Download PDF - A stationary phase formula for exponential sums over $ℤ/p^{m}ℤ$ and applications to GL(3)-Kloosterman sumsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

A stationary phase formula for exponential sums over ℤpmℤ and applications to GL(3)-Kloosterman sums

Authors 1, 1

Affiliations

  1. Department of Mathematics, Columbia University, New York, New York 10027, U.S.A.

Keywords

ENG
exponential sums, Fourier coefficients of automorphic forms

Bibliography

  1. [B] N. Bourbaki, Commutative Algebra, Chapters 1-7, Springer, Berlin, 1989.
  2. [B-F-G] D. Bump, S. Friedberg and D. Goldfeld, Poincaré series and Kloosterman sums for SL(3,ℤ), Acta Arith. 50 (1988), 31-89.
  3. [D-R] R. Dąbrowski and M. Reeder, Kloosterman sets in reductive groups, J. Number Theory, to appear.
  4. [Da] H. Davenport, Multiplicative Number Theory, 2nd ed., Springer, Berlin, 1980.
  5. [D] P. Deligne, Applications de la formule des traces aux sommes trigonométriques, in: SGA 4 1/2, Lecture Notes in Math. 569, Springer, Berlin, 1977.
  6. [D-G] M. Demazure and P. Gabriel, Introduction to Algebraic Geometry and Algebraic Groups, North-Holland, Amsterdam, 1980.
  7. [Fi] B. Fisher, A note on Hensel's lemma in several variables, Proc. Amer. Math. Soc., to appear.
  8. [F] S. Friedberg, Poincaré series for GL(n): Fourier expansion, Kloosterman sums and algebreo-geometric estimates, Math. Z. 196 (1987), 165-188.
  9. [G] M. Greenberg, Rational points in Henselian discrete valuation rings, Publ. Math. IHES 31 (1966), 59-64.
  10. [EGA] A. Grothendieck et al., Éléments de Géométrie Algébrique, EGA Chapter IV, part 4, Publ. Math. IHES 32, IHES, Paris, 1967.
  11. [SGA] A. Grothendieck, Séminaire de Géométrie Algébrique du Bois-Marie, SGA 1 Chapter II, Lecture Notes in Math. 224, Springer, Berlin, 1971.
  12. [H] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer, Berlin, 1983.
  13. [K1] N. Katz, Travaux de Laumon, Astérisque 161-162 (1988), 105-132.
  14. [K2] N. Katz, Gauss Sums, Kloosterman Sums and Monodromy Groups, Ann. of Math. Stud. 116, Princeton Univ. Press, Princeton, 1988.
  15. [K3] N. Katz, Exponential Sums and Differential Equations, Ann. of Math. Stud. 124, Princeton Univ. Press, Princeton, 1990.
  16. [Kl] H. D. Kloosterman, On the representations of a number in the form ax² + by² + cz² + dt², Acta Math. 49 (1926), 407-464.
  17. [L] M. Larsen, Appendix in [B-F-G].
  18. [Lo-Sm1] J. H. Loxton and R. A. Smith, Estimates for multiple exponential sums, J. Austral. Math. Soc. Ser. A 33 (1982), 125-134.
  19. [Lo-Sm2] J. H. Loxton and R. A. Smith, On Hua's estimate for exponential sums, J. London Math. Soc. 26 (1982), 15-20.
  20. [Lo-V] J. H. Loxton and R. C. Vaughan, The estimation of complete exponential sums, Canad. Math. Bull. 28 (1985), 440-454.
  21. [M-H] J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Springer, Heidelberg, 1973.
  22. [M] D. Mumford, The Red Book of Varieties and Schemes, Lecture Notes in Math. 1358, Springer, Berlin, 1988.
  23. [Sa] H. Salié, Über die Kloostermanschen Summen S(u,v;q), Math. Z. 34 (1931), 91-109.
  24. [Se] A. Selberg, On the estimation of Fourier coefficients of modular forms, in: Proc. Sympos. Pure Math. 8, Amer. Math. Soc., 1965, 1-15.
  25. [Sm1] R. A. Smith, On n-dimensional Kloosterman sums, J. Number Theory 11 (1979), 324-343.
  26. [Sm2] R. A. Smith, Estimates for exponential sums, Proc. Amer. Math. Soc. 79 (1980), 365-368.
  27. [S] G. Stevens, Poincaré series on GL(r) and Kloosterman sums, Math. Ann. 277 (1987), 25-51.
  28. [W1] A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204-207.
  29. [W2] A. Weil, Sur certains groupes d'opérateurs unitaires, Acta Math. 111 (1964), 143-211.
Acta Arithmetica
1997/80/1
Export to PBN, Opens in new tab
Pages:
1-48
Main language of publication
English
Received
1995-12-12
Accepted
1996-09-06
Published
1997
Exact and natural sciences