Deligne's reciprocity for function fields
type: Item
instance of: scholarly article
title: Deligne's reciprocity for function fields
zbMATH Open document ID: 0932.11040
author: Samarendra K. Sinha
published in: Journal of Number Theory
publication date: 1999-03-17T00:00:00Z
review text: In 1982, guided by conjectures on special values of \(L\)-functions, \textit{P. Deligne} [Lect. Notes Math. 900, 9-100 (1982; Zbl 0537.14006)]\ proved the reciprocity law for the Euler \(\Gamma\)-function as an application of results on Hodge cycles on abelian varieties. In this paper the author presents an analogue of this reciprocity law in the context of the arithmetic of function fields and the theory of Drinfeld modules. Here the reciprocity law is proved for the geometric \(\Gamma\)-function, based on the Carlitz exponential, and uses Anderson's theory of characteristic \(p\)-solitons. The author also obtains a finer version of the reciprocity law, which addresses questions of rationality and is the analogue of one of the conjectures in \textit{P. Deligne}'s paper [Proc. Symp. Pure Math. 33, No. 2, 313-346 (1979; Zbl 0449.10022)].
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 998502
zbMATH Keywords: reciprocity law
zbMATH Keywords: arithmetic of function fields
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: geometric \(\Gamma\)-function
zbMATH Keywords: Carlitz exponential
zbMATH Keywords: characteristic \(p\)-solitons
reviewed by: Barry W. Green
MaRDI profile type: Publication
full work available at URL: jnth.1997.2082
OpenAlex ID: W2007013861
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DOI: 10.1006/JNTH.1997.2082
description: scientific article; zbMATH DE number 998502