turkmath.org

This talk is about the number of rational points N(C) of an algebraic curve C over a finite field. The classical theorem by Hasse-Weil gives sharp upper and lower bounds for N(C) in terms of the genus g(C) and the cardinality q of the underlying field. However, these bounds can be improved if the genus is large with respect to q. This leads to the concept of asymptotic bounds. We discuss results on asymptotic bounds, and give a partial answer to the question, for which pairs of integers (N,g) there exist curves with N(C)=N and g(C)=g.