Explicit Construction of the Hilbert Class Fields of Imaginary Quadratic Fields by Integer Lattice Reduction

Erich Kaltofen and Noriko Yui
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Erich Kaltofen: Rensselaer Polytechnic Institute, Department of Computer Science
Noriko Yui: Queen’s University, Department of Mathematics

Chapter 8 in Number Theory, 1991, pp 149-202 from Springer

Abstract: Abstract Motivated by a constructive realization of generalized dihedral groups as Galois groups over Q and by Atkin’s primality test, we present an explicit construction of the Hilbert class fields (ring class fields) of imaginary quadratic fields (orders). This is done by first evaluating the singular moduli of level one for an imaginary quadratic order, and then constructing the “genuine” (i.e., level one) class equation. The equation thus obtained has integer coefficients of astronomical size, and this phenomenon leads us to the construction of the “reduced” class equations, i.e., the class equations of the singular moduli of higher levels. These, for certain levels, turn out to define the same Hilbert class field (ring class field) as the level one class equation, and to have coefficients of small size (e.g., seven digits). The construction of the “reduced” class equations was carried out on MACSYMA, using a refinement of the integer lattice reduction algorithm of Lenstra-Lenstra-Lavász, implemented on the Symbolics 3670 at Rensselaer Polytechnic Institute.

Keywords: Hilbert class fields; Ring class fields; Class equations; Singular moduli; Weber’s class invariants; Generalized dihedral groups; Atkin’s primality test; Integer lattice reduction algorithm (search for similar items in EconPapers)
Date: 1991
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DOI: 10.1007/978-1-4757-4158-2_8

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