Available in electronic format
Available in print format		 Mathematics of Computation
Journal of the American Mathematical Society
ISSN 1088-6842(e) ISSN 0025-5718(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

$ \mathbf{Li}^{\boldsymbol{(p)}}$-service? An algorithm for computing $ \boldsymbol{p}$-adic polylogarithms

Author(s): Amnon Besser; Rob de Jeu.
Journal: Math. Comp. 77 (2008), 1105-1134.
MSC (2000): Primary 11Y16, 11G55; Secondary 11S80
Posted: November 5, 2007
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: We describe an algorithm for computing Coleman's $ p$-adic polylogarithms up to a given precision.

References:

1.
A. A. Beilinson.
Higher regulators and values of $ {L}$-functions.
J. Sov. Math., 30:2036-2070, 1985. MR 760999 (86h:11103)

2.
A. Besser.
Syntomic regulators and $ p$-adic integration I: rigid syntomic regulators.
Israel Journal of Math., 120:291-334, 2000. MR 1809626 (2002c:14035)

3.
A. Besser.
Finite and $ p$-adic polylogarithms.
Compositio Math., 130(2):215-223, 2002. MR 1883819 (2002m:11058)

4.
A. Besser, P. Buckingham, R. de Jeu, and X.-F. Roblot.
On the $ p $-adic Beilinson conjecture for number fields.
To appear in the special volume of the Pure and Applied Mathematics Quarterly in honour of the eightieth birthday of Jean-Pierre Serre.

5.
A. Besser and R. de Jeu.
The syntomic regulator for the $ {K} $-theory of fields.
Annales Scientifiques de l'École Normale Supérieure, 36(6):867-924, 2003. MR 2032529 (2005f:11133)

6.
A. Borel.
Cohomologie de $ {\rm SL}\sb{n}$ et valeurs de fonctions zêta aux points entiers.
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 4(4):613-636, 1977.
Errata in vol. 7, p. 373 (1980). MR 0506168 (58:22016)

7.
R. Coleman.
Dilogarithms, regulators, and $ p$-adic $ L$-functions.
Invent. Math., 69:171-208, 1982. MR 674400 (84a:12021)

8.
R. Coleman and J. Teitelbaum.
Numerical solution of the $ p$-adic hypergeometric equation.
In $ p$-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), pages 53-62. Amer. Math. Soc., Providence, RI, 1994. MR 1279601 (95d:11079)

9.
R. de Jeu.
Zagier's conjecture and wedge complexes in algebraic $ {K}$-theory.
Compositio Mathematica, 96:197-247, 1995. MR 1326712 (96h:19005)

10.
J. Fresnel and M. van der Put.
Rigid analytic geometry and its applications, volume 218 of Progress in Mathematics.
Birkhäuser Boston Inc., Boston, MA, 2004. MR 2014891 (2004i:14023)

11.
C. Koç and T. Acar.
Montgomery multiplication in $ {\rm GF}(2\sp k)$.
Des. Codes Cryptogr., 14(1):57-69, 1998. MR 1608220 (99k:11189)

12.
P. Montgomery.
Modular multiplication without trial division.
Math. Comp., 44(170):519-521, 1985. MR 777282 (86e:11121)

13.
P. Schneider.
Introduction to the Beilinson Conjectures.
In Beilinson's Conjectures on Special Values of $ L$-Functions, pages 1-35. Academic Press, Boston, MA, 1988. MR 944989 (89g:11053)

14.
The Magma group, Sydney.
Magma.
Available from http://magma.maths.usyd.edu.au/.

15.
J. von zur Gathen and J. Gerhard.
Modern computer algebra.
Cambridge University Press, New York, 1999. MR 1689167 (2000j:68205)

16.
L. C. Washington.
Introduction to cyclotomic fields, volume 83 of Graduate Texts in Mathematics.
Springer-Verlag, New York, second edition, 1997. MR 1421575 (97h:11130)

17.
D. Zagier.
Polylogarithms, Dedekind Zeta Functions and the Algebraic $ {K}$-theory of Fields.
In Arithmetic algebraic geometry (Texel, 1989), pages 391-430. Birkhäuser Boston, Boston, MA, 1991. MR 1085270 (92f:11161)

Similar Articles:

Retrieve articles in Mathematics of Computation with MSC (2000): 11Y16, 11G55, 11S80

Retrieve articles in all Journals with MSC (2000): 11Y16, 11G55, 11S80

Additional Information:

Amnon Besser
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Be'er-Sheva 84105, Israel

Rob de Jeu
Affiliation: Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham DH1 3LE, United Kingdom
Address at time of publication: Faculteit Exacte Wetenschappen, Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands

DOI: 10.1090/S0025-5718-07-02027-3
PII: S 0025-5718(07)02027-3
Keywords: Computational number theory, Coleman integration, $p$-adic polylogarithm
Received by editor(s): June 19, 2006
Received by editor(s) in revised form: December 18, 2006
Posted: November 5, 2007
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google