Abstract Let K be an imaginary quadratic field of discriminant d K , and let n be a nontrivial integral ideal of K in which N is the smallest positive integer. Let Q N ( d K ) be the set of primitive positive definite binary quadratic forms of discriminant d K whose leading coefficients are relatively prime to N. We adopt an equivalence relation ∼ n on Q N ( d K ) so that the set of equivalence classes Q N ( d K ) / ∼ n can be regarded as a group isomorphic to the ray class group of K modulo n . We further establish an explicit isomorphism of Q N ( d K ) / ∼ n onto Gal ( K n / K ) in terms of Fricke invariants, where K n 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.

中文翻译：

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组合律和复数乘法

摘要 令K 为判别式d K 的虚二次域，令n 为K 的非平凡积分理想，其中N 为最小正整数。令 QN ( d K ) 为首项系数与 N 互质的判别式 d K 的原始正定二元二次型集合。我们在 QN ( d K ) 上采用等价关系 ∼ n 使得等价类集合 QN ( d K ) / ∼ n 可以看作是 K 模 n 的射线类群的同构群。我们进一步根据 Fricke 不变量建立 QN ( d K ) / ∼ n 到 Gal ( K n / K ) 的显式同构，其中 K n 表示 K 模 n 的射线类场。这将是由高斯和狄利克雷起源和发展的二元二次型经典合成理论的某种扩展。