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

Jared Asuncion 1, 2, 3
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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Preprints, Working Papers, ...
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https://hal.inria.fr/hal-03210279
Contributor : Jared Asuncion Connect in order to contact the contributor
Submitted on : Tuesday, April 27, 2021 - 8:28:56 PM
Last modification on : Wednesday, February 2, 2022 - 3:54:33 PM

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  • HAL Id : hal-03210279, version 1
  • ARXIV : 2104.13639

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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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