The general idea is: it can be shown that the Picard functor of a scheme $X/S$ is represented by a group algebraic space over $S$. If $X$ is an abelian scheme, then it can be shown that $Pic^0(X/S)$ is an abelian algebraic space. A Theorem of Raynaud shows that any such algebraic space is automatically a scheme. I think I recall that BLR only proves this fact for certain types of schemes $S$.