A note on optimal towers over finite fields
type: Item
instance of: scholarly article
title: A note on optimal towers over finite fields
zbMATH Open document ID: 1185.11072
author: Takehiro Hasegawa
published in: Tokyo Journal of Mathematics
publication date: 2008-07-21T00:00:00Z
review text: Let \(B_n(F)\) be the number of places of degree \(n\) of \(F/\mathbb{F}_q\) and \(g(F)\) be the genus of \(F\). Let \(F=(F_0,F_1,F_2,\cdots)\) be a tower of a function fields over \(\mathbb{F}_q\) and \[ \lambda(F)=\lambda(F/\mathbb{F}_q)=\lim _{i \rightarrow \infty} B_1(F_i)/g(F_i). \] The author defines \(\Delta_n(F)=\Delta_n(F/\mathbb{F}_q)=\lim _{i \rightarrow \infty} B_n(F_i)/g(F_i)\) and studies the exact values of \(\Delta_n(F)\), and upper and lower bounds for \(\Delta_n(F)\). Using the results on \(\Delta_n(F)\), it is proved that for all \(n \geq 2\), \(\Delta_n(F/\mathbb{F}_q)=0\) and \(\lambda_n(F\mathbb{F}_{q^n}/\mathbb{F}_{q^n})=\sqrt{q}-1\) if \(q\) is a square and \(F\) is an optimal tower function fields over \(\mathbb{F}_q\) where \(F\mathbb{F}_{q^n}=(F_0\mathbb{F}_{q^n},F_1\mathbb{F}_{q^n},F_2\mathbb{F}_{q^n},\cdots)\) is defined to be composite tower of \(F\) over \(\mathbb{F}_{q^n}\). In particular, it is shown that the composite tower \(F\mathbb{F}_{q^n}\)is not optimal. Moreover, simple criterion whether a tower is optimal or not and many new recursive towers of finite ramification type are given.
reviewed by: Ferruh Özbudak
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14H05
zbMATH DE Number: 5302213
zbMATH Keywords: finite fields
zbMATH Keywords: towers of function fields
zbMATH Keywords: congruence zeta functions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1202136690
OpenAlex ID: W2048252823
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DOI: 10.3836/TJM/1202136690
description: scientific article; zbMATH DE number 5302213