Exponential sums and polynomial congruences along p-adic submanifolds

In this article, we consider the estimation of exponential sums along the points of the reduction mod p^m of a p-adic analytic submanifold of Z"p^n. More precisely, we extend Igusa@?s stationary phase method to this type of exponential sums. We also study the number of solutions of a polynomial congruence along the points of the reduction mod p^m of a p-adic analytic submanifold of Z"p^n. In addition, we attach a Poincare series to these numbers, and establish its rationality. In this way, we obtain geometric bounds for the number of solutions of the corresponding polynomial congruences.

[1]  C. Wall,et al.  Lie Algebras And Lie Groups , 1967, The Mathematical Gazette.

[2]  G. I. Perel'muter.  Estimation of a sum along an algebraic curve , 1969 .

[3]  Jan Denef,et al.  Arithmetic and Geometric Applications of Quantifier Elimination for Valued Fields , 2000 .

[4]  Jan Denef,et al.  Report on Igusa's local zeta function , 1991 .

[5]  Réduction modulopn des sous-ensembles analytiques fermés deZpN , 1982 .

[6]  W. A. Zúñiga-Galindo.  Local zeta function for nondegenerate homogeneous mappings , 2005 .

[7]  W. A. Zúñiga-Galindo.  Exponential sums along p-adic curves , 2002, math/0207201.

[8]  B. M. Fulk.  MATH , 1992 .

[9]  V. Arnold,et al.  Mathematics: Frontiers and Perspectives , 2000 .

[10]  Marcus du Sautoy,et al.  AN INTRODUCTION TO THE THEORY OF LOCAL ZETA FUNCTIONS (AMS/IP Studies in Advanced Mathematics 14) , 2001 .

[11]  W. A. Zuniga-Galindo,et al.  Local zeta functions and Newton polyhedra , 2002, Nagoya mathematical journal.

[12]  Johannes Buchmann,et al.  Coding Theory, Cryptography and Related Areas , 2000, Springer Berlin Heidelberg.

[13]  G. I. Perel'muter.  Smallest polynomial nonresidue and incomplete Gaussian sums in finite fields , 1993 .

[14]  ZETA FUNCTIONS FOR ANALYTIC MAPPINGS, LOG-PRINCIPALIZATION OF IDEALS, AND NEWTON POLYHEDRA , 2006, math/0601336.

[15]  Nicholas M. Katz.  Travaux de Laumon , 1988 .

[16]  W. A. Zúñiga-Galindo.  Local Zeta Functions Supported on Analytic Submanifolds and Newton Polyhedra , 2009, 0903.2289.

[17]  H. Hironaka.  Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: II , 1964 .

[18]  W. Li,et al.  Character Sums overp-adic Fields , 1999 .

[19]  J. Denef,et al.  Weights of exponential sums, intersection cohomology, and Newton polyhedra , 1991 .

[20]  J. Bourgain.  On Exponential Sums in Finite Fields , 2010 .

[21]  Enrico Bombieri,et al.  On Exponential Sums in Finite Fields , 1966 .

[22]  Alexander Grothendieck,et al.  Formule de Lefschetz et rationalité des fonctions $L$ , 1966 .

[23]  D. Gorenstein,et al.  Algebraic approximation of structures over complete local rings , 1969 .

[24]  곽순섭,et al.  Generalized Functions , 2006, Theoretical and Mathematical Physics.

[26]  S. Sperber,et al.  Twisted exponential sums and Newton polyhedra. , 1993 .

[27]  André Weil,et al.  Basic number theory , 1967 .

[28]  Jun-ichi Igusa,et al.  An Introduction to the Theory of Local Zeta Functions , 2007 .

[29]  N. Hashimoto.  Asymptotic expansion of an oscillating integral on a hypersurface , 1995 .

[30]  Carlos J. Moreno.  Exponential Sums and Stationary Phase (I) , 2000 .

[31]  Jan Denef,et al.  The rationality of the Poincaré series associated to thep-adic points on a variety , 1984 .

[32]  P. Deligne,et al.  Applications de la formule des traces aux sommes trigonométrigues , 1977 .

[33]  Jan Denef,et al.  P-adic and real subanalytic sets , 1988 .