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$\newcommand{\Spec}{\mathrm{Spec}\ }$ Let $(P)$ be a property of rings. I call $(P)$ local when $(P)$ satisfy these two conditions: if $A$ is a ring satisfying $(P)$, then the distinguished rings $A …

There is another obvious obstruction: by definition two twists are isomorphics above an extension, but are not isomorphic above their field of definition (and also not isogenous). I believe that thes …

$\def\Pic{\mathcal{Pic}}\def\Gm{\mathbb{G}_m}\def\Hom{\mathop{Hom}}\def\HOM{\mathcal{Hom}}$ If $A/S$ is an abelian scheme, the fppf sheaf $\Pic^0_{A/S}$ is representable by an abelian scheme $\hat{A}$ …

As outlined by the other answers, the Jacobian $J_X$ of a curve $X$ defined over $\mathbb{F}_q$ indeed encapsulates all cohomology information of $X$. In particular one can read the zeta function $\ze …

For a scheme $X$, say that $X$ is topologically unibranch at $x$ if $\mathop{Spec} O_{X,x}$ is geometrically unibranch (meaning that $O_{X,y}$ is geometrically unibranch at all generisations $y$ of $x …

The general reference for this sort of questions is Waterhouse, Abelian varieties over finite fields. Your question is answered in: Theorem 7.2. If $A$ is ordinary (and simple), then $\mathop{End}(A)$ …