Let $A$ be an abelian scheme over some base scheme $S$. Let $A^\vee$ be the dual abelian scheme, defined as $\text{Pic}^0_{A/S}$ where $\text{Pic}_{A/S}(T)=\text{Pic}(A_T)/\text{Pic}(A)$. (maybe some assumptions are needed for this actually to be an abelian scheme again.)
My main question:
- I "know" that $A^\vee$ can be described as (or is isomorphic to) $\underline{\text{Ext}}(A,\mathbb G_m)$, at least that's what I've read quite a few times. I've never seen a proof and I don't know a reference. I don't even know how to define $\underline{\text{Ext}}(A,\mathbb G_m)$. Is it the right derived functor of $\underline{\text{Hom}}(A,\mathbb G_m)$? Could someone explain the mentioned isomorphism or give a reference?
my follow up questions (which might be superfluous depending on the answer to my first question)
Is it possible to define $\underline{\text{Ext}}(A,\mathbb G_m)$ in a more explicit way? What is $\underline{\text{Ext}}(A,\mathbb G_m)(T)$ for an $S$-scheme $T$? Is there a relation between $\underline{\text{Ext}}(A,\mathbb G_m)(T)$ and $\text{Ext}(A_T,\mathbb G_{m,T})$ or $\text{Ext}(A(T),\mathbb G_m(T)$? (not necessarily equality but maybe a morphism between 2 of them?) (I was hoping to deduce the finiteness of $\text{Ext}(A_T,\mathbb G_{m,T})$ if $A^\vee(T)=\underline{\text{Ext}}(A,\mathbb G_m)(T)$ is finite)
If 2. is too difficult: Is it possible to say something about the global sections? What is $\underline{\text{Ext}}(A,\mathbb G_m)(S)$?
What about base changes? The relative Picard functor behaves well w.r.t. base changes. For example if $A_K$ is the generic fiber of $A$ then the generic fiber of $\text{Pic}^0_{A/S}$ is $\text{Pic}^0_{A_K/K}$. What happens on the corresponding "Ext-side"? The generic fiber of $\underline{\text{Ext}}(A,\mathbb G_m)$ should be $\underline{\text{Ext}}(A_K,\mathbb G_m).$ Is the latter just the corresponding tensor product $\underline{\text{Ext}}(A,\mathbb G_m)\otimes K$ of $O$-modules?
Any help is appreciated, I don't expect anyone to answer all questions, of course.
