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Stark's conjecture over complex cubic number fields

Authors: David S. Dummit, Brett A. Tangedal and Paul B. van Wamelen
Journal: Math. Comp. 73 (2004), 1525-1546
MSC (2000): Primary 11R42; Secondary 11Y40, 11R37, 11R16
Published electronically: August 26, 2003
MathSciNet review: 2047099
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Abstract: Systematic computation of Stark units over nontotally real base fields is carried out for the first time. Since the information provided by Stark's conjecture is significantly less in this situation than the information provided over totally real base fields, new techniques are required. Precomputing Stark units in relative quadratic extensions (where the conjecture is already known to hold) and coupling this information with the Fincke-Pohst algorithm applied to certain quadratic forms leads to a significant reduction in search time for finding Stark units in larger extensions (where the conjecture is still unproven). Stark's conjecture is verified in each case for these Stark units in larger extensions and explicit generating polynomials for abelian extensions over complex cubic base fields, including Hilbert class fields, are obtained from the minimal polynomials of these new Stark units.

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Additional Information

David S. Dummit
Affiliation: Department of Mathematics and Statistics, University of Vermont, Burlington, Vermont 05401-1455

Brett A. Tangedal
Affiliation: Department of Mathematics, College of Charleston, Charleston, South Carolina 29424-0001

Paul B. van Wamelen
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803-4918

Keywords: Algebraic number fields, Stark's conjecture
Received by editor(s): November 14, 2000
Received by editor(s) in revised form: January 3, 2003
Published electronically: August 26, 2003
Additional Notes: The first author was supported in part by NSF Grant DMS-9624057 and NSA Grant MDA-9040010024
Article copyright: © Copyright 2003 American Mathematical Society