An Upper Bound for the Garcia-Stichtenoth Numbers of Towers

Abstract

We give several examples of towers $\mathcal{F}=(F_{0}, F_{1}, F_{2}, \dotsb)$ of function fields of one variable over a finite field $\mathbb{F}_{q}$, for which the Garcia-Stichtenoth number $$ \lambda(\mathcal{F})=\lim_{m \to \infty}\frac{\text{number of $\mathbb{F}_{q}$-rational places of $F_{m}$}}{\text{genus of $F_{m}$}} $$ is zero. Moreover, we study an upper bound for the limit $\lambda(\mathcal{F})$.