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Binary quadratic forms and ray class groups

Part of: Arithmetic algebraic geometry Algebraic number theory: global fields Forms and linear algebraic groups

Published online by Cambridge University Press:  23 January 2019

Ick Sun Eum ,
Ja Kyung Koo  and
Dong Hwa Shin
Ick Sun Eum
Affiliation:
Department of Mathematics Education, Dongguk University-Gyeongju, Gyeongju-si, Gyeongsangbuk-do 38066, Republic of Korea (zandc@dongguk.ac.kr)
Ja Kyung Koo
Affiliation:
Department of Mathematical Sciences, KAIST Daejeon 34141, Republic of Korea (jkkoo@math.kaist.ac.kr)
Dong Hwa Shin
Affiliation:
Department of Mathematics, Hankuk University of Foreign Studies, Yongin-si, Gyeonggi-do 17035, Republic of Korea (dhshin@hufs.ac.kr)
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Abstract

Let K be an imaginary quadratic field different from $\open{Q}(\sqrt {-1})$ and $\open{Q}(\sqrt {-3})$. For a positive integer N, let KN be the ray class field of K modulo $N {\cal O}_K$. By using the congruence subgroup ± Γ1(N) of SL2(ℤ), we construct an extended form class group whose operation is basically the Dirichlet composition, and explicitly show that this group is isomorphic to the Galois group Gal(KN/K). We also present an algorithm to find all distinct form classes and show how to multiply two form classes. As an application, we describe Gal(KNab/K) in terms of these extended form class groups for which KNab is the maximal abelian extension of K unramified outside prime ideals dividing $N{\cal O}_K$.

Keywords

Binary quadratic formsclass field theoryclass groupscomplex multiplication

MSC classification

Primary: 11R29: Class numbers, class groups, discriminants
Secondary: 11E16: General binary quadratic forms 11G15: Complex multiplication and moduli of abelian varieties 11R37: Class field theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 695 - 720
Copyright
Copyright © Royal Society of Edinburgh 2019

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