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Let $\pi:E\to Y$ be a universal elliptic curve over an open modular curve $Y$. Take a prime $\ell$ and take $\mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$ where the dual, $(-)^\vee$, means the internal dual in the category of constructible $\mathbb{Q}_\ell$-sheaves. For integers $r\geq0$, define $S^r=\mathrm{Sym}^r\mathcal{H}$. From time to time while I read papers, I found that the first (etale) cohomology groups of $Y_{\overline{\mathbb{Q}}}$ with coefficients $S^r$ or $S^r(1)$, i.e. $H^1_{et}(Y_{\overline{\mathbb{Q}}},S^r)$ or $H^1_{et}(Y_{\overline{\mathbb{Q}}},S^r(1))$, have been studied extensively so that authors usually do not give any reference or proofs on the vanishing or decompositions of them.

Could anyone guide me to the right statements and references including their proofs? It would be greatly helpful for me. Thank you in advance.

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  • $\begingroup$ This question is a little vague but you could check out Deligne's original paper on the Ramanujan conjecture. I think he does everything by hand in that one. $\endgroup$
    – Will Sawin
    Commented May 8, 2019 at 0:47
  • $\begingroup$ @WillSawin Thank you, what you mean by Deligne's paper on the Ramanujan conjecture is Weil I? $\endgroup$
    – User0829
    Commented May 8, 2019 at 0:49
  • $\begingroup$ @WillSawin It should be Formes modulaires et représentations ℓ-adiques, right? $\endgroup$
    – User0829
    Commented May 8, 2019 at 0:50
  • $\begingroup$ Yes, that's the one I'm thinking of. $\endgroup$
    – Will Sawin
    Commented May 8, 2019 at 2:12
  • $\begingroup$ Also relevant is Scholl's paper Motives for modular forms. $\endgroup$
    – François Brunault
    Commented May 8, 2019 at 11:50

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