TITLE

# CONSTRUCTION OF CLASS FIELDS OVER IMAGINARY QUADRATIC FIELDS AND APPLICATIONS

AUTHOR(S)
BUMKYU CHO; JA KYUNG KOO
PUB. DATE
June 2010
SOURCE
Quarterly Journal of Mathematics;Jun2010, Vol. 61 Issue 2, p199
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
Let K be an imaginary quadratic field, Hî”® the ring class field of an order î”® in K and K(N) be the ray class field modulo N over K for a positive integer N. In this paper we provide certain general techniques of finding Hî”® and K(N) by using the theory of Shimura's canonical models via his reciprocity law, from which we partially extend some results of Schertz (Remark 4.2), Chen-Yui (Remark 4.2, Corollary 4.4), Coxâ€“McKayâ€“Stevenhagen (Corollary 4.5) and Caisâ€“Conrad (Remark 5.3). And, we further reilluminate the classical result of Hasse by means of such a method (Corollary 5.4), and discover how to get one ray class invariant over K from Hasse's two generators (Corollary 5.5) which is different from Ramachandra's invariant [K. Ramachandra, Some applications of Kronecker's limit formulas, Ann. Math. 80 (1964), 104â€“148].
ACCESSION #
50640829

## Related Articles

• Some Results Connected with the Class Number Problem in Real Quadratic Fields. Grytczuk, Aleksander; Grytczuk, Jarosław // Acta Mathematica Sinica;Oct2005, Vol. 21 Issue 5, p1107

We investigate arithmetic properties of certain subsets of square-free positive integers and obtain in this way some results concerning the class number h( d) of the real quadratic field $$\mathbb{Q}{\left( {{\sqrt d }} \right)}$$. In particular, we give a new proof of the result of Hasse,...

• Weighted Distribution of the 4-rank of Class Groups and Applications. Fouvry, Étienne; Klüners, Jürgen // IMRN: International Mathematics Research Notices;Aug2011, Vol. 2011 Issue 16, p3618

We prove that the distribution of the values of the 4-rank of ideal class groups of quadratic fields is not affected when it is weighted by a divisor type function. We then give several applications concerning a new lower bound of the sums of class numbers of real quadratic fields with...

• Subgroups of Class Groups of Algebraic Quadratic Function Fields. Wang, Kunpeng; Zhang, Xianke // Chinese Annals of Mathematics;Jul2003, Vol. 24 Issue 3, p315

Ideal class groups H(K) of algebraic quadratic function fields K are studied. Necessary and sufficient condition is given for the class group H(K) to contain a cyclic subgroup of any order n, which holds true for both real and imaginary fields K. Then several series of function fields K,...

• ON THE CLASS OF ALL RECIPROCAL BASES FOR INTEGERS. Šalát, T.; Tomanová, J. // Acta Mathematica Universitatis Comenianae;2007, Vol. 76 Issue 2, p257

In this paper the structure of the class of all reciprocal bases of N is investigated from metric and topological point of view. For this purpose the method of dyadic values of infinite subsets of N will be applied.

• JAZ volume 100 Issue 1 Cover and Back matter.  // Journal of the Australian Mathematical Society;Feb2016, Vol. 100 Issue 1, pb1

The article offers information on preparation of manuscripts for the periodical and copying along with the table of contents.

• A characterization of the family of external lines to a quadratic cone of PG(3, q), q. Di Gennaro, R.; Durante, N.; Olanda, D. // Journal of Geometry;2010, Vol. 96 Issue 1/2, p63

In this paper we characterize the family of external lines to a quadratic cone of PG(3, q), q odd, by their intersection properties with points and planes of the space.

• EXACT COVERING SYSTEMS IN NUMBER FIELDS. Jiang, Yupeng; Deng, Yingpu // Quarterly Journal of Mathematics;Mar2014, Vol. 65 Issue 1, p211

It is well known that, in an exact covering system in â„¤, the biggest modulus must be repeated. Very recently, Kim gave an analogous result for certain quadratic fields, and Kim also conjectured that it must hold in any algebraic number field. In this paper, we prove Kim's conjecture. In...

• Transitive parabolic unitals in semifield planes. Johnson, Norman // Journal of Geometry;2006, Vol. 85 Issue 1/2, p61

Every semifield plane with spread in PG(3, K), where K is a field admitting a quadratic extension K +, is shown to admit a transitive parabolic unital.

• Products of Involutions in Steinberg Group over Skew Fields*. Jizhu Nan; Hong You // Chinese Annals of Mathematics;Apr2007, Vol. 28 Issue 2, p253

Consider the stable Steinberg group St( K) over a skew field K. An element x is called an involution if x 2 = 1. In this paper, an involution is allowed to be the identity. The authors prove that an element A of GL n ( K) up to conjugation can be represented as BC, where B is lower triangular...

Share

## Read the Article

Courtesy of THE LIBRARY OF VIRGINIA

Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library