Proceedings of the Japan Academy, Series A, Mathematical Sciences

Absolute zeta functions

Anton Deitmar, Shin-ya Koyama, and Nobushige Kurokawa

Full-text: Open access

Abstract

Two new concepts of zeta functions for schemes over the field of one element are proposed. A localization formula and an explicit formula in the affine case are given. This allows for a computation for every scheme.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 84, Number 8 (2008), 138-142.

Dates
First available in Project Euclid: 6 October 2008

Permanent link to this document
https://projecteuclid.org/euclid.pja/1223299521

Digital Object Identifier
doi:10.3792/pjaa.84.138

Mathematical Reviews number (MathSciNet)
MR2457801

Zentralblatt MATH identifier
1225.11113

Subjects
Primary: 11G25: Varieties over finite and local fields [See also 14G15, 14G20]

Keywords
Zeta function field of one element

Citation

Deitmar, Anton; Koyama, Shin-ya; Kurokawa, Nobushige. Absolute zeta functions. Proc. Japan Acad. Ser. A Math. Sci. 84 (2008), no. 8, 138--142. doi:10.3792/pjaa.84.138. https://projecteuclid.org/euclid.pja/1223299521


Export citation

References

  • A. Deitmar, Schemes over $\mathbf{F}_{1}$, in Number fields and function fields–-two parallel worlds, 87–100, Progr. Math., 239, Birkhäuser, Boston, Boston, MA, 2005.
  • A. Deitmar, Remarks on zeta functions and $K$-theory over $\mathbf{F}_{1}$, Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 8, 141–146.
  • F. Denef, Report on Igusa's local zeta function (Séminaire Bourbaki Exp. No. 741), Astérisque, 201–203 (1991), 359–386.
  • J. Denef, The rationality of the Poincaré series associated to the $p$-adic points on a variety, Invent. Math. 77 (1984), no. 1, 1–23.
  • J. Igusa, An introduction to the theory of local zeta functions, Amer. Math. Soc., Providence, RI, 2000.
  • N. Kurokawa, H. Ochiai and M. Wakayama, Absolute derivations and zeta functions, Doc. Math. 2003 (2003), Extra Vol., 565–584. (Electronic).
  • N. Kurokawa, Analyticity of Dirichlet series over prime powers, in Analytic number theory (Tokyo, 1988), 168–177, Lecture Notes in Math., 1434, Springer, Berlin.
  • N. Kurokawa, Zeta functions over $\mathbf{F}_{1}$, Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 10, 180–184.
  • Y. Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa), Astérisque 228 (1995), 121–163.
  • C. Soulé, Les variétés sur le corps à un élément, Mosc. Math. J. 4 (2004), no. 1, 217–244, 312.