Exponential Sums and Polynomial Congruences Along p-adic Submanifolds

2009/10/10 by Dirk Segers, Segers, Dirk, W. A. Zúñiga‐Galindo +2
Mathematics · #11D79 #11L05 #11S40 #14G20 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #advanced mathematical theories #math.AG #math.NT #msc:11D79 #msc:11L05 #msc:11S40 #msc:14G20

paper · pdf · doi:10.48550/arxiv.0910.1887

Several typos were corrected.To Appear in Finite Fields and its Applications

openalex publication_date 2009/10/10 · arxiv created 2011/01/18 · arxiv updated 2011/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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_pⁿ. 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_pⁿ. 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.

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