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

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- by Amnon Besser and Rob de Jeu;
- Math. Comp.
**77**(2008), 1105-1134 - DOI: https://doi.org/10.1090/S0025-5718-07-02027-3
- Published electronically: November 5, 2007
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## Abstract:

We describe an algorithm for computing Coleman’s $p$-adic polylogarithms up to a given precision.## References

- A. A. Beĭlinson,
*Higher regulators and values of $L$-functions*, Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 181–238 (Russian). MR**760999** - Amnon Besser,
*Syntomic regulators and $p$-adic integration. I. Rigid syntomic regulators*. part B, Proceedings of the Conference on $p$-adic Aspects of the Theory of Automorphic Representations (Jerusalem, 1998), 2000, pp. 291–334. MR**1809626**, DOI 10.1007/BF02834843 - Amnon Besser,
*Finite and $p$-adic polylogarithms*, Compositio Math.**130**(2002), no. 2, 215–223. MR**1883819**, DOI 10.1023/A:1013727116183 - 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.
- Amnon Besser and Rob de Jeu,
*The syntomic regulator for the $K$-theory of fields*, Ann. Sci. École Norm. Sup. (4)**36**(2003), no. 6, 867–924 (2004) (English, with English and French summaries). MR**2032529**, DOI 10.1016/j.ansens.2003.01.003 - Armand Borel,
*Cohomologie de $\textrm {SL}_{n}$ et valeurs de fonctions zeta aux points entiers*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**4**(1977), no. 4, 613–636 (French). MR**506168** - Robert F. Coleman,
*Dilogarithms, regulators and $p$-adic $L$-functions*, Invent. Math.**69**(1982), no. 2, 171–208. MR**674400**, DOI 10.1007/BF01399500 - Robert Coleman and Jeremy Teitelbaum,
*Numerical solution of the $p$-adic hypergeometric equation*, $p$-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991) Contemp. Math., vol. 165, Amer. Math. Soc., Providence, RI, 1994, pp. 53–62. MR**1279601**, DOI 10.1090/conm/165/01603 - Rob De Jeu,
*Zagier’s conjecture and wedge complexes in algebraic $K$-theory*, Compositio Math.**96**(1995), no. 2, 197–247. MR**1326712** - Jean Fresnel and Marius van der Put,
*Rigid analytic geometry and its applications*, Progress in Mathematics, vol. 218, Birkhäuser Boston, Inc., Boston, MA, 2004. MR**2014891**, DOI 10.1007/978-1-4612-0041-3 - Çetin K. Koç and Tolga Acar,
*Montgomery multiplication in $\textrm {GF}(2^k)$*, Des. Codes Cryptogr.**14**(1998), no. 1, 57–69. MR**1608220**, DOI 10.1023/A:1008208521515 - Peter L. Montgomery,
*Modular multiplication without trial division*, Math. Comp.**44**(1985), no. 170, 519–521. MR**777282**, DOI 10.1090/S0025-5718-1985-0777282-X - Peter Schneider,
*Introduction to the Beĭlinson conjectures*, Beĭlinson’s conjectures on special values of $L$-functions, Perspect. Math., vol. 4, Academic Press, Boston, MA, 1988, pp. 1–35. MR**944989** - The Magma group, Sydney.
*Magma*. Available from http://magma.maths.usyd.edu.au/. - Joachim von zur Gathen and Jürgen Gerhard,
*Modern computer algebra*, Cambridge University Press, New York, 1999. MR**1689167** - 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 - Don Zagier,
*Polylogarithms, Dedekind zeta functions and the algebraic $K$-theory of fields*, Arithmetic algebraic geometry (Texel, 1989) Progr. Math., vol. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 391–430. MR**1085270**, DOI 10.1007/978-1-4612-0457-2_{1}9

## Bibliographic Information

**Amnon Besser**- Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Be’er-Sheva 84105, Israel
- MR Author ID: 364540
**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
- Received by editor(s): June 19, 2006
- Received by editor(s) in revised form: December 18, 2006
- Published electronically: November 5, 2007
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**77**(2008), 1105-1134 - MSC (2000): Primary 11Y16, 11G55; Secondary 11S80
- DOI: https://doi.org/10.1090/S0025-5718-07-02027-3
- MathSciNet review: 2373194