Euler system obtained from congruences between Hilbert modular forms
\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.
scientific article; zbMATH DE number 5376044
scientific article