$l$-adic representations from Shimura curves

Question

This question may be kind of vague. And we use the same notations as in Carayol's papers:

H. Carayol, Sur les représentations l-adiques associées aux formes modulaires de Hilbert;

H. Carayol, Sur la mauvaise réduction des courbes de Shimura.

We know Carayol constructed l-adic representation $\sigma$ of $Gal(\bar{F}/F)$ "in" the étale cohomology group of "quaternionic" Shimura curves, roughly speaking, by taking some Hecke "eigenspace" of $H_{ét}^1(M_K\times_F \bar{F}, \mathcal{F}_{\lambda})$.

My question is could we get this Galois representation from some unitary Shimura curves? More precisely, does $\sigma|_{Gal(\bar{E}/E)}$ appear in $H_{ét}^1(M_{K'}'\times_E \bar{E},\mathcal{F}_{\lambda}')$ for some $K'$, and some locally constant sheaf $\mathcal{F}_{\lambda}'$?

Thanks!

Answer 1

Yes, this can be done. In recent years, Clozel, Harris, Taylor and others have shown how to attach Galois representations to sufficiently nice automorphic representations of $GL_n$ (for arbitrary $n$) over totally real and CM fields. Very very roughly, when the base is a CM field the necessary Galois representations are constructed using unitary Shimura varieties; and when the base is totally real, the Galois representations are constructed by patching representations of "enough" imaginary CM extensions of the given totally real field.

See for instance Taylor's review article http://www.math.ias.edu/~rtaylor/longicm02.pdf, where it is sketched in the proof of Theorem 3.6 how to get representations of $Gal(\overline{\mathbb{Q}} / \mathbb{Q})$ by gluing together representations of $Gal(\overline{L} / L)$ for all imaginary quadratic fields $L$ in which some auxilliary prime $p$ is split.