The conductors of Eisenstein characters in cyclotomic number fields
type: Item
instance of: scholarly article
title: The conductors of Eisenstein characters in cyclotomic number fields
zbMATH Open document ID: 0833.11050
author: David R. Hayes
published in: Finite Fields and their Applications
publication date: 1996-03-17T00:00:00Z
review text: To summarize this paper one can do little better than to paraphrase (part of) the author's excellent introduction: Let \({\mathbf k}= \mathbb{F}_q (T)\) be the field of rational functions over the finite field \(\mathbb{F}_q\) and let \(M\in \mathbb{F}_q [T]\) be monic. Let \({\mathbf k} (M)\) be the cyclotomic function field obtained by adjoining the \(M\)-division points of the Carlitz module; thus \({\mathbf k} (M)/{\mathbf k}\) is abelian and shares many properties with \(\mathbb{Q} (\zeta_n)/ \mathbb{Q}\). In ``Hecke characters and Eisenstein reciprocity in function fields'' [J. Number Theory 43, 251-292 (1993; Zbl 0774.11028)]\ the author constructed a Hecke character \(\Phi_K\) for any subextension \(K/{\mathbf k}\) of \({\mathbf k} (M)/{\mathbf k}\) (in fact, the author's construction works quite generally for sub- extensions of cyclotomic function fields over arbitrary base fields). The character \(\Phi_{{\mathbf k} (M)}\) is an analog of the Gauss sum Hecke character of \textit{A. Weil} constructed in ``La cyclotomie jadis et naguére'' [Enseign. Math., II. Ser. 20, 247-263 (1974; Zbl 0352.12006)]\ and ``Sommes de Jacobi et charactères de Hecke'' [Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. Kl. 1974, 1-14 (1974; Zbl 0367.10035)]. (N.B.: In \textit{A. Weil}'s first paper ``Jacobi sums as `Grössencharaktere'\ '' [Trans. Am. Math. Soc. 73, 487-495 (1952; Zbl 0048.270)], Hecke characters were constructed only out the Jacobi sums. In the Göttingen paper (loc. cit.) Weil observed that the Jacobi sum is a product of Gauss sums and so he attempted to find the most general combination of Gauss sums that lie in a given subfield and make a Hecke character. It is found that in \(\mathbb{Q} (\zeta_n)\), the \(n\)-th -- \(2n\)- th if \(n\) is odd -- power of the (normalized) Gauss sum can be made into a Hecke character. This is what the author calls a ``Gauss sum Hecke character''. In general, the Brumer-Stark units in a subfield make a Hecke character that the author …
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11L05
zbMATH DE Number: 796751
zbMATH Keywords: Gauss sum Hecke character
zbMATH Keywords: cyclotomic function field
zbMATH Keywords: Carlitz module
zbMATH Keywords: Brumer-Stark conjecture
zbMATH Keywords: exact calculation of the conductor
zbMATH Keywords: Eisenstein reciprocity
reviewed by: David Goss
MaRDI profile type: Publication
full work available at URL: ffta.1995.1023
OpenAlex ID: W1983872261
DOI: 10.1006/FFTA.1995.1023
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description: scientific article; zbMATH DE number 796751