On some extensions of Gauss' work and applications

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Let K be an imaginary quadratic field of discriminant d(K) with ring of integers O-K, and let tau(K) be an element of the complex upper half plane so that O-K = [tau(K), 1]. For a positive integer N, let Q(N)(d(K)) be the set of primitive positive definite binary quadratic forms of discriminant d(K) with leading coefficients relatively prime to N. Then, with any congruence subgroup G of SL2(Z) one can define an equivalence relation (similar to)(Gamma) on Q(N)(d(K)). Let F-Gamma,F-Q denote the field of meromorphic modular functions for G with rational Fourier coefficients. We show that the set of equivalence classes Q(N)(d(K))/(similar to)(Gamma) can be equipped with a group structure isomorphic to Gal(KF Gamma,Q (tau(K))/K) for some Gamma, which generalizes the classical theory of form class groups.
Publisher
DE GRUYTER POLAND SP Z O O
Issue Date
2020-12
Language
English
Article Type
Article
Citation

OPEN MATHEMATICS, v.18, pp.1915 - 1934

ISSN
2391-5455
DOI
10.1515/math-2020-0126
URI
http://hdl.handle.net/10203/281165
Appears in Collection
MA-Journal Papers(저널논문)
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