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Pure L-Functions from Algebraic Geometry over Finite Fields

  • Conference paper
Finite Fields and Applications

Abstract

This survey gives a concrete and intuitively self-contained introduction to the theory of pure L-functions arising from a family of algebraic varieties defined over a finite field of characteristic p. The standard fundamental questions in any theory of L-functions include the meromorphic continuation, functional equation, Riemann hypothesis (RH for short), order of zeros at special points and their special values. Our emphasis here will be on the meromorphic continuation and the RH. These two questions can be described in a general setup without introducing highly technical terms.

The construction of the pure L-function depends on the choice of an absolute value of the rational number field Q. In the case that the absolute value is the complex or ℓ-adic absolute value (ℓ ≠ p), the rationality of the pure L-function requires the full strength of the ℓ-adic cohomology including Deligne’s main theorem on the Weil conjectures. In the case that the absolute value is the p-adic absolute value, the pure L-function is no longer rational but conjectured by Dwork to be p-adic meromorphic. This conjecture goes beyond all existing p-adic cohomology theories. Its truth opens up several new directions including a possible p-adic RH for such pure L-functions.

The guiding principle of our exposition in this paper is to describe all theorems and problems as simple as possible, directly in terms of zeta functions and L-functions without using cohomological terms. In the case that this is not easy to do so, we simply give an intuitive discussion and try to convey a little feeling. Along the way, a number of natural open questions and conjectures are raised, some of them may be accessible to certain extent but others may be somewhat wild due to the lack of sufficient evidences.

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References

  1. A. Adolphson and S. Sperber, Exponential sums and Newton polyhedra, Ann. Math., 130(1989), 367–406.

    Article  MATH  MathSciNet  Google Scholar 

  2. P. Berthelot, Géométrie rigide et cohomologie des variétés algébriques de caractéristique p, Mém. Soc. Math. France (N.S.) No. 23(1986), 3, 7–32.

    MathSciNet  Google Scholar 

  3. E. Bombieri, On exponential sums in finite fields, II, Invent. Math., 47(1978), 29–39.

    MATH  MathSciNet  Google Scholar 

  4. R. Coleman, p-adic Banach spaces and families of modular forms. Invent. Math., 127(1997), 417–479.

    MATH  Google Scholar 

  5. R. Coleman and B. Mazur, The eigencurve, in Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996), London Math. Soc. Lecture Note Ser., 254, 1998, 1–113.

    Article  MathSciNet  Google Scholar 

  6. P. Deligne, La Conjecture de Weil, I, Publ. Math., IHES, 43(1974), 273–307.

    MathSciNet  Google Scholar 

  7. P. Deligne, La Conjecture de Weil, II, Publ. Math., IHES, 52(1980), 137–252.

    MATH  MathSciNet  Google Scholar 

  8. P. Deligne, Fonctions L modulo ℓn et modulo p, in: Cohomologie étale (SGA 41/2), 110–128, Lecture Notes in Math. 569, Springer-Verlag, 1977.

    Google Scholar 

  9. P. Deligne, Formes modulaires et representations ℓ-adiques, Sém Bourbaki, 355, Lecture Notes in Math., Vol. 179(1971), 136–172.

    Google Scholar 

  10. A.J. de Jong and F. Oort, Purity of the stratification by Newton polygons, J. Amer. Math. Soc., 13(2000), 209–241.

    Article  MATH  MathSciNet  Google Scholar 

  11. B. Dwork, On the rationality of the zeta function of an algebraic variety, Amer. J. Math., 82(1960), 631–648.

    Article  MATH  MathSciNet  Google Scholar 

  12. B. Dwork, p-adic cycles, Puhl. Math., IHES, 37(1969), 27–115.

    MATH  MathSciNet  Google Scholar 

  13. B. Dwork, On Hecke polynomials, Invent. Math., 12(1971), 249–256.

    Article  MATH  MathSciNet  Google Scholar 

  14. B. Dwork, Normalized period matrices, Ann. Math., 94(1971), 337–388.

    Article  MATH  MathSciNet  Google Scholar 

  15. B. Dwork, Normalized period matrices II, Ann. Math., 98(1973), 1–57.

    Article  MATH  MathSciNet  Google Scholar 

  16. B. Dwork and S. Sperber, Logarithmic decay and overconvergence of the unit root and associated zeta functions, Ann. Sci. Ecole Norm. Sup. 24(1991), 575–604.

    MATH  MathSciNet  Google Scholar 

  17. G. Faltings, F-isocrystals on open varieties: results and conjectures. The Grothendieck Festschrift, Vol II (1990), 219–248.

    MathSciNet  Google Scholar 

  18. F. Gouvêa and B. Mazur, Families of modular eigenforms, Math. Comp., 58(1992), 793–806.

    Article  MATH  MathSciNet  Google Scholar 

  19. [Gr]A. Grothendieck, Formule de Lefschetz et rationalité des fonctions L, Séminaire Bourbaki, exposé 279, 1964/65.

