choice of local system in Deligne's construction of $l$-adic Galois representations - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T22:57:15Z http://mathoverflow.net/feeds/question/66987 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66987/choice-of-local-system-in-delignes-construction-of-l-adic-galois-representatio choice of local system in Deligne's construction of $l$-adic Galois representations unknown 2011-06-05T20:52:35Z 2011-06-05T21:01:17Z <p>Hello,</p> <p>Deligne famously constructed $l$-adic representations of $G_\mathbf Q = Gal(\overline{\mathbf Q}/\mathbf Q)$ starting form cusp modular forms of weight $k$ by looking inside the cohomology group $H^1_{et}(M_n, Sym^k R^1f_{n*}\mathbf Q_l)$, where $M_n$ is the moduli space of elliptic curves with full level-$n$ structure and $f_n: E_n\to M_n$ is the universal elliptic curve.</p> <p>Question 1: How did Deligne know that he had to look at $Sym^k R^1f_{n*}\mathbf Q_l$ and not something else?</p> <p>Question 2: What happens if one chooses some other local system (which is suitably invariant under $GL_2$). Is it possible to say anything about the resulting Galois representation?</p> <p>Thanks</p>