Zeta functions of Kuga fiber varieties
type: Item
instance of: scholarly article
title: Zeta functions of Kuga fiber varieties
zbMATH Open document ID: 0666.14005
DOI: 10.1215/S0012-7094-88-05715-8
author: Salman Abdulali
published in: Duke Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: Let \(A_ 1\to V\) and \(A_ 2\to V\) be two families of abelian varieties of Kuga's type [cf. \textit{M. Kuga}, Proc. Symp. Pure Math., 9, 338-346 (1966; Zbl 0173.489)] with same base V, both defined over a number field k. The author obtains relations between the zeta functions of \(A_ 1\) and \(A_ 2\) by showing that any rational cycle on a fiber of a Kuga family which is invariant under monodromy is a Hodge cycle, and then applying Deligne's theorem on absolute Hodge cycles. An interesting illustration of his result, based on constructions of Kuga, Ohta and Addington, is provided by a non-split family over a product of two Shimura curves.
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14K05
zbMATH DE Number: 4089699
zbMATH Keywords: abelian varieties of Kuga's type
zbMATH Keywords: zeta functions
zbMATH Keywords: absolute Hodge cycles
zbMATH Keywords: product of two Shimura curves
reviewed by: Daniel Bertrand
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4089699