Tate classes and \(L\)-functions on a product of a quaternionic Shimura surface and a Picard modular surface
type: Item
instance of: scholarly article
title: Tate classes and \(L\)-functions on a product of a quaternionic Shimura surface and a Picard modular surface
zbMATH Open document ID: 1179.11019
author: Cristian Virdol
published in: Journal of Number Theory
publication date: 2008-06-09T00:00:00Z
review text: The author determines the space of Tate classes on the product of a quaternionic Shimura surface and a Picard modular surface in terms of automorphic representations. More precisely, let \(S\) be the quasi-projective Shimura surface associated with a quaternion algebra \(D\), split at exactly \(2\) infinite places, over a totally real field \(F\), and some level structure. Let \(\mathbf S\) be the compactified Picard modular surface associated with the quasi-split unitary similitude group over \(\mathbb Q\) associated with a quadratic imaginary extension \(L\) of \(\mathbb Q\) and a Hermitian inner product on \(L^3\) of signature \((2,1)\). The author describes the space of Tate classes of the product \(S_M\times \mathbf S_M\), where \(M\) is the composite of the reflex field of \(S\) and of \(L\), in terms of the decomposition, resulting from the Hecke algebra operation, of the relevant cohomology group into automorphic representations. The fields of definition of the Tate classes are determined. In special cases, it is proved that the dimension of the space of Tate classes is given by the order at \(s=3\) of the corresponding \(L\)-function, as predicted by the Tate conjecture.
reviewed by: Ulrich Görtz
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 5285723
zbMATH Keywords: Tate conjecture
zbMATH Keywords: quaternionic Shimura surface
zbMATH Keywords: Picard modular surface
MaRDI profile type: Publication
full work available at URL: j.jnt.2007.12.005
OpenAlex ID: W2001681230
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DOI: 10.1016/J.JNT.2007.12.005
description: scientific article; zbMATH DE number 5285723