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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.

Question 1: How did Deligne know that he had to look at $Sym^k R^1f_{n*}\mathbf Q_l$ and not something else?

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?


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Q1 one answer is that the Eichler-Shimura isomorphism, or some standard Hodge--de Rham spectral sequence, relates modular forms to this cohomology group. In fact you can really compute the space with its Hecke action (not its Galois action) using entirely transcendental methods and just see the modular forms coming out. Q2 the answer is I guess that up to twist $Symm^k$ are the only representations of $GL(2)$ so it's really hard to imagine any other possibilities other than the ones Deligne uses. Forget about etale cohomology and think about it from the... – Kevin Buzzard Jun 5 '11 at 21:18
...point of view of complex geometry. These sheaves that Deligne uses are just all the sheaves coming from all the algebraic representations of $GL(2)$. – Kevin Buzzard Jun 5 '11 at 21:19
Thanks for your answer Kevin. Do you know where can I find the transcendental computation that you mention in your comment (i.e. where can I find a proof of the Eichler-Shimura isomorphism, which is only quoted in Deligne's paper)? – Nicolás Jun 5 '11 at 22:03
I learnt it from Shimura's book. I'm not sure I'd recommend that to you though :-( I don't remember it being in Miyake. Is it in Diamond-Shurman? Let me stick to weight 2. You're asking what relationship there is between $H^1_et(Y(\Gamma),Q_\ell)$ and weight 2 modular forms for $\Gamma$. But there's this standard map from $H^0(\Omega^1)$ (coherent coh) to $H^1(\mathbf{C})$ (singular coh) given by integration -- that's Eichler-Shimura in the weight 2 case. Maybe Shimura's book is the best place to read about it? It's just an integral though, and then an application of Stokes' theorem... – Kevin Buzzard Jun 5 '11 at 22:30
B. Conrad's (still unpublished?) book on Ramanujan's conjecture is a (the?) modern exposition on Deligne's 1969 Bourbaki talk. It has all detailed computations. Check Brian's webpage. – shenghao Sep 3 '12 at 7:51

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