On the estimation of certain exponential sums

Abstract

I. Let k = Fq be a finite field of characteristic p, let kn = Fqn be the unique extension of k of degree n, let V be a quasi-projective variety defined over k and let f ∈ k(V ) be a rational function on V , also defined over k. As usual, V (kn) will denote the set of points of V defined over kn. We also denote by K an algebraic closure of k and for a scheme X over k we denote by XK the scheme XK = X ⊗K, i.e. X after base change from k to K. In what follows we shall assume that f is defined everywhere on V and has no poles on V , so that f : V → Ak is a morphism. Let ψ0 be a non-trivial additive character on Fp and let ψn = ψ0 ◦ Trkn/ p be the corresponding character on kn; every additive character on kn can be written as ψn(lx) for some l ∈ kn. Also, we shall write ψ instead of ψ1 for the character induced on k. The exponential sum associated with V, f, kn and the character ψn is by definition

Locations

• Acta Arithmetica - View - PDF