Stark’s conjecture over complex cubic number fields
HTML articles powered by AMS MathViewer
- by David S. Dummit, Brett A. Tangedal and Paul B. van Wamelen PDF
- Math. Comp. 73 (2004), 1525-1546 Request permission
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
- Henri Cohen, Advanced topics in computational number theory, Graduate Texts in Mathematics, vol. 193, Springer-Verlag, New York, 2000. MR 1728313, DOI 10.1007/978-1-4419-8489-0
- Henri Cohen and Xavier-François Roblot, Computing the Hilbert class field of real quadratic fields, Math. Comp. 69 (2000), no. 231, 1229–1244. MR 1651747, DOI 10.1090/S0025-5718-99-01111-4
- Samit Dasgupta, Stark’s conjectures, Honors thesis, Harvard University, 1999.
- 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 1415801, DOI 10.1090/S0025-5718-97-00852-1
- David S. Dummit and Brett A. Tangedal, Computing the lead term of an abelian $L$-function, Algorithmic number theory (Portland, OR, 1998) Lecture Notes in Comput. Sci., vol. 1423, Springer, Berlin, 1998, pp. 400–411. MR 1726088, DOI 10.1007/BFb0054879
- Eduardo Friedman, Hecke’s integral formula, Séminaire de Théorie des Nombres, 1987–1988 (Talence, 1987–1988) Univ. Bordeaux I, Talence, [1988?], pp. Exp. No. 5, 23. MR 993106
- C. Batut, K. Belabas, D. Bernardi, H. Cohen, and M. Olivier, User’s guide to PARI/GP version 2.0.20, 2000.
- Erich Hecke, Lectures on the theory of algebraic numbers, Graduate Texts in Mathematics, vol. 77, Springer-Verlag, New York-Berlin, 1981. Translated from the German by George U. Brauer, Jay R. Goldman and R. Kotzen. MR 638719
- A. F. Lavrik, Functional equations of the Dirichlet functions, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 431–442 (Russian). MR 0213310
- Edmund Landau, Über Ideale und Primideale in Idealklassen, Math. Zeit. 2 (1918), 52–154.
- Robert L. Long, Algebraic number theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 41, Marcel Dekker, Inc., New York-Basel, 1977. MR 0469888
- M. Pohst and H. Zassenhaus, Algorithmic algebraic number theory, Encyclopedia of Mathematics and its Applications, vol. 30, Cambridge University Press, Cambridge, 1989. MR 1033013, DOI 10.1017/CBO9780511661952
- 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.
- Xavier-François Roblot, Stark’s conjectures and Hilbert’s twelfth problem, Experiment. Math. 9 (2000), no. 2, 251–260. MR 1780210
- H. M. Stark, Values of $L$-functions at $s=1$. I. $L$-functions for quadratic forms, Advances in Math. 7 (1971), 301–343 (1971). MR 289429, DOI 10.1016/S0001-8708(71)80009-9
- H. M. Stark, $L$-functions at $s=1$. II. Artin $L$-functions with rational characters, Advances in Math. 17 (1975), no. 1, 60–92. MR 382194, DOI 10.1016/0001-8708(75)90087-0
- H. M. Stark, $L$-functions at $s=1$. III. Totally real fields and Hilbert’s twelfth problem, Advances in Math. 22 (1976), no. 1, 64–84. MR 437501, DOI 10.1016/0001-8708(76)90138-9
- Harold M. Stark, $L$-functions at $s=1$. IV. First derivatives at $s=0$, Adv. in Math. 35 (1980), no. 3, 197–235. MR 563924, DOI 10.1016/0001-8708(80)90049-3
- H. M. Stark, Class fields for real quadratic fields and $L$-series at $1$, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 355–375. MR 0453703
- H. M. Stark, Hilbert’s twelfth problem and $L$-series, Bull. Amer. Math. Soc. 83 (1977), no. 5, 1072–1074. MR 441923, DOI 10.1090/S0002-9904-1977-14389-9
- John Tate, Les conjectures de Stark sur les fonctions $L$ d’Artin en $s=0$, Progress in Mathematics, vol. 47, Birkhäuser Boston, Inc., Boston, MA, 1984 (French). Lecture notes edited by Dominique Bernardi and Norbert Schappacher. MR 782485
- Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7
Additional Information
- David S. Dummit
- Affiliation: Department of Mathematics and Statistics, University of Vermont, Burlington, Vermont 05401-1455
- Email: dummit@math.uvm.edu
- Brett A. Tangedal
- Affiliation: Department of Mathematics, College of Charleston, Charleston, South Carolina 29424-0001
- MR Author ID: 612497
- Email: tangedalb@cofc.edu
- Paul B. van Wamelen
- Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803-4918
- Email: wamelen@math.lsu.edu
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Math. Comp. 73 (2004), 1525-1546
- MSC (2000): Primary 11R42; Secondary 11Y40, 11R37, 11R16
- DOI: https://doi.org/10.1090/S0025-5718-03-01586-2
- MathSciNet review: 2047099