Exponential Sums Along p-adic Curves

2002/07/23 by W. A. Zúñiga‐Galindo, W. A. Zuniga-Galindo, Zuniga-Galindo, W. A.
Computer Science · Mathematics · #11T55 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Primary 11L40 #math.AG #math.NT #msc:11L40 #msc:11T55

paper · pdf · doi:10.48550/arxiv.math/0207201

9 pages. Accepted in Finite Fields and Their Applications

arxiv created 2002/07/23 · openalex publication_date 2002/07/23 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a p-adic field, R the valuation ring of K, and P the maximal ideal of R. Let Y subseteq R² be a non-singular closed curve, and Y_m its image in R/P^m times R/P^m, i.e. the reduction modulo P^m of Y. We denote by Psi an standard additive character on K. In this paper we discuss the estimation of exponential sums of type S_m(z,Psi,Y,g):= sum_x in Y_m Psi(zg(x)), with z in K, and g a polynomial function on Y. We show that if the p-adic absolute value of z is big enough then the complex absolute value of S_m(z,Psi,Y,g) is O(q^{m(1-beta(f,g))}), for a positive constant beta(f,g) satisfying 0

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