Is every abelian scheme $\mathcal{A}/X$ under suitable conditions on $X$ a quotient of a Picard scheme of a curve $\mathcal{C}/X$? I need it for $X/\mathbf{F}_q$ smooth projective.
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Let me sketch an idea why there are not enough Jacobians. At some points I have asked for references/proofs in []. Edit: Didn't work out as hoped. 


$X$
is the spectrum of a finite field. – Matthieu Romagny Nov 15 '12 at 17:10