A stationary phase formula for exponential sums over /pm and applications to GL(3)-Kloosterman sums
Romuald Dąbrowski ; Benji Fisher
Acta Arithmetica, Tome 80 (1997), p. 1-48 / Harvested from The Polish Digital Mathematics Library
Publié le : 1997-01-01
EUDML-ID : urn:eudml:doc:207025
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     author = {Romuald D\k abrowski and Benji Fisher},
     title = {A stationary phase formula for exponential sums over $$\mathbb{Z}$/p^{m}$\mathbb{Z}$$ and applications to GL(3)-Kloosterman sums},
     journal = {Acta Arithmetica},
     volume = {80},
     year = {1997},
     pages = {1-48},
     zbl = {0893.11032},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav80i1p1bwm}
}
Romuald Dąbrowski; Benji Fisher. A stationary phase formula for exponential sums over $ℤ/p^{m}ℤ$ and applications to GL(3)-Kloosterman sums. Acta Arithmetica, Tome 80 (1997) pp. 1-48. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav80i1p1bwm/

[000] [B] N. Bourbaki, Commutative Algebra, Chapters 1-7, Springer, Berlin, 1989. | Zbl 0666.13001

[001] [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. | Zbl 0647.10020

[002] [D-R] R. Dąbrowski and M. Reeder, Kloosterman sets in reductive groups, J. Number Theory, to appear. | Zbl 0919.11055

[003] [Da] H. Davenport, Multiplicative Number Theory, 2nd ed., Springer, Berlin, 1980.

[004] [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. | Zbl 0349.10031

[005] [D-G] M. Demazure and P. Gabriel, Introduction to Algebraic Geometry and Algebraic Groups, North-Holland, Amsterdam, 1980. | Zbl 0431.14015

[006] [Fi] B. Fisher, A note on Hensel's lemma in several variables, Proc. Amer. Math. Soc., to appear.

[007] [F] S. Friedberg, Poincaré series for GL(n): Fourier expansion, Kloosterman sums and algebreo-geometric estimates, Math. Z. 196 (1987), 165-188. | Zbl 0612.10020

[008] [G] M. Greenberg, Rational points in Henselian discrete valuation rings, Publ. Math. IHES 31 (1966), 59-64. | Zbl 0142.00901

[009] [EGA] A. Grothendieck et al., Éléments de Géométrie Algébrique, EGA Chapter IV, part 4, Publ. Math. IHES 32, IHES, Paris, 1967.

[010] [SGA] A. Grothendieck, Séminaire de Géométrie Algébrique du Bois-Marie, SGA 1 Chapter II, Lecture Notes in Math. 224, Springer, Berlin, 1971.

[011] [H] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer, Berlin, 1983. | Zbl 0521.35001

[012] [K1] N. Katz, Travaux de Laumon, Astérisque 161-162 (1988), 105-132.

[013] [K2] N. Katz, Gauss Sums, Kloosterman Sums and Monodromy Groups, Ann. of Math. Stud. 116, Princeton Univ. Press, Princeton, 1988. | Zbl 0675.14004

[014] [K3] N. Katz, Exponential Sums and Differential Equations, Ann. of Math. Stud. 124, Princeton Univ. Press, Princeton, 1990.

[015] [Kl] H. D. Kloosterman, On the representations of a number in the form ax² + by² + cz² + dt², Acta Math. 49 (1926), 407-464.

[016] [L] M. Larsen, Appendix in [B-F-G].

[017] [Lo-Sm1] J. H. Loxton and R. A. Smith, Estimates for multiple exponential sums, J. Austral. Math. Soc. Ser. A 33 (1982), 125-134. | Zbl 0493.10041

[018] [Lo-Sm2] J. H. Loxton and R. A. Smith, On Hua's estimate for exponential sums, J. London Math. Soc. 26 (1982), 15-20. | Zbl 0474.10030

[019] [Lo-V] J. H. Loxton and R. C. Vaughan, The estimation of complete exponential sums, Canad. Math. Bull. 28 (1985), 440-454. | Zbl 0575.10033

[020] [M-H] J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Springer, Heidelberg, 1973. | Zbl 0292.10016

[021] [M] D. Mumford, The Red Book of Varieties and Schemes, Lecture Notes in Math. 1358, Springer, Berlin, 1988. | Zbl 0658.14001

[022] [Sa] H. Salié, Über die Kloostermanschen Summen S(u,v;q), Math. Z. 34 (1931), 91-109.

[023] [Se] A. Selberg, On the estimation of Fourier coefficients of modular forms, in: Proc. Sympos. Pure Math. 8, Amer. Math. Soc., 1965, 1-15.

[024] [Sm1] R. A. Smith, On n-dimensional Kloosterman sums, J. Number Theory 11 (1979), 324-343. | Zbl 0409.10024

[025] [Sm2] R. A. Smith, Estimates for exponential sums, Proc. Amer. Math. Soc. 79 (1980), 365-368. | Zbl 0434.10026

[026] [S] G. Stevens, Poincaré series on GL(r) and Kloosterman sums, Math. Ann. 277 (1987), 25-51. | Zbl 0597.12017

[027] [W1] A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204-207. | Zbl 0032.26102

[028] [W2] A. Weil, Sur certains groupes d'opérateurs unitaires, Acta Math. 111 (1964), 143-211. | Zbl 0203.03305