Exponential sums along p-adic curves

Abstract

Let K be a p-adic field, R the valuation ring of K, and P the maximal ideal of R. Let Y⊆ R 2 be a non-singular closed curve, and Y m its image in R/ P m × R/ P m , i.e. the reduction modulo P m of Y. We denote by Ψ an standard additive character on K. In this paper we discuss the estimation of exponential sums of type S m ( z, Ψ, Y, g)≔∑ x∈ Y m Ψ( zg( x)), with z∈ 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, Ψ, Y, g) is O( q m(1− β( f, g)) ), for a positive constant β( f, g) satisfying 0< β( f, g)<1.