Kernel of restriction in étale cohomology of curves over number fields

Question

Let $X$ be a smooth projective curve defined over a number field $K$. Let $\overline{K}$ denote the algebraic closure of $K$, and set $\overline{X} := X\otimes \overline{K}$. Denote by $\iota: \overline{X}\longrightarrow X$ the canonical morphism.

Let $\mathcal{F}$ denote an étale local system on $X$, and let $\iota^*\mathcal{F}$ denote its pullback to $\overline{X}$.

Is there a "Poitou–Tate duality"-type theorem, which gives an isomorphism between the $\mathbb{Q}_p$-vector groups:

$$\operatorname{Ker}(H^1_\text{ét}(X,\mathcal{F})\longrightarrow H^1_\text{ét}(\overline{X},\iota^*\mathcal{F})),$$ and $$\operatorname{Ker}(H^2_\text{ét}(X,\mathcal{F})\longrightarrow H^2_\text{ét}(\overline{X},\iota^*\mathcal{F})),$$ potentially replacing $\mathcal{F}$ by a twist of its dual if necessary?

Answer 1

Even for 0-dimensional varieties over number fields, the statement of Poitou–Tate duality is much more subtle than this: it's not enough just to compare the kernels of base-extension to $\overline{K}$, you need to bring in sums over the completions $K_v$. It will be more complicated still in the 1-dimensional case.

I recommend you take a look at Milne's "Arithmetic Duality Theorems".