Coverings of Curves with Asymptotically many Rational Points

Abstract

Ihara defined the quantity A( q), which is the lim sup as g approaches ∞ of the ratio N q ( g)/ g, where N q ( g) is the maximum number of rational points a curve of genus g defined over a finite field F q may have. A( q) is of great relevance for applications to algebraic–geometric codes. It is known that A( q)⩽√q−1 and equality holds when q is a square. By constructing class field towers with good parameters, in this paper we present improvements of lower bounds of A( q) for q an odd power of a prime.