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)

Bibliography

1

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

2

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.

3

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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