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Preprints, Working Papers, ... Year : 2021

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

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 ∈ 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.
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Dates and versions

hal-03210279 , version 1 (27-04-2021)

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Jared Asuncion. Computing the Hilbert Class Fields of Quartic CM Fields Using Complex Multiplication. 2021. ⟨hal-03210279⟩
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