Some Aspects of the Arithmetic Theory of Polynomials

  • Jun-ichi Igusa
Part of the Progress in Mathematics book series (PM, volume 67)


This is an expository paper based on our memos of two lectures, one at the A.M.S. annual meeting at Cincinnati in January of 1982 with Professor Mostow presiding and another at the conference at Yale in honor of his 60th birthday. We have emphasized the universality or the uniformity of results and problems for all local fields; and consequently we have included certain material which is usually considered as analysis rather than arithmetic. On the other hand, as the title suggests, we have covered only certain parts of the arithmetic theory of polynomials. For instance the arithmetic theory of polynomials over a finite field is almost entirely left out. Also we have not given enough explanation to the recent results of Barlet [3] and Heath-Brown [11], which were mentioned at the time of the conference respectively by Professors Deligne and Tamagawa. Nevertheless it is our hope that this paper will give a fair and useful survey of certain parts of the arithmetic theory of polynomials appropriate for the Festschrift.


Number Field Theta Series Meromorphic Continuation Gauge Form Arithmetic Theory 
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  1. [1]
    H. Ariturk, The Siegel-Weil formula for orthogonal groups, Thesis, Johns Hopkins, 1975.Google Scholar
  2. [2]
    M.F. Atiyah, Resolution of singularities and division of distributions, Comm. pure and appl. math. 23 (1970), 145–150.CrossRefGoogle Scholar
  3. [3]
    D. Barlet, Contribution effective de la monodromie aux développements asymptotiques, preprint.Google Scholar
  4. [4]
    I.N. Bernshtein and S.I. Gel’fand, Meromorphic property of the functions Pλ, Functional Analysis and its Applications 3 (1969), 68–69.CrossRefGoogle Scholar
  5. [5]
    I.N. Bernshtein, The analytic continuation of generalized functions with respect to a parameter, Functional Analysis and its Applications 6 (1972), 273–285.CrossRefGoogle Scholar
  6. [6]
    A. Borel, Some finiteness properties of adele groups over number fields, Pub. math. I.H.E.S. 16 (1963), 5–30.Google Scholar
  7. [7]
    B.A. Datskovsky, On zeta functions associated with the space of binary cubic forms with coefficients in a function field, Thesis, Harvard, 1984.Google Scholar
  8. [8]
    P. Deligne, La conjecture de Weil. I, Pub. math. I.H.E.S. 43 (1974), 273–307.Google Scholar
  9. [9]
    J. Denef, The rationality of the Poincaré series associated to the p-adic points on a variety, preprint.Google Scholar
  10. [10]
    S.J. Haris, An equality of distributions associated to families of theta series, Nagoya Math. J. 59 (1975), 153–168.Google Scholar
  11. [11]
    D.R. Heath-Brown, Cubic forms in ten variables, London Math. Soc. 47 (1983), 225–257.Google Scholar
  12. [12]
    H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I-II, Ann. Math. 79 (1964), 109–326.CrossRefGoogle Scholar
  13. [13]
    J. Igusa, On certain representations of semi-simple algebraic groups and the arithmetic of the corresponding invariants. I, Invent. math. 12 (1971), 62–94.CrossRefGoogle Scholar
  14. [14]
    J. Igusa, Geometry of absolutely admissible representations, Number Theory, Algebraic Geometry and Commutative Algebra, Kinokuniya, Tokyo (1973), 373–452.Google Scholar
  15. [15]
    J. Igusa, Complex powers and asymptotic expansions, I. Crelles J. Math. 268/269 (1974), 110–130; II. ibid. 278-279 (1975), 307-321; or Forms of Higher Degree, Tata Inst. Lect. Notes 59, Springer-Verlag (1978).Google Scholar
  16. [16]
    J. Igusa, On the first terms of certain asymptotic expansions, Complex Analysis and Algebraic Geometry, Iwanami Shoten, Tokyo (1977), 357–368.CrossRefGoogle Scholar
  17. [17]
    J. Igusa, Exponential sums associated with a Freudenthal quartic, J. Fac. Sci. Univ. Tokyo 24 (1977), 231–246.Google Scholar
  18. [18]
    J. Igusa, On Lie algebras generated by two differential operators, Manifolds and Lie groups, Progress in Math. 14, Birkhäuser (1981), 187-195.Google Scholar
  19. [19]
    J. Igusa, Some results on p-adic complex powers, to appear in the American Journal of Mathematics.Google Scholar
  20. [20]
    P. Jeanquartier, Développement asymptotique de la distribution de Dirac attachée à une fonction analytique, C.R. 271 (1970), 1159–1161.Google Scholar
  21. [21]
