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Zbl 1234.11149
Longo, Matteo
Euler system obtained from congruences between Hilbert modular forms. (English)
[J] Rend. Semin. Mat. Univ. Padova 118, 1-34 (2007). ISSN 0041-8994

{\it M. Bertolini} and {\it 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 $\Bbb Q$. 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 $\Bbb Q$.\par 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 $\Bbb Q$ 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\par (i) generalization of the argument given by Bertolini and Darmon to the context of the Iwasawa main conjecture for Hilbert modular forms;\par (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.
[Andrzej Dabrowski (Szczecin)]

MSC 2000:: *11R23 Iwasawa theory
11F33 Congruences for (p-adic) modular forms
11F41 Hilbert modular forms and surfaces
11G05 Elliptic curves over global fields
11G40 L-functions of varieties over global fields

Keywords: Hilbert modular form; Euler system; modular Abelian variety; Galois representation Selmer group; Shimura curve; Iwasawa main conjecture

Citations: Zbl 1093.11037; Zbl 1152.11028

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