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Galois representations for Hilbert forms. (English) Zbl 0684.12013
Let F be a totally real number field and let \({\mathbb{A}}_ F\) be its adele ring. Let \(\pi\) be a cuspidal automorphic representation of \(GL_ 2({\mathbb{A}}_ F)\) such that for each infinite place v, \(\pi_ v\) is a discrete series representation of weight \(k_ v\) and central character \(t\to t^{-w}\), where w is an integer independent of v. Then \(\pi\) corresponds to a holomorphic Hilbert modular new form of weight \((k_ v).\)
In this paper the authors prove the following: Suppose all \(k_ v\equiv w mod 2.\) Then there exists a number field \(T\subset {\mathbb{C}}\) and a collection \(\rho (\pi)=\{\rho_{\lambda}\}\), where for each \(\ell\)-adic completion \(T_{\lambda}\) of T, \(\rho_{\lambda}\) is a continuous representation of Gal\((\bar F/F)\) in \(GL_ 2(T_{\lambda})\) such that the L-functions \(L_ v(s,\rho_{\lambda})=L_ v(s,\pi)\) for all finite places v prime to \(\ell\) of F at which \(\pi_ v\) is unramified. Furthermore, the system \(\rho\) (\(\pi)\) is motivic.
The main part of the proof is to construct, for each quadratic CM extension E/F, the \(\ell\)-adic representations associated to the L- function of the base change to \(GL_ 2({\mathbb{A}}_ E)\) of certain cuspidal representations of the quasi-split unitary group in two variables relative to E/F in the étale cohomology of local systems on a Shimura surface associated to a certain unitary group in three variables.
Recently, R. Taylor [in Automorphic forms, Shimura varieties, and L-functions II, Academic Press, Boston, 323-336 (1990)] obtained a similar result, but the construction is quite different.
Reviewer: T.Takeuchi

11R56 Adèle rings and groups
11R42 Zeta functions and \(L\)-functions of number fields
11R39 Langlands-Weil conjectures, nonabelian class field theory
14G25 Global ground fields in algebraic geometry
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F70 Representation-theoretic methods; automorphic representations over local and global fields
Full Text: DOI
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