Let $A$ be an abelian variety over a p-adic field $K$. Let $I$ be the inertia group of $K$. There is a Yoneda pairing $$H^n(\hat{\mathbb{Z}},A^I) \times Ext^{2-n}_{\hat{\mathbb{Z}}}(A^I,\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z} \quad (*)$$ Is this pairing non-degenerate ?

What I know is that

For $n=2$, both groups are 0.

For $n=1$, $H^1(\hat{\mathbb{Z}},A^I)$ is a finite group.

The pairing $(*)$ is non-degenerate if $A^I$ is replaced by a finitely generated abelian group.

All these facts are available in the book Arithmetic Duality Theorems by Milne. But that is about all I know. I find the groups $Ext^{2-n}_{\hat{\mathbb{Z}}}(A^I,\mathbb{Z})$ very hard to understand. I don't even know if $Ext^{1}_{\hat{\mathbb{Z}}}(A^I,\mathbb{Z})$ is finite or not.

Any help is much appreciated. Thank you very much.


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