A note on tamely ramified towers of global function fields
type: Item
instance of: scholarly article
title: A note on tamely ramified towers of global function fields
zbMATH Open document ID: 1028.11038
author: Bruno Anglès
author: Christian Maire
published in: Finite Fields and their Applications
publication date: 2002-08-15T00:00:00Z
review text: If \(N_q(g)\) denotes the maximum number of \(\mathbb F_q\)-rational points that a smooth irreducible curve with genus \(g\) can have, then \(A(q)=\lim\sup N_q (g)/g\) is an important asymptotic measure of \(\mathbb F_q\)-rational points. The Drinfel'd-Vlădut bound \(A(q)\leq q^{\frac{1}{2}}-1\) is attained when \(q\) is a square but the value of \(A(q)\) is not known for non-square \(q\). The authors extend previous work by \textit{R. Schoof} [J. Number Theory 41, 6-14 (1992; Zbl 0762.11026)] and \textit{H. Niederreiter} and \textit{C. Xing} [Math. Nachr. 195, 171-186 (1998; Zbl 0920.11039)] to construct \(T\)-tamely ramified and \(S\)-decomposed Hilbert class field towers \(F_{\infty}(S,T)/F\) (where \(S\) and \(T\) are disjoint finite sets of places of \(F\) and \(S\neq\emptyset\)) and obtain a lower bound for \(A(q)\) when the tower is infinite. The key result is Theorem 7, which gives a sufficient condition on the \(l\)-rank of the \((S,T)\)-divisor class group for the tower to be infinite. Simple examples are then constructed which give \(A(3)\geq\frac{8}{17}\) and \(A(5)\geq\frac{8}{11}\), which improve the lower bounds obtained by Niederreiter and Xing using unramified Hilbert class field towers with decomposition. Reviewer's remark: The paper contains a few misprints, for example: on p. 208, line 3, change ``lower'' to ``upper''; on p. 213 third line from bottom, change \(\geq\) to \(\leq\).
reviewed by: M. Rafiq Omar
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14H25
zbMATH DE Number: 1781828
zbMATH Keywords: global function fields
zbMATH Keywords: rational places
zbMATH Keywords: curves over finite fields
zbMATH Keywords: lower bound for \(A(2), A(3)\)
zbMATH Keywords: class field towers
zbMATH Keywords: Drinfel'd-Vlădut bound
MaRDI profile type: Publication
full work available at URL: ffta.2000.0336
OpenAlex ID: W1967907500
cites work: Q5532660
cites work: Number of points of an algebraic curve
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cites work: Tamely ramified towers and discriminant bounds for number fields. II.
cites work: Q4748267
cites work: Towers of Global Function Fields with Asymptotically Many Rational Places and an Improvement on the Gilbert - Varshamov Bound
cites work: Tours ramifiées infinies de corps de classes. (Infinite ramified class field towers)
cites work: Algebraic curves over \({\mathbb{F}}_ 2\) with many rational points
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DOI: 10.1006/FFTA.2000.0336
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description: scientific article; zbMATH DE number 1781828