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.