Construction of class fields over imaginary biquadratic fields

TL;DR

This paper constructs specific class fields over imaginary biquadratic fields using Siegel-Ramachandra invariants and demonstrates their equivalence to ray class fields for most cases.

Contribution

It introduces a new method to explicitly generate class fields over imaginary biquadratic fields via Siegel-Ramachandra invariants, linking them to ray class fields.

Findings

Class fields are generated by Siegel-Ramachandra invariants.

The constructed fields coincide with ray class fields for almost all parameters.

Provides explicit descriptions 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 $N>0$, $\mu\geq 0$ 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 $\widetilde{K_{N,p,\mu}^{1,2}}$ 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 $K_{(Np^{\mu+1})}=\widetilde{K_{N,p,\mu}^{1,2}}$ for almost all $\mu$.