Computing the Hilbert Class Fields of Quartic CM Fields Using Complex Multiplication

TL;DR

This paper presents an explicit method and an algorithm to compute the Hilbert class fields of quartic CM fields using complex multiplication, improving computational efficiency over existing methods.

Contribution

It makes explicit the integer m in Shimura's family of abelian extensions and provides an algorithm to compute defining polynomials for Hilbert class fields more efficiently.

Findings

Algorithm computes defining polynomials faster than generic Kummer methods.

Explicit m allows direct construction of Hilbert class fields.

Proof-of-concept implementation demonstrates improved performance.

Abstract

Let $K$ be a quartic CM field, that is, a totally imaginary quadratic extension of a real quadratic number field. In a 1962 article titled On the classfields obtained by complex multiplication of abelian varieties, Shimura considered a particular family $\{F_K(m) : m \in \mathbb{Z} >0 \}$ of abelian extensions of $K$, and showed that the Hilbert class field $H_K$ of $K$ is contained in $F_K(m)$ for some positive integer m. We make this m explicit. We then give an algorithm that computes a set of defining polynomials for the Hilbert class field using the field $F_K(m)$. Our proof-of-concept implementation of this algorithm computes a set of defining polynomials much faster than current implementations of the generic Kummer algorithm for certain examples of quartic CM fields.