ASYMPTOTICALLY GOOD TOWERS OF CURVES OVER FINITE FIELDS
ERNEST CHRISTIAAN L¨OTTER
Department of Mathematics, University of Stellenbosch. Email: ecl@sun.ac.za

For curves $C$ defined over a finite field $\mathbb{F}_{q}$, we define

$$N_{q}\left( g\right) =\max\left\{ \char93 C\left( \mathbb{F}_{q}\right) \right\}$$

where $C$ runs through all curves of genus $g$ over $\mathbb{F}_{q}$.

In this talk a survey of results on asymptotic bounds on $N_{q}\left( g\right)$ is given, in particular the Drinfeld-Vladut upper bound. For square $q$ this bound is attained, as shown by Ihara. It is however not known how good this bound is for non-square $q$.

Recently Van der Geer and Van der Vlugt have constructed an asymptotically good explicit tower of Artin-Schreier curves defined over $\mathbb{F}_{8}$, by building it up recursively by a simple equation and calculating the genus of each curve and number of points on each curve. We draw attention to this work as well as an open problem it suggests.

SAMS subject classification: 3, 6