### Question


Given an abelian variety $V$ and an integer $n$, is there a natural abelian category with a natural object $X$ and natural coefficients $F$ so that $V\simeq H^n (X,F)$?


### Motivation


Studying abelian varieties is awesome. Studying objects in long exact sequences is awesome. How do (somewhat forcefully) combine these two? I mean without taking cohomology of the variety like everyone else does...


### Possible answers


The abelian variety is a $G$-module, where $G=Gal(\bar{k}/k)$, $k$ the field over which the variety is defined. So, maybe there is an interesting $G$-module that answers the above? The cases of abelian varieties over number fields and finite fields are the most interesting, so $G$ is assumed to be interesting as well (i.e. not trivial).


Maybe it arises as the $n$-th cohomology of some interesting sheaf of some interesting related variety?



Can an abelian variety be represented as the cohomology of some other object?
Let $\mathcal{A}/S$ be an Abelian scheme. Then the dual Abelian scheme is given by $R^1\pi\_\*\mathcal{A}$, if I remember correctly. Also, $\mathcal{A}^\vee(V) = \mathrm{Ext}^1\_V(\mathcal{A},\mathbf{G}\_m)$.