    V.G. Kac, Infinite dimensional Lie algebras: an introduction, Progress in Math. 44, Birkhäuser (1983).Google Scholar
  22. [22]
    T. Kimura, The b-functions and holonomy diagram of irreducible regular prehomogeneous vector spaces, Nagoya Math. J. 85 (1982), 1–80.Google Scholar
  23. [23]
    R.P. Langlands, Orbital integrals on forms of SL(3). I, Amer. J. Math. 105 (1983), 465–506.CrossRefGoogle Scholar
  24. [24]
    Lê Dũng Trang, Sur les noeuds algébriques, Comp. Math. 25 (1972), 281–321.Google Scholar
  25. [25]
    B. Lichtin, Some algebro-geometric formulae for poles of f(x,y) s, to appear in the American Journal of Mathematics.Google Scholar
  26. [26]
    A. Macintyre, On definable subsets of p-adic fields, J. Symb. Logic, 41 (1976), 605–610.CrossRefGoogle Scholar
  27. [27]
    B. Malgrange, Intégrales asymptotiques et monodromie, Ann. Éc. Norm. Sup. 7 (1974), 405–430.Google Scholar
  28. [28]
    J.G.M. Mars, Les nombres de Tamagawa de certains groupes exceptionnels, Bull. Soc. Math. France 94 (1966), 97–140.Google Scholar
  29. [29]
    J.G.M. Mars, The Tamagawa number of 2An, Ann. Math. 89 (1969), 557–574.CrossRefGoogle Scholar
  30. [30]
    D. Meuser, On the rationality of certain generating functions, Math. Ann. 256 (1981), 303–310.CrossRefGoogle Scholar
  31. [31]
    D. Meuser, On the poles of a local zeta function for curves, Invent. math. 73 (1983), 445–465.CrossRefGoogle Scholar
  32. [32]
    G.D. Mostow, Self-adjoint groups, Ann. Math. 62 (1955), 44–55.CrossRefGoogle Scholar
  33. [33]
    T. Ono, On the relative theory of Tamagawa numbers, Ann. Math. 82 (1965), 88–111.CrossRefGoogle Scholar
  34. [34]
    T. Ono, An integral attached to a hypersurface, Amer. J. Math. 90 (1968), 1224–1236.CrossRefGoogle Scholar
  35. [35]
    I. Ozeki, On the micro-local structure of the regular prehomogeneous vector space associated with SL(5) × GL(4). I, Proc. Japan Acad. 55 (1979), 37–40.Google Scholar
  36. [36]
    M. Rosenlicht, Some rationality questions on algebraic groups, Annali di Mat. 43 (1957), 25–50.CrossRefGoogle Scholar
  37. [37]
    P.J. Sally and M.H. Taibleson, Special functions on locally compact fields, Acta Math. 116 (1966), 279–309.CrossRefGoogle Scholar
  38. [38]
    F. Sato, Zeta functions in several variables associated with prehomogeneous vector spaces. I, Tôhoku Math. J. 34 (1982), 437–483.CrossRefGoogle Scholar
  39. [39]
    M. Sato and T. Shintani, On zeta functions associated with prehomogeneous vector spaces, Ann. Math. 100 (1974), 131–170.CrossRefGoogle Scholar
  40. [40]
    M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1–155.Google Scholar
  41. [41]
    J.-P. Serre, Cohomologie Galoisienne, Lect. Notes in Math. 5, Springer-Verlag (1965).Google Scholar
  42. [42]
    J.-P. Serre, Arbres, amalgames, SL2, astérisque 46 (1977).Google Scholar
  43. [43]
    J.-P. Serre, Quelques applications de théorème de densitè de Chebotarev, Pub. math. I.H.E.S. 54 (1981), 123–201.Google Scholar
  44. [44]
    T. Shintani, On Dirichlet series whose coefficients are class numbers of integral binary cubic forms, J. Math. Soc. Japan 24 (1972), 132–188.CrossRefGoogle Scholar
  45. [45]
    L. Strauss, Poles of a two-variable p-adic complex power, Trans. Amer. Math. Soc. 278 (1983), 481–493.Google Scholar
  46. [46]
    T. Tamagawa, Adèles, Proc. Symp. pure Math. 9 (1966), 113–121.CrossRefGoogle Scholar
  47. [47]
    J. Tate, Fourier analysis in number fields and Hecke’s zeta-functions, Thesis, Princeton, 1950; Algebraic Number Theory, Acad. Press (1967), 305-347.Google Scholar
  48. [48]
    A. Weil, Adeles and algebraic groups, Institute for Advanced Study, 1961; Progress in Math. 23, Birkhäuser (1982).Google Scholar
  49. [49]
    A. Weil, Sur la formule de Siegel dans la théorie des groupes classiques, Acta Math. 113 (1965), 1–87; Collected Papers III, Springer-Verlag (1979), 71-157.CrossRefGoogle Scholar
  50. [50]
    A. Weil, Fonction zêta et distributions, Sém. Bourbaki 312 (1966), 1–9; Collected Papers III, Springer-Verlag (1979), 158-163.Google Scholar
  51. [51]
    D.J. Wright, Dirichlet series associated with the space of binary cubic forms with coefficients in a number field, Thesis, Harvard, 1982.Google Scholar

Copyright information

© Springer Science+Business Media New York 1987

Authors and Affiliations

  • Jun-ichi Igusa
    • 1
  1. 1.The Johns Hopkins UniversityBaltimoreUSA

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