Rational Points on Curves over Finite Fields

Preface These notes treat the problem of counting the number of rational points on a curve defined over a finite field. The notes are an extended version of an earlier set of notes Aritmetisk Algebraisk Geometri – Kurver by Johan P. Hansen [Han] on the same subject. In Chapter 1 we summarize the basic notions of algebraic geometry, especially rational points and the Riemann-Roch theorem. For the convenience of the unexperienced algebraic geometer, the chapter uses the language of classical algebraic geometry as e.g. in [Ful69]. In Appendix A the readers familiar with [Har77] may find a scheme/sheaf-theoretic formulation of Chapter 1. Moreover Appendix A contains proofs of many of the results stated in Chapter 1 without proof. In Chapter 2 we introduce the Zeta function associated to a curve defined over F q – a function containing information on the number of rational points on the curve over all finite field extensions of F q. We prove that the Zeta function is a rational function obeying a certain functional equation. Furthermore we see how the Riemann hypothesis implies the Weil bound (Corollary 2.6) on the number of rational points on the curve. When first familiar with the notions of rational functions and the Riemann-Roch theorem , Chapter 2 is rather straightforward. In contrast to this, Chapter 3 is more technical and assumes knowledge of field theory, Galois theory and the intimate relation between a smooth projective curve and its function field. Via this connection to field theory the Zeta function as defined in Chapter 2, is in the beginning of Chapter 3 put into a wider context. Afterwards we show the Riemann hypothesis for curves. In Appendix B the Weil bound (Corollary 2.6) is improved considerably. In Appendix C we give Weil's original proof of the Weil bound.