Tate conjecture for twisted Picard modular surfaces
type: Item
instance of: scholarly article
title: Tate conjecture for twisted Picard modular surfaces
zbMATH Open document ID: 1283.11086
author: Cristian Virdol
published in: Journal of Number Theory
publication date: 2011-04-20T00:00:00Z
review text: This paper starts with a clear and brief statement of the Tate conjecture on algebraic cycles. Now let \(F\) be a totally real number field. Let \(E\) be a quadratic imaginary extension of \(F\). Let \(S_K= S_{K,G}\) be the Picard modular surface associated to a unitary similitude group \(G = \text{GU}(3)\) defined relative to \(E/F\) and a compact subgroup \(K\) of \(G(A_{F,f})\), where \(A_{F,f}\) is the ring of finite adeles of \(F\). The paper considers a twist \(S'_K\) of \(S_K\), and proves that the first part of the Tate conjecture for \(S'_K\) (the Tate classes of \(S'_K\) are algebraic) is true since it is known for \(S_K\). Further, results on the second part of the Tate conjecture (meromorphic continuation of the relevant \(L\)-function, functional equation, order of pole) are obtained in some cases for the same twist \(S'_K\).
reviewed by: Yuval Z. Flicker
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R80
zbMATH DE Number: 5880855
zbMATH Keywords: Picard modular surfaces
zbMATH Keywords: potential modularity
zbMATH Keywords: Tate cycles
zbMATH Keywords: poles of \(L\)-functions
MaRDI profile type: Publication
full work available at URL: j.jnt.2010.12.005
OpenAlex ID: W2149387698
Wikidata QID: Q123224115
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DOI: 10.1016/J.JNT.2010.12.005
description: scientific article; zbMATH DE number 5880855