Hecke characters and Eisenstein reciprocity in function fields
type: Item
instance of: scholarly article
title: Hecke characters and Eisenstein reciprocity in function fields
zbMATH Open document ID: 0774.11028
author: David R. Hayes
published in: Journal of Number Theory
publication date: 1993-06-29T00:00:00Z
review text: In the famous paper ``Jacobi sums as `Grössencharaktere','' Trans. Am. Math. Soc. 73, 487-495 (1952; Zbl 0048.270), \textit{A. Weil} provided the first family of curves over number fields (Fermat curves) whose \(L\)- functions had a meromorphic continuation to all of \(\mathbb{C}\). This was accomplished by showing that such \(L\)-functions could be factored according to the \(L\)-functions of certain generalized characters (``Grössencharakter'') of the idèle class group (or the associated ``Hecke character'' on the idèle group itself). Standard techniques then show that the \(L\)-functions of these Grössencharakters have meromorphic continuations and functional equations. The main part of all this is to show that Jacobi sums -- which are the key to the above factorization -- can be used to define such Grössencharakters. In the paper ``Sommes de Jacobi et caractères de Hecke'', Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. Kl. 1974, 1-14 (1974; Zbl 0367.10035), \textit{A. Weil} showed how to construct such characters for general abelian extensions of \(\mathbb{Q}\) but now used Gauss sums instead of Jacobi sums. Now let \(k\) be a global function field. The Brumer-Stark conjecture is known to be true over any global function field by a theorem of Deligne [see, e.g., Chapter V of \textit{J. Tate}, Les conjectures de Stark sur les fonctions \(L\) d'Artin en \(s=0\) (Prog. Math. 47) (Birkhäuser, 1984; Zbl 0545.12009)], and the author has given a proof of this using Drinfeld modules in [``Stickelberger elements in function fields'', Compos. Math. 55, 209-239 (1985; Zbl 0569.12008)]. A main result of the beautiful and important paper being reviewed is that the Brumer-Stark units in any finite abelian extension \(K/k\) can be aligned into a Hecke character \(\Phi\). When \(K/k\) is a cyclotomic function field (so coming directly from sign normalized rank one Drinfeld modules), \(\Phi\) is an analogue of the Gauss sum Hecke character described by Weil. The main tool used b…
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 222008
zbMATH Keywords: global function field
zbMATH Keywords: Brumer-Stark units
zbMATH Keywords: finite abelian extension
zbMATH Keywords: Hecke character
zbMATH Keywords: cyclotomic function field
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: Eisenstein reciprocity law
reviewed by: David Goss
MaRDI profile type: Publication
full work available at URL: jnth.1993.1022
OpenAlex ID: W2047997660
DOI: 10.1006/JNTH.1993.1022
description: scientific article; zbMATH DE number 222008