Algebraic Curves Over Finite Fields with many Rational Points and their Applications

Algebraic curves over finite fields with many rational points have received a lot of attention in recent years. We present a survey of this subject covering both the case of fixed genus and the asymptotic theory. A strong impetus in the asymptotic theory has come from a thorough exploitation of the method of infinite class field towers. On the other hand, we show by a counterexample that Perret’s conjecture on infinite class field towers is wrong, and so Perret’s method of infinite ramified class field towers breaks down. In the last two sections of the paper we discuss applications of algebraic curves over finite fields with many rational points to coding theory and to the construction of low-discrepancy sequences.