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 $H^1_{et}(Y_{\overline{\mathbb{Q}}},S^r(1))$, has 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.

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.

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, $(-)^\vee$, is taken in the category of constructible $\ell$-adic 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 $H^1_{et}(Y_{\overline{\mathbb{Q}}},S^r(1))$, has 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.

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 $H^1_{et}(Y_{\overline{\mathbb{Q}}},S^r(1))$, has 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.