Henning Stichtenoth: Towers of algebraic function fields

For a function field $ F / \mathbb{F}_q$ we denote by $ g(F)$, resp. $ N(F)$, the genus, resp. the number of $ \mathbb{F}_q$ - rational places of $ F$. The Hasse-Weil theorem gives an upper bound for $ N(F)$ in terms of $ q$ and $ g(F)$. For large genus, this bound is improved essentially by the Drinfeld-Vladut bound. We present some towers $ F_0 \subseteq F_1 \subseteq \ldots$ of function fields $ F_n / \mathbb{F}_q$ such that the number of rational places of $ F_n$ is close to the Drinfeld-Vladut bound as $ n \to \infty$. The function fields $ F_n$ are described explicitly by very simple equations. (joint work with Arnaldo Garcia and others)


Juscelino Bezerra, Arnaldo Garcia, and Henning Stichtenoth.
An explicit tower of function fields over cubic finite fields and Zink's lower bound.
(preprint), 2004.

Arnaldo Garcia and Henning Stichtenoth.
A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound.
Invent. Math., 121(1):211-222, 1995.

Arnaldo Garcia, Henning Stichtenoth, and Hans-Georg Rück.
On tame towers over finite fields.
J. Reine Angew. Math., 557:53-80, 2003.

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