds products on $X(2) \times X(2)$. We then use the big CM value formula of Bruinier, Kudla, and Yang to give explicit factorization formulas for the norms of $\lambda (\frac{d+\sqrt d}2)$, $1-\lambda (\frac{d+\sqrt d}2)$, and $\lambda (\frac{d_1+\sqrt{d_1}}2) -\lambda (\frac{d_2+\sqrt{d_2}}2)$, with the latter under the condition $(d_1, d_2)=1$. Finally, we use these results to show that $\lambda (\frac{d+\sqrt d}2)$ is always an algebraic integer and can be easily used to construct units in the ray class field of ${\mathbb{Q}}(\sqrt{d})$ of modulus $2$. In the process, we also give explicit formulas for a whole family of local Whittaker functions, which are of independent interest.

scholarly journals A note on ray class fields of global fields

1990 ◽  
Vol 120 ◽  
pp. 61-66 ◽  
Author(s):  
Franz Halter-Koch
Keyword(s):  
Field Theory ◽  
Function Field ◽  
Class Field ◽  
Relative Version ◽  
Class Fields ◽  
Ray Class Fields ◽  
Definition Of ◽  
Ray Class Field ◽  
Hilbert Class Field ◽  
Function Field Case

The notion of a ray class field, which is fundamental in Takagi’s class field theory, has no immediate analogon in the function field case. The reason for this lies in the lacking of a distinguished maximal order. In this paper I overcome this difficulty by a relative version of the notion of ray class fields to be defined for every holomorphy ring of the field. The prototype for this new notion is M. Rosen’s definition of a Hilbert class field for function fields [6].

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