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December 2007 A Note on Optimal Towers over Finite Fields
Takehiro HASEGAWA
Tokyo J. Math. 30(2): 477-487 (December 2007). DOI: 10.3836/tjm/1202136690

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.

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Takehiro HASEGAWA. "A Note on Optimal Towers over Finite Fields." Tokyo J. Math. 30 (2) 477 - 487, December 2007. https://doi.org/10.3836/tjm/1202136690

Information

Published: December 2007
First available in Project Euclid: 4 February 2008

zbMATH: 1185.11072
MathSciNet: MR2376523
Digital Object Identifier: 10.3836/tjm/1202136690

Subjects:
Primary: 11R58
Secondary: 14G15, 14H05

Rights: Copyright © 2007 Publication Committee for the Tokyo Journal of Mathematics

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Vol.30 • No. 2 • December 2007
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