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 $N>0$, $\mu \ge 0$ and an odd prime $p$ with $\gcd (N,p)=1$, let $K_{(N{p}^{\mu })}$ and $(K_i{)}_{(N{p}^{\mu })}$ for $i=1,2$ be the ray class fields of $K$ and $K_i$, respectively, modulo $N{p}^{\mu }$. We first present certain class fields $\widetilde{K_{N,p,\mu }^{1,2}}$ of $K$, in the sense of Hilbert, which are generated by Siegel-Ramachandra invariants of $(K_i{)}_{(N{p}^{\mu +1})}$ for $i=1,2$ over $K_{(N{p}^{\mu })}$ and show that $K_{(N{p}^{\mu +1})}=\widetilde{K_{N,p,\mu }^{1,2}}$ for almost all $\mu$.

21 pages, Due to its gigantic length we had some difficulties, so it has been broken into two papers