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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

##### MSC:
 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
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##### References:
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