Search

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. [ABSTRACT FROM AUTHOR]

Abstract Let L be an extended ring class field of an imaginary quadratic field K other than Q (− 1) and Q (− 3). We show that there is a form class group induced from a congruence subgroup which describes the Galois group of L over K in a concrete way. We also construct a primitive generator of L over K as a real algebraic integer which can be applied to certain quadratic Diophantine equations. [ABSTRACT FROM AUTHOR]

Let K be an imaginary quadratic field and O K be its ring of integers. Let h E be the Weber function on a certain elliptic curve E with complex multiplication by O K . We show that if N (>1) is an integer prime to 6, then the function h E alone generates the ray class field modulo N O K over K when evaluated at some N -torsion point of E , which would be a partial answer to the question mentioned in [10, p. 105] . [ABSTRACT FROM AUTHOR]

We give some explicit conditions under which the singular values of β-quotients generate ring class fields over imaginary quadratic fields. [ABSTRACT FROM AUTHOR]

Abstract: Let g be a principal modulus with rational Fourier coefficients for a discrete subgroup of lying in between and for a positive integer N. Let K be an imaginary quadratic field. We introduce a relatively simple proof, without using Shimuraʼs canonical model, of the fact that the singular value of g generates the ray class field modulo N or the ring class field of the order of conductor N over K. Further, we construct a primitive generator of the ray class field of arbitrary modulus over K from Hasseʼs two generators. [Copyright &y& Elsevier]