Etale cohomology of relative elliptic curve

Question

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.

Answer 1

This will follow from the conjunction of three fundamental facts:

(1) The pushforward of a (locally) constant $\mathbb Q_p$-sheaf along a smooth proper morphism of schemes where $p$ is invertible is locally constant.

(2) Proper base change: The stalk of the pushforward of a sheaf along a proper morphism is the cohomology of the geometric fiber.

(3) The first ($\mathbb Q_p$)-cohomology of an elliptic curve over a separably closed field (of characteristic not $p$) has rank two.

Indeed, by (1), $R^1 f_* \mathbb Q_p$ is locally constant, so it remains to determine the rank. The rank is the rank of the stalk at any point which by (2) and (3) is two.

(1) follows from a combination of proper base change and one of the two closely related results of smooth base change or the local acyclicity of smooth morphisms. In Milne's lecture on étale cohomology it is Theorem 20.2. In the stacks project the argument is given in tag 0GKD. You should be able to find corresponding statements in almost every introduction to étale cohomology.

(2) is the usual statement of proper base change.

(3) is the usual computation of the cohomology of curves, specialized to curves of genus one. This is usually done via Kummer theory.