Graphs and recursively defined towers of function fields
type: Item
instance of: scholarly article
title: Graphs and recursively defined towers of function fields
zbMATH Open document ID: 1080.11081
published in: Journal of Number Theory
publication date: 2005-04-28T00:00:00Z
review text: A tower of function fields over \(\mathbb F_{q}\) defined recursively by \(f(x,y) \in \mathbb F_{q}[x,y]\) is given by adjoining to \(F_{n-1}\) a root \(x_{n}\) of \(f(x_{n-1},y)\), where \(F_{0} = \mathbb F_{q}(x_{0})\), \(x_{0}\) is an indeterminant, and \(f(x,y)\) is appropriately irreducible. Such towers have been of great use in the literature; here, the author associates to each a directed graph, whose vertices are places of the rational function field of \(\bar \mathbb F_{q}\) and whose directed edges are \(( \alpha, \beta)\) with \(f(\alpha, \beta) = 0\). Interest focuses on sets such as all \(\alpha\) that are the first entry in a directed edge. Under appropriate hypotheses, having in particular to do with ramification, the number of \(\mathbb F_{q}\)-rational places \(Q\) in \(F_{n}\) with \(x_{0}(Q)\) belonging to some directed edge equals the number of paths of length \(n\) in the graph. As stated in the abstract, interesting results are achieved.
reviewed by: Thomas A. Schmidt
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 2162038
zbMATH Keywords: function field towers
author: Peter Beelen
MaRDI profile type: Publication
full work available at URL: j.jnt.2004.05.011
OpenAlex ID: W2084928144
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DOI: 10.1016/J.JNT.2004.05.011
description: scientific article; zbMATH DE number 2162038