Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [Co] Henri Cohen, Advanced topics in computational number theory, Springer-Verlag, New York, 2000. MR 2000k:11144
  • [CR] Henri Cohen and Xavier-François Roblot, Computing the Hilbert class field of real quadratic fields, Math. Comp. 69 (2000), no. 231, 1229-1244. MR 2000j:11200
  • [Da] Samit Dasgupta, Stark's conjectures, Honors thesis, Harvard University, 1999.
  • [DST] David S. Dummit, Jonathan W. Sands, and Brett A. Tangedal, Computing Stark units for totally real cubic fields, Math. Comp. 66 (1997), no. 219, 1239-1267. MR 97i:11110
  • [DT] David S. Dummit and Brett A. Tangedal, Computing the lead term of an abelian ${L}$-function, Lecture Notes in Comput. Sci. 1423, Springer, Berlin, 1998, 400-411. MR 2001d:11110
  • [F] Eduardo Friedman, Hecke's integral formula, Séminaire de Théorie des Nombres Univ. Bordeaux I, Talence (1987-88), Exposé n$^\circ$ 5, 23 pp. MR 90i:11136
  • [GP] C. Batut, K. Belabas, D. Bernardi, H. Cohen, and M. Olivier, User's guide to PARI/GP version 2.0.20, 2000.
  • [H] Erich Hecke, Lectures on the theory of algebraic numbers, Springer-Verlag, New York, 1981. MR 83m:12001
  • [L] A. F. Lavrik, On functional equations of Dirichlet functions, English translation in Math. USSR-Izvestija 1 (1967), 421-432. MR 35:4174
  • [La] Edmund Landau, Über Ideale und Primideale in Idealklassen, Math. Zeit. 2 (1918), 52-154.
  • [Lo] Robert L. Long, Algebraic number theory, Marcel Dekker, Inc., New York, Basel, 1977. MR 57:9668
  • [PZ] Michael Pohst and Hans Zassenhaus, Algorithmic algebraic number theory, Cambridge University Press, Cambridge, 1989. MR 92b:11074
  • [R1] Xavier-François Roblot, Algorithmes de factorisation dans les extensions relatives et applications de la conjecture de Stark à la construction des corps de classes de rayon, Thèse, Université Bordeaux I, 1997.
  • [R2] -, Stark's conjectures and Hilbert's twelfth problem, Experiment. Math. 9 (2000), 251-260. MR 2001g:11174
  • [St1] Harold M. Stark, Values of ${L}$-functions at $s=1$. I. ${L}$-functions for quadratic forms, Advances in Math. 7 (1971), 301-343. MR 44:6620
  • [St2] -, ${L}$-functions at $s=1$. II. Artin ${L}$-functions with rational characters, Advances in Math. 17 (1975), 60-92. MR 52:3082
  • [St3] -, ${L}$-functions at $s=1$. III. Totally real fields and Hilbert's twelfth problem, Advances in Math. 22 (1976), 64-84. MR 55:10427
  • [St4] -, ${L}$-functions at $s=1$. IV. First derivatives at $s=0$, Advances in Math. 35 (1980), 197-235. MR 81f:10054
  • [St5] -, Class fields for real quadratic fields and ${L}$-series at $1$, Algebraic number fields (A. Fröhlich, ed.), Academic Press, London, 1977, 355-375. MR 56:11963
  • [St6] -, Hilbert's twelfth problem and ${L}$-series, Bull. Amer. Math. Soc. 83 (1977), 1072-1074. MR 56:314
  • [Ta] John Tate, Les conjectures de Stark sur les fonctions ${L}$ d'Artin en $s=0$, Birkhäuser, Boston, 1984. Lecture notes edited by Dominique Bernardi and Norbert Schappacher. MR 86e:11112
  • [Wa] Lawrence C. Washington, Introduction to cyclotomic fields, 2nd Ed., Springer-Verlag, New York, 1997. MR 97h:11130

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11R42, 11Y40, 11R37, 11R16

Retrieve articles in all journals with MSC (2000): 11R42, 11Y40, 11R37, 11R16

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

American Mathematical Society