There are plenty of sources discussing dual abelian varieties, but I'm looking for a reference that discusses the construction and properties of the dual abelian scheme. I'm willing to accept general theory that the Picard scheme of an abelian scheme exists, but why is it an abelian scheme? 



the dual abelian scheme
You can also look at chapter 6 of Mumford's "Geometric Invariant Theory", together with part of Kleiman's article in "FGA Explained" (section 9.6) for the construction of $Pic\_{X/S}^\tau$ from $Pic\_{X/S}$.


