Euler system obtained from congruences between Hilbert modular forms
type: Item
instance of: scholarly article
title: Euler system obtained from congruences between Hilbert modular forms
zbMATH Open document ID: 1234.11149
author: Matteo Longo
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2008-12-01T00:00:00Z
full work available at URL: https://eudml.org/doc/108723
review text: \textit{M. Bertolini} and \textit{H. Darmon} [Ann. Math. (2) 162, No. 1, 1--64 (2005; Zbl 1093.11037)] have used the theory of congruences between modular forms to construct an Euler system which they use to prove one of the divisibility properties predicted by the Iwasawa main conjecture for elliptic curves over \(\mathbb Q\). This construction can be generalized to the context of Hilbert modular forms of parallel weight 2 with rational eigenvalues under the action of the Hecke algebra. This generalization is used by the author in [Ann. Inst. Fourier 56, No. 3, 689--733 (2006; Zbl 1152.11028)] to prove the Birch and Swinnerton-Dyer conjecture for modular elliptic curves with everywhere good reduction, analytic rank zero, without CM and defined over a totally real number field of even degree over \(\mathbb Q\). In this article, the author provides a generalization of the Euler system constructed by Bertolini and Darmon. Let \(K\) be a totally imaginary quadratic extension of a totally real number field \(F\). Let \(F\) be a prime ideal of the ring of integers of the finite extension of \(\mathbb Q\) generated by the eigenvalues of the Hecke algebra on the Hilbert modular newform \(\phi\). Assume that there exists an Abelian variety \(A\) over \(F\) of \(\text{GL}_2\)-type such that \(L_K(A,s)= L_K(\phi,s)\). Under certain assumptions on \(K\) and \(F\), the author proves that there exists an Euler system relative to \(A/K\) and \(F^n\) for all \(n\). (Theorem 1.4). He gives two applications (i) generalization of the argument given by Bertolini and Darmon to the context of the Iwasawa main conjecture for Hilbert modular forms; (ii) extension of his own results [Ann. Inst. Fourier 56, No. 3, 689--733 (2006; Zbl 1152.11028)] to the context of abelian varieties of \(\text{GL}_2\)-type.
reviewed by: Andrzej Dąbrowski
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 5376044
zbMATH Keywords: Hilbert modular form
zbMATH Keywords: Euler system
zbMATH Keywords: modular Abelian variety
zbMATH Keywords: Galois representation Selmer group
zbMATH Keywords: Shimura curve
zbMATH Keywords: Iwasawa main conjecture
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5376044