Analytic Number Theory Henryk Iwaniec and Emmanuel Kowalski Author addresses: Rutgers University E-mail address:
[email protected] ETH Zrich E-mail address:
[email protected] 1991 Mathematics Subject Classification. Primary 11Fxx, 11Lxx, 11Mxx, 11Nxx, 11T23, 11T24, 11R42. Key words and phrases. Analytic number theory, distribution of prime numbers, automorphic forms, L-functions, exponential sums. CHAPTER 11 SUMS OVER FINITE FIELDS 11.1. Introduction. In this chapter we consider a special type of exponential and character sums, called sometimes \complete sums", which can be seen as sums over the elements of a finite field. Although the methods of Chapter 8 can still be applied to the study of such sums, disregarding this special feature, the deepest understanding and the strongest results are obtained when the finite field aspect is taken into account and the powerful techniques of algebraic geometry are brought to bear. We have already encountered in the previous chapters some examples of expo- nential sums which can be interpreted as sums over finite fields, for example, the quadratic Gauss sums X x ax G (p) = e a p p x mod p or the Kloosterman sums (1.56) X? ax + bx¯ S(a; b; p) = e : p x mod p In this chapter we will study these sums in particular. The culminating point of our presentation is the elementary method of Stepanov which we apply for proving Weil's bound for Kloosterman sums p jS(a; b; p)j 6 2 p and Hasse's bound for the number of points of an elliptic curve over a finite field.