    Google Scholar 

  20. L. Illusie, Ordinarité des intersections complètes générales, in Grothendieck Festschrift, Volume II (1990), 375–405.

    Google Scholar 

  21. N. Katz, On the differential equations satisfied by period matrices, Publ. Math., IHES., 35(1968), 223–258.

    Google Scholar 

  22. N. Katz, Une formule de congruence pour la fonction ζ, in: Groupes de monodromie en géométrie algébrique (SGA 7 II), Exp. XXII, 401–438, Lecture Notes in Math. 340, Springer-Verlag, 1973

    Article  Google Scholar 

  23. N. Katz, Travaux de Dwork, Séminaire Bourbaki, exposé 409(1971/72), Lecture Notes in Math., No. 317, 1973, 167–200.

    Google Scholar 

  24. N. Katz, Slope filtration of F-crystals, Astérisque, 63(1979), 113–164.

    MATH  Google Scholar 

  25. N. Katz, Gauss sums, Kloosterman sums, and monodromy groups. Annals of Mathematics Studies, 116, 1988, Princeton University Press.

    Google Scholar 

  26. B. Lian and S.T. Yau, Integrality of certain power series, Proceedings of the International Conference on Algebra and Geometry, ed, M.-C. Kang, International Press, 1998, 213–225.

    Google Scholar 

  27. S. Lubkin, A p-adic proof of Weil’s conjectures, Ann. Math., 87(1968), 105–225.

    Article  MATH  MathSciNet  Google Scholar 

  28. B. Mazur, Frobenius and the Hodge filtration, Bull. Amer. Math. Soc., 78(1972), 653–667.

    Article  MATH  MathSciNet  Google Scholar 

  29. B. Mazur, Frobenius and the Hodge filtration (estimates), Ann. of Math., 98(1973), 58–95.

    Article  MATH  MathSciNet  Google Scholar 

  30. P. Monsky, Formal cohomology III, Ann. Math., 93(1971), 315–343.

    Article  MATH  MathSciNet  Google Scholar 

  31. P. Monsky and G. Washnitzer, Formal cohomology I, Ann. Math., 88(1968), 181–217.

    Article  MATH  MathSciNet  Google Scholar 

  32. F. Oort, Some questions in algebraic geometry, preprint.

    Google Scholar 

  33. J.-P. Serre, Endomorphismes complétements continues des espaces de Banach p-adiques, Publ. Math., IHES., 12(1962), 69–85.

    MATH  Google Scholar 

  34. J. Tate, Algebraic cycles and poles of zeta functions, 1965 Arithmetic Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), 93–110.

    Google Scholar 

  35. D. Wan, Zeta functions of algebraic cycles, Manuscripta Math., 74(1992), 413–444.

    MATH  Google Scholar 

  36. D. Wan, Newton polygons of zeta functions and L-functions, Ann. Math., 137(1993), 247–293.

    Article  Google Scholar 

  37. D. Wan, Meromorphic continuation of L-functions of p-adic representations, Ann. Math., 143(1996), 469–498.

    Article  MATH  Google Scholar 

  38. D. Wan, Dimension variation of classical and p-adic modular forms. Invent. Math., 133(1998), 449–463.

    Article  MATH  MathSciNet  Google Scholar 

  39. D. Wan, Computing zeta functions over finite fields, Contemporary Mathematics, 225(1999), 131–141.

    Google Scholar 

  40. D. Wan, A quick introduction to Dwork’s conjecture. Contemporary Mathematics, 245(1999), 147–163.

    Google Scholar 

  41. D. Wan, An introduction to the theory of Newton polygons for L-functions of exponential sums, to appear.

    Google Scholar 

  42. D. Wan, Dwork’s conjecture on unit root zeta functions, Ann. Math., 150(1999), 867–927.

    Article  MATH  Google Scholar 

  43. D. Wan, Rank one case of Dwork’s conjecture, J. Amer. Math. Soc., to appear.

    Google Scholar 

  44. D. Wan, Higher rank case of Dwork’s conjecture, J. Amer. Math. Soc., to appear.

    Google Scholar 

  45. D. Wan, Partial zeta functions of algebraic varieties over finite fields, Finite Fields & Applications, to appear.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Irvine, CA, 92697-3875

    Daqing Wan

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  1. Daqing Wan
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Editors and Affiliations

  1. Lehrstuhl für Diskrete Mathematik, Optimierung und Operations Research, Universität Augsburg, 86135, Augsburg, Germany

    Dieter Jungnickel

  2. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore, 117543, Republic of Singapore

    Harald Niederreiter

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© 2001 Springer-Verlag Berlin Heidelberg

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Wan, D. (2001). Pure L-Functions from Algebraic Geometry over Finite Fields. In: Jungnickel, D., Niederreiter, H. (eds) Finite Fields and Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56755-1_34

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  • DOI: https://doi.org/10.1007/978-3-642-56755-1_34

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-62498-8

  • Online ISBN: 978-3-642-56755-1

  • eBook Packages: Springer Book Archive

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