On the Birch and Swinnerton-Dyer conjecture for abelian varieties attached to Hilbert modular forms
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.
scientific article; zbMATH DE number 5859973
scientific article