A formula for constructing curves over finite fields with many rational points
type: Item
instance of: scholarly article
title: A formula for constructing curves over finite fields with many rational points
zbMATH Open document ID: 1044.11054
published in: Journal of Number Theory
publication date: 1999-01-01T00:00:00Z
review text: In previous work [Lect. Notes Comput. Sci. 1122, 187--195 (1996; Zbl 0935.11022)], the author extended ideas of Serre and used class field theory to construct curves over finite fields with many rational points. Here, the author gives a formula for computing the order of the Galois group of an abelian extension of \(\mathbb F_q(T)\) in which all but one of the places of degree one are split. The proof involves transforming the multiplicative structure of this group into an additive one using the additive group of the ring of generalized Witt vectors. Tables are given for different values of \(q\) and \(k\) of the genus and number of rational points on the curve obtained by splitting all \(q\) rational points of the projective line over \(F_q\) different from a given one \(P\) in the ray class field of conductor \(kP\). Related results may be found in the work of \textit{C. P. Xing} and \textit{H. Niederreiter} [C. R. Acad. Sci., Paris, Sér. I, Math. 322, 651--654 (1996; Zbl 0853.11051)].
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 1271765
author: Kristin Lauter
reviewed by: Robert F. Lax
MaRDI profile type: Publication
full work available at URL: jnth.1998.2312
OpenAlex ID: W2031935849
cites work: Q5555033
cites work: Q4375611
cites work: Deligne-Lusztig curves as ray class fields
cites work: Q4878386
cites work: Q3837809
cites work: Q3277020
cites work: Q3324075
cites work: How to Construct Curves over Finite Fields With Many Points
DOI: 10.1006/JNTH.1998.2312
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description: scientific article; zbMATH DE number 1271765