$l$-adic representations from Shimura curves - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T05:45:25Z http://mathoverflow.net/feeds/question/92155 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92155/l-adic-representations-from-shimura-curves $l$-adic representations from Shimura curves e1 2012-03-25T11:41:04Z 2012-03-25T12:55:48Z <p>This question may be kind of vague. And we use the <strong>same</strong> notations as in Carayol's papers:</p> <p><em>H. Carayol, Sur les représentations l-adiques associées aux formes modulaires de Hilbert;</em></p> <p><em>H. Carayol, Sur la mauvaise réduction des courbes de Shimura.</em></p> <p>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 $H_{ét}^1(M_K\times_F \bar{F}, \mathcal{F}_{\lambda})$.</p> <p>My question is could we get this Galois representation from some unitary Shimura curves? More precisely, does <code>$\sigma|_{Gal(\bar{E}/E)}$</code> appear in <code>$H_{ét}^1(M_{K'}'\times_E \bar{E},\mathcal{F}_{\lambda}')$</code> for some <code>$K'$</code>, and some locally constant sheaf <code>$\mathcal{F}_{\lambda}'$</code>?</p> <p>Thanks!</p> http://mathoverflow.net/questions/92155/l-adic-representations-from-shimura-curves/92164#92164 Answer by David Loeffler for $l$-adic representations from Shimura curves David Loeffler 2012-03-25T12:55:48Z 2012-03-25T12:55:48Z <p>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.</p> <p>See for instance Taylor's review article <a href="http://www.math.ias.edu/~rtaylor/longicm02.pdf" rel="nofollow">http://www.math.ias.edu/~rtaylor/longicm02.pdf</a>, 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.</p>