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Stability and New Non-Abelian Zeta Functions

  • Lin Weng
Part of the Developments in Mathematics book series (DEVM, volume 8)

Abstract

In this paper, we first use classification of uni-modular lattices as a motivation to introduce semi-stable lattices. Then, as an integration over moduli spaces of semi-stable lattices, using a new arithmetic cohomology, we define a new type of non-abelian zeta functions for num­ber fields. This is a natural generalization of what Iwasawa and Tate did for Dedekind functions. Basic facts for these zeta functions such as functional equations, singularities and residues at simple poles are discussed. Finally we introduce and study new non-abelian zeta func­tions for curves over finite fields, as a natural generalization of Artin’s (abelian) zeta functions, using moduli spaces of semi-stable bundles. Some interesting examples are also given here.

Keywords

arithmetic cohomology adelic moduli spaces reciprocity law stability zeta functions 

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References

  1. [1]
    A.N. Andrianov, Euler products that correspond to Siegel’s modular forms of genus 2, Russian Math. Surveys 29: 3 (1974), 45–116.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    E. Artin, Quadratische Körper im Gebiete der höheren Kongruenzen, I,II, Math. Zeit, 19 153–246 (1924)MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    J.H. Conway N.J.A. Sloane, Sphere packings, lattices and groups, Springer-Verlag, 1993.Google Scholar
  4. [4]
    Ch. Deninger, Motivic L-functions and regularized determinants, Proc. Sympos. Pure Math, 55 (1), AMS, (1994), 707–743.Google Scholar
  5. [5]
    U.V. Desale S. Ramanan, Poincaré polynomials of the variety of stable bundles, Math. Ann 216, 233–244 (1975).MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    G. van der Geer R. Schoof, Effectivity of Arakelov Divisors and the Theta Divisor of a Number Field, math.AG/9802121Google Scholar
  7. [7]
    D.R. Grayson, Reduction theory using semistability. Comment. Math. Helv. 59 (1984), no. 4, 600–634.MathSciNetCrossRefGoogle Scholar
  8. [8]
    D.R. Grayson, Reduction theory using semistability. II, Comment. Math. Helv. 61 (1986), no. 4, 661–676.MathSciNetCrossRefGoogle Scholar
  9. [9]
    G. Harder M.S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles over curves, Math Ann. 212 (1975), 215–248.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    H. Hasse, Mathematische Abhandlungen, Walter de Gruyter, 1975.Google Scholar
  11. [11]
    K. Iwasawa, Letter to Dieudonné, April 8, 1952, in Advanced Studies in Pure Math. 21 (1992), 445–450.MathSciNetGoogle Scholar
  12. [12]
    W. Kohnen, this proceeding.Google Scholar
  13. [13]
    S. Lang, Algebraic Number Theory, Springer-Verlag, 1986.Google Scholar
  14. [14]
    S. Lang, Fundamentals on Diophantine Geometry, Springer-Verlag, 1983.Google Scholar
  15. [15]
    A. Moriwaki, Stable sheaves on arithmetic curves, personal note dated in 1992.Google Scholar
  16. [16]
    D. Mumford, Geometric Invariant Theory, Springer-Verlag, 1965.Google Scholar
  17. [17]
    J. Neukirch, Algebraic Number Theory, Springer-Verlag, 1999.Google Scholar
  18. [18]
    C. S. Seshadri, Fibrés vectoriels sur les courbes algébriques, Asterisque 96, 1982.Google Scholar
  19. [19]
    U. Stuhler, Eine Bemerkung zur Reduktionstheorie quadratischer Formen, Arch. Math. (Basel) 27 (1976), no. 6, 604–610.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    U. Stuhler, Zur Reduktionstheorie der positiven quadratischen Formen. II, Arch. Math. (Basel) 28 (1977), no. 6, 611–619.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    J. Tate, Fourier analysis in number fields and Hecke’s zeta functions, Thesis, Princeton University, 1950Google Scholar
  22. [22]
    A. Weil, Sur les courbes algébriques et les variétés qui s’en déduisent, Herman, 1948Google Scholar
  23. [23]
    L. Weng, Riemann-Roch theorem, stability and new zeta functions for number fields, math. AG/0007146Google Scholar
  24. [24]
    L. Weng, Constructions of new non-abelian zeta functions for curves, math. AG/0102064Google Scholar
  25. [25]
    L. Weng, Refined Brill-Noether locus and non-abelian zeta functions for elliptic curves, math.AG/0101183Google Scholar
  26. [26]
    L. Weng, A Program for Geometric Arithmetic, math. AG/0111241Google Scholar
  27. [27]
    L. Weng, A note on arithmetic cohomologies for number fields, math.AG/0112164Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Lin Weng
    • 1
  1. 1.Graduate School of MathematicsKyushu UniversityFukuokaJapan

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