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Computing Stark units for totally real
cubic fields

Authors: David S. Dummit, Jonathan W. Sands and Brett A. Tangedal
Journal: Math. Comp. 66 (1997), 1239-1267
MSC (1991): Primary 11R42; Secondary 11Y40, 11R37, 11R16
MathSciNet review: 1415801
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Abstract | References | Similar Articles | Additional Information

Abstract: A method for computing provably accurate values of partial zeta functions is used to numerically confirm the rank one abelian Stark Conjecture for some totally real cubic fields of discriminant less than 50000. The results of these computations are used to provide explicit Hilbert class fields and some ray class fields for the cubic extensions.

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  • 1. T. Arakawa, Generalized eta-functions and certain ray class invariants of real quadratic fields, Math. Ann. 260 (1982), 475-494. MR 84b:12016
  • 2. B. L. J. Braaksma, Asymptotic expansions and analytic continuations for a class of Barnes-integrals, Comp. Math. 15 (1963), 239-341. MR 29:4923
  • 3. D. S. Dummit and D. Hayes, Checking the refined $\mathfrak {p}$-adic Stark Conjecture when $\mathfrak {p}$ is Archimedean, Algorithmic Number Theory, Proceedings ANTS 2, Talence, France, Lecture Notes in Computer Science 1122 (Henri Cohen, ed.), Springer-Verlag, Berlin-Heidelberg-New York, 1996, pp. 91-97.
  • 4. V. Ennola and R. Turunen, On totally real cubic fields, Math. Comp 44 (170) (1985), 495-518. MR 86e:11100
  • 5. C. Fogel, personal communication.
  • 6. E. Friedman, Hecke's Integral Formula, Séminaire de Théorie des Nombres de Bordeaux No. 5 (1987-88) (1989). MR 90i:11136
  • 7. F. Hajir, Elliptic units of cyclic unramified extensions of complex quadratic fields, Acta Arithmetica 64 (1993), 69-85. MR 94h:11102
  • 8. F. Hajir, Unramified Elliptic Units, M.I.T. Thesis, 1993.
  • 9. F. Hajir (with F. Rodriguez-Villegas), Explicit Elliptic Units I, Duke Math. J. (to appear).
  • 10. D.R. Hayes, Brumer elements over a real quadratic base field, Expositiones Mathematicae 8 (1990), 137-184. MR 92a:11142
  • 11. D.R. Hayes, The partial zeta functions of a real quadratic field evaluated at $s=0$, Number Theory (Richard A. Mollin, ed.), Walter de Gruyter, Berlin, 1990, pp. 207-226. MR 92j:11134
  • 12. D.R. Hayes, Base Change for the Brumer-Stark Conjecture, preprint.
  • 13. J. Nakagawa, On the Stark-Shintani conjecture and cyclotomic $\mathbb Z_{p}$-extensions of class fields over real quadratic fields, J. Math. Soc. Japan 36 (4) (1984), 577-588. MR 87a:11108a
  • 14. K. Rubin, A Stark Conjecture ``over ${\mathbb {Z}}$'' for abelian $L$-functions with multiple zeros, Ann. Inst. Fourier, Grenoble 46 (1996), 33-62. CMP 96:11
  • 15. J. W. Sands, Abelian fields and the Brumer-Stark conjecture, Comp. Math. 53 (1984), 337-346. MR 86c:11102
  • 16. J. W. Sands, Galois groups of exponent two and the Brumer-Stark conjecture, J. Reine Angew. Math. 349 (1984), 129-135. MR 85i:11098
  • 17. J. W. Sands, Two cases of Stark's conjecture, Math. Ann. 272 (1985), 349-359. MR 87a:11117
  • 18. T. Shintani, On a Kronecker limit formula for real quadratic fields, J. Fac. Sci. Univ. Tokyo 24 (1977), 167-199. MR 57:277
  • 19. T. Shintani, On certain ray class invariants of real quadratic fields, J. Math. Soc. Japan 30 (1978), 139-167. MR 58:16599
  • 20. H. M. Stark, Class fields for real quadratic fields and $L-$series at 1, Algebraic Number Fields (A. Fröhlich, ed.), Academic Press, London, 1977, pp. 355-375. MR 56:11963
  • 21. H. M. Stark, Hilbert's twelfth problem and $L$-series, Bull. A.M.S. 83 (5) (1977), 1072-1074. MR 56:314
  • 22. H. 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
  • 23. H. M. Stark, Values of $L$-functions at $s=1$. II. Artin $L$-functions with rational characters, Advances in Math. 17 (1975), 60-92. MR 52:3082
  • 24. H. M. Stark, $L$-functions at $s=1$. III. Totally real fields and Hilbert's twelfth problem, Advances in Math. 22 (1976), 64-84. MR 55:10427
  • 25. H. M. Stark, $L$-functions at $s=1$ IV. First derivatives at $s=0$, Advances in Math. 35 (1980), 197-235. MR 81f:10054
  • 26. J. T. Tate, Les conjectures de Stark sur les fonctions $L$ d'Artin en $s=0$, Birkhäuser, Boston, 1984. MR 86e:11112
  • 27. F. Y. Wang, Conductors of fields arising from Stark's conjecture, Ph.D. Thesis, MIT, 1991.
  • 28. A. Wiles, On a conjecture of Brumer, Annals of Math. 131 (1990), 555-565. MR 91i:11164

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Additional Information

David S. Dummit
Affiliation: Department of Mathematics and Statistics, University of Vermont, Burlington, Vermont 05405

Jonathan W. Sands
Affiliation: Department of Mathematics and Statistics, University of Vermont, Burlington, Vermont 05405

Brett A. Tangedal
Affiliation: Department of Mathematics and Statistics, University of Vermont, Burlington, Vermont 05405

Received by editor(s): February 9, 1996
Received by editor(s) in revised form: May 15, 1996
Additional Notes: Research supported by grants from the NSA and the NSF
Article copyright: © Copyright 1997 American Mathematical Society

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