On Deligne's conjecture for Hilbert motives over totally real number fields (Q974785)

Let \(M\) be a motive defined over a number field \(F\) with coefficients in a number field \(E\). One can associate to \(M\) an \(L\)-function \(\mathbb L(M,s)\) having values in \(E\otimes_{\mathbb Q} \mathbb C\). From the properties of the restriction of scalars, one knows that \(\mathbb L(M, s)= \mathbb L(\text{Res}_{F/\mathbb Q}M,s)\). When \(M\) is critical one has the +-period defined by Deligne \(c^+(\text{Res}_{F/\mathbb Q}M) \in E\otimes_{\mathbb Q} \mathbb C\). Then Deligne's conjecture states that: Conjecture 1.1. If \(M\) is a critical motive defined over \(F\) with coefficients in \(E\), then: \[ \mathbb L(M; 0)=c^+(\text{Res}_{F/\mathbb Q}M)\in E\otimes 1\subset E\otimes_{\mathbb Q} \mathbb C. \] This conjecture is known to be true for rank 1 motives if \(F\) is either totally real or a CM field (Blasius) and for motives associated to classical modular forms of \(\text{GL}(2)=\mathbb Q\) (Deligne). In this paper, we prove that if Deligne's conjecture holds for motives associated to Hilbert modular forms of weight at least 3, then Deligne's conjecture holds for arbitrary base change to totally real number fields of motives associated to Hilbert modular forms of weight at least 3.