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Hello,

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 $Sym^k$ of the local system $Sym^k R^1f_{n*}\mathbf Q_l$ 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?

Thanks
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choice of local system in Deligne's construction of $l$-adic Galois representations

Hello,

Deligne famously constructed $l$-adic representations of $G_\mathbf Q = Gal(\overline{\mathbf Q}/\mathbf Q)$ starting form cusp modular forms 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 $Sym^k$ of the local system $R^1f_{n*}\mathbf Q_l$ 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?

Thanks