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kindasorta
kindasorta
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Let $E_a: y^2 = x(x-1)(x-a)$ be a smooth proper relative elliptic curve over $\text{Spec}(A)$, with $a\in A$, and assume $\text{Spec}(A)$ is a $\text{Spec}(\mathbb{Q}_p)$-scheme.

Let $R^1f_*E_a$$R^1f_*\mathbb{Q}_p$ be the higher direct image of the etale cohomology of $E_a$, with respect to the structure morphism $f: E_a\longrightarrow \text{Spec}(A)$. Regarding $R^1f_*E_a$$R^1f_*\mathbb{Q}_p$ as a sheaf over the etale site of $\text{Spec}(A)$. Is this sheaf free/locally free of rank $2$ over $\text{Spec}(A)$? I am interested in some direction as to how to prove this fact, but will settle for a reference.

Let $E_a: y^2 = x(x-1)(x-a)$ be a smooth proper relative elliptic curve over $\text{Spec}(A)$, with $a\in A$, and assume $\text{Spec}(A)$ is a $\text{Spec}(\mathbb{Q}_p)$-scheme.

Let $R^1f_*E_a$ be the higher direct image of the etale cohomology of $E_a$, with respect to the structure morphism $f: E_a\longrightarrow \text{Spec}(A)$. Regarding $R^1f_*E_a$ as a sheaf over the etale site of $\text{Spec}(A)$. Is this sheaf free/locally free of rank $2$ over $\text{Spec}(A)$? I am interested in some direction as to how to prove this fact, but will settle for a reference.

Let $E_a: y^2 = x(x-1)(x-a)$ be a smooth proper relative elliptic curve over $\text{Spec}(A)$, with $a\in A$, and assume $\text{Spec}(A)$ is a $\text{Spec}(\mathbb{Q}_p)$-scheme.

Let $R^1f_*\mathbb{Q}_p$ be the higher direct image of the etale cohomology of $E_a$, with respect to the structure morphism $f: E_a\longrightarrow \text{Spec}(A)$. Regarding $R^1f_*\mathbb{Q}_p$ as a sheaf over the etale site of $\text{Spec}(A)$. Is this sheaf free/locally free of rank $2$ over $\text{Spec}(A)$? I am interested in some direction as to how to prove this fact, but will settle for a reference.

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kindasorta
kindasorta
  • 2.9k
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  • 14

Etale cohomology of relative elliptic curve

Let $E_a: y^2 = x(x-1)(x-a)$ be a smooth proper relative elliptic curve over $\text{Spec}(A)$, with $a\in A$, and assume $\text{Spec}(A)$ is a $\text{Spec}(\mathbb{Q}_p)$-scheme.

Let $R^1f_*E_a$ be the higher direct image of the etale cohomology of $E_a$, with respect to the structure morphism $f: E_a\longrightarrow \text{Spec}(A)$. Regarding $R^1f_*E_a$ as a sheaf over the etale site of $\text{Spec}(A)$. Is this sheaf free/locally free of rank $2$ over $\text{Spec}(A)$? I am interested in some direction as to how to prove this fact, but will settle for a reference.