Preprint A286/2004
On Curves over Finite Fields
Arnaldo Garcia
Keywords: algebraic curves | finite fields | rational points | genus | linear codes | asymptotics | towers of curves
In these notes we present some basic results of the Theory of Curves over Finite Fields.Assuming a famous theorem of A.Weil, which bounds the number of solutions in a finite field(i.e., number of rational points) in terms of the genus and the cardinality of the finite field, we then prove several other related bounds(bounds of Serre,Ihara,Stohr-Voloch,etc.).We then treat Maximal Curves(classification and genus spectrum).Maximal curves are the curves attaining the upper bound of A.Weil.If the genus of the curve is large with respect to the cardinality of the finite field,Ihara noticed that Weil's bound cannot be reached and he introduced then a quantity A(q) for the study of the asymptotics of curves over a fixed finite field.This leads to towers of curves and we devote special attention to the so-called recursive towers of curves.We present several examples of recursive towers with good asymptotic behaviour, some of them attaining the Drinfeld-Vladut bound.The connection with the asymptotics of linear codes is a result of Tsfasman-Vladut-Zink which is obtained via Goppa's construction of codes from algebraic curves over finite fields.