Tokyo Journal of Mathematics

A Note on Optimal Towers over Finite Fields

Takehiro HASEGAWA

Full-text: Open access

Abstract

Recently, under the influence of Elkies' conjecture [2], the optimal recursive towers of algebraic function fields (of one variable) over the finite fields with square cardinality are studied [3, 5, 6]. In this paper, we define the limit $$ \lim_{i \to \infty} \text{(the number of places of degree $n$ in $F_{i}/\mathbf{F}_{q}$)/(genus of $F_{i}$)} $$ of a tower $F_{0} \subseteq F_{1} \subseteq F_{2} \subseteq \cdots$ over the finite field $\F_{q}$. Using this limit, we prove that all the proper constant field extensions of all the optimal towers over the finite fields with square cardinality are not optimal, and we show a simple criterion whether a tower is optimal or not. Moreover, we give many new recursive towers of finite ramification type.

Article information

Source
Tokyo J. Math., Volume 30, Number 2 (2007), 477-487.

Dates
First available in Project Euclid: 4 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1202136690

Digital Object Identifier
doi:10.3836/tjm/1202136690

Mathematical Reviews number (MathSciNet)
MR2376523

Zentralblatt MATH identifier
1185.11072

Subjects
Primary: 11R58: Arithmetic theory of algebraic function fields [See also 14-XX]
Secondary: 14G15: Finite ground fields 14H05: Algebraic functions; function fields [See also 11R58]

Citation

HASEGAWA, Takehiro. A Note on Optimal Towers over Finite Fields. Tokyo J. Math. 30 (2007), no. 2, 477--487. doi:10.3836/tjm/1202136690. https://projecteuclid.org/euclid.tjm/1202136690


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