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Cyclotomic function fields, Hilbert class fields, and global function fields with many rational places. (English) Zbl 0891.11057

Let \(K\) be a global field with positive characteristic, that is the rational function field of a projective smooth algebraic curve over a finite field with \(q\) elements. It is of use for geometric coding theory [H. Stichtenoth, Algebraic function fields and codes (Springer, Berlin, 1993; Zbl 0816.14011)] and for construction of low-discrepancy sequences [H. Niederreiter and C. Xing, Finite Fields Appl. 2, No. 3, 241–273 (1996; Zbl 0893.11029)]] to describe explicitly such fields having the maximum (or at least, a number of places of degree 1 as close as possible to the) number \(N_q(g)\) of places of degree 1 for such fields of given genus \(g\).
In the present paper, the authors give explicit examples of such fields over the finites fields with 3, 4 and 5 elements, for any genus less than or equal to 15, 15 (with a hole for genus 14) and 12 respectively. Such fields are given either in the form \(K=\mathbb{F}_{3,4\text{ or }5}(x,y_1,y_2)\) (where \(y_i\) is an algebraic expression of \(x\) given either by a Kummer or an Artin-Schreier extension), either as a Hayes cyclotomic extension of \(\mathbb{F}_{3,4\text{ or }5}(x)\).
Note that other papers of the same authors give similar examples for binary, ternary, and quinary fields, in [Acta Arith. 75, 383–396 (1996; Zbl 0877.11065); in: Number theory (Ed. Győry, Pethő, Sos), de Gruyter (1998; Zbl 0923.11093); , Acta Arith. 83, No. 1, 65–86 (1998; Zbl 1102.11312); Demonstratio Math. 30, No. 4, 919–930 (1997; Zbl 0958.11075)].
Reviewer: Marc Perret (Lyon)

MSC:

11R58 Arithmetic theory of algebraic function fields
11R37 Class field theory
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