On the Birch and Swinnerton-Dyer conjecture for abelian varieties attached to Hilbert modular forms
type: Item
instance of: scholarly article
title: On the Birch and Swinnerton-Dyer conjecture for abelian varieties attached to Hilbert modular forms
zbMATH Open document ID: 1226.11070
author: Cristian Virdol
published in: Journal of Number Theory
publication date: 2011-03-03T00:00:00Z
review text: Given a totally real number field \(F\) and a Hilbert newform \(f\) of parallel weight \(2\) of \(\text{GL}(2)/F\) with trivial central character and with field of coefficients a number field \(M\), it is conjectured that there exists an abelian variety \(A\) defined over \(F\), of dimension \([M:{\mathbb Q}]\), such that the \(L\)-function \(L(s,A)\) is equal to \(\prod_{\sigma:M\to{\mathbb C}}L(s,f^\sigma)\) modulo the factors at places dividing the level of \(f\). Assume that the Birch and Swinnerton-Dyer conjecture is true for all totally real number fields \(F\) and all abelian varieties \(B/F\) associated as above to Hilbert newforms of parallel weight \(2\) with trivial central character of \(\text{GL}(2)/F\). The author proves that the Birch and Swinnerton-Dyer conjecture is true for all abelian varieties \(A/F' \), where \(A/F\) is an abelian variety associated as above to a Hilbert newform of parallel weight \(2\) with trivial central character of \(\text{GL}(2)/F\) and \(F'/F\) is an arbitrary totally real finite extension.
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11R80
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 5859973
zbMATH Keywords: abelian variety
zbMATH Keywords: totally real number field
zbMATH Keywords: Birch and Swinnerton-Dyer conjecture
zbMATH Keywords: Hilbert modular forms
MaRDI profile type: Publication
full work available at URL: j.jnt.2010.11.008
OpenAlex ID: W2094017441
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Wikidata QID: Q122872734
DOI: 10.1016/J.JNT.2010.11.008
description: scientific article; zbMATH DE number 5859973