Construction of class fields over imaginary biquadratic fields

Abstract

Let K be an imaginary biquadratic field and K-1, K-2 be its imaginary quadratic subfields. For integers N > 0, mu >= 0, and an odd prime p with gcd(N, p) = 1, let K-(Np mu) and (K-i)((Np mu)) for i = 1, 2 be the ray class fields of K and K-i, respectively, modulo Np-mu. We first present certain class fields <(K-N,p,mu(1,2))over tilde> of K, in the sense of Hilbert, which are generated by Siegel-Ramachandra invariants of (K-i)((Np mu+1)) for i = 1, 2 over K-(Np mu), and show that K(Np mu+1) = <(K-N(,p,mu)1,2)over tilde> for almost all mu.