Counting Points on Curves over Finite Fields

Abstract

In this chapter we apply the technique we have worked out earlier to prove the Riemann hypothesis for the zeta-function ζ (X, s) of a curve X defined over a finite field F _q .This result was proved for the first time by Hasse (in the case of elliptic curves) and Weil (in the general case) using the correspondence theory on X. Here we give an elementary proof based essentially on using only the Riemann—Roch theorem (see Stepanov [184, 185, 187], Bombieri [17], Schmidt [159] and Stöhr and Voloch [200]).