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Composition law and complex multiplication

Title
Composition law and complex multiplication
Authors
Eum I.S.Jung H.Y.Koo J.K.Shin D.H.
Ewha Authors
정호윤
Issue Date
2020
Journal Title
Journal of Number Theory
ISSN
0022-314XJCR Link
Citation
Journal of Number Theory vol. 209, pp. 396 - 420
Keywords
Binary quadratic formsClass field theoryComplex multiplicationModular functions
Publisher
Academic Press Inc.
Indexed
SCI; SCIE; SCOPUS WOS scopus
Document Type
Article
Abstract
Let K be an imaginary quadratic field of discriminant dK, and let n be a nontrivial integral ideal of K in which N is the smallest positive integer. Let QN(dK) be the set of primitive positive definite binary quadratic forms of discriminant dK whose leading coefficients are relatively prime to N. We adopt an equivalence relation ∼n on QN(dK) so that the set of equivalence classes QN(dK)/∼n can be regarded as a group isomorphic to the ray class group of K modulo n. We further establish an explicit isomorphism of QN(dK)/∼n onto Gal(Kn/K) in terms of Fricke invariants, where Kn denotes the ray class field of K modulo n. This would be a certain extension of the classical composition theory of binary quadratic forms, originated and developed by Gauss and Dirichlet. © 2019 Elsevier Inc.
DOI
10.1016/j.jnt.2019.09.005
Appears in Collections:
ETC > ETC
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