Rational points on curves over finite fields (Q2718876)

These expository notes consist of three chapters and three appendices. The first chapter reviews varieties, curves, divisors, and the Riemann-Roch Theorem. The second chapter deals with the zeta function of a curve over a finite field. The third chapter gives the Bombieri-Stepanov proof of the Riemann hypothesis for curves over finite fields, as one may find in the books of \textit{O. Moreno} [Algebraic curves over finite fields. Cambridge Tracts in Mathematics, 97. Cambridge etc.: Cambridge University Press (1991; Zbl 0733.14025)] and \textit{S. Stepanov} [Codes on algebraic curves. New York, NY: Kluwer Academic/Plenum Publishers (1999; Zbl 0997.94027)]. The first appendix gives scheme-theoretic formulations of the ideas in Chapter 1 and includes the usual proof of the Riemann-Roch Theorem using sheaf cohomology. The second appendix presents Weil's explicit formulas and Oesterlé's use of them to obtain better bounds for the maximum number of points on a curve of a given genus over a finite field. Examples using a Maple program are also given. The third appendix consists of Weil's original proof of the Hasse-Weil bound.