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Motives for Hilbert modular forms. (English) Zbl 0829.11028
In this paper the authors attach motives to collections \(\pi_1, \ldots, \pi_n\) of holomorphic Hilbert modular forms over a totally real field \(F\) of discrete series type at infinity. The method they used is based on the construction of certain sub-motives of the cohomology of fiber systems over Picard modular surfaces. Let \(U(3)\) be a unitary group in three variables defined relative to a quadratic extension \(E = KF\) that determines a Picard modular surface, where \(K\) is an imaginary quadratic extension of \(\mathbb{Q}\). Then the sub-motives of interest are attached to endoscopic automorphic forms on \(U(3)\) by passing from \(\pi_1, \ldots, \pi_n\) to Picard modular surfaces through a base change to \(GL(2)_E\), a twisting and a descent to \(U(2)_{E/F}\), an extension to \(U(2) \times U(1)\), an endoscopic transfer to quasi-split \(U(3)\), and finally a transfer to an inner form of \(U(3)\). Producing motives themselves involves the construction of algebraic correspondences, and this is achieved by verifying the Hodge conjecture for the generic fibers of the fiber systems of abelian varieties carried by the Picard modular surfaces.
As an application the authors obtain compatible systems of \(\ell\)-adic representations of \(\text{Gal} (\overline F/F)\) of degree \(2^n\) whose \(L\)-function coincides with the tensor product \(L\)-functions \(L(s, \pi_1 \otimes \cdot \otimes \pi_n)\) at almost all places.

11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
14G35 Modular and Shimura varieties
11G18 Arithmetic aspects of modular and Shimura varieties
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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