muter UDC 513.6Let F be an algebraic curve determined over a finite field k = [q]; e, X are subsidiary additive and multiplicative characters of the field k; ~o, r are functions in F determined over k and satisfying some natural conditions. If P passes through the points of curve F, rational over k, then I Xp~re(eP(P)) X(~(P)) I ~ C ]/'q , where constant C depends only on the powers of F, ~, r Let the algebraic curve F be determined over the finite field k = [ql; ~o and r are rational functions in F, also determined over k. Denoting the additive and multiplicative characters of the field k by e and X, respectively, let us examine the sum S = ~,eErme (a,~ (q~ (P))) % (Nm (• (P))),where F m is the point set of curve F, rational over the field km = [qm], m = 1, 2 .... , Nm, a m are the norm and sign of km,respeetively , ink. The term corresponding to the pole, if only one of the functions ~, is assumed equal to zero.A. Well laid the general foundation for the estimation of the sum (1), studying a case when F is a straight line [1]. If one of the functions ~, ~b is constant, then, as follows from [2], the method of estimating the corresponding sum can be based on the study of some cyclic covering of curve F. In just this way, Bombieri, studying the covering of Artin-Schreier in [3], obtained an estimation of the exponential sum corresponding to a case of ~b = 1 in (1). We will follow A. Well's idea.
