Exponential sums along p-adic curves

AbstractLet K be a p-adic field, R the valuation ring of K, and P the maximal ideal of R. Let Y⊆R2 be a non-singular closed curve, and Ym its image in R/Pm×R/Pm, i.e. the reduction modulo Pm of Y. We denote by Ψ an standard additive character on K. In this paper we discuss the estimation of exponential sums of type Sm(z,Ψ,Y,g)≔∑x∈YmΨ(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 Sm(z,Ψ,Y,g) is O(qm(1−β(f,g))), for a positive constant β(f,g) satisfying 0<β(f,g)