Theorem 1.5 of M. Olsson's paper

*Hom stacks and restriction of scalars*(Duke Math. J. 134 (2006), 139-164.) gives a general criterion for a Weil restriction to be representable by an algebraic space, which applies here to show that $\mathfrak{R}_{S'/S}(A)$ is representable by an algebraic space over $S$.The arguments used to prove Proposition 7.6.5 (f) and (h) of

*Néron Models*show that $\mathfrak{R}_{S'/S}(A)$ is smooth and proper as an algebraic space. Moreover the formation of the Weil restriction commutes with base change and its fibers are connected by Proposition A.5.9 of the book*Pseudo-reductive Groups*by Conrad,Gabber and Prasad.By 1) and 2), $\mathfrak{R}_{S'/S}(A)$ is an abelian algebraic space in the sense of Section I.1 of the book

*Degeneration of abelian varieties*by Chai and Faltings. By Theorem 1.9 of loc. cit. (due to Raynaud) it is actually an abelian scheme.~~Theorem 1.5 of M. Olsson's paper~~*Hom stacks and restriction of scalars*(Duke Math. J. 134 (2006), 139-164.) gives a general criterion for a Weil restriction to be representable by an algebraic space, which applies here to show that $\mathfrak{R}_{S'/S}(A)$ is representable by an algebraic space over $S$.

By passing to a Galois cover, $\mathfrak{R}_{S'/S}(A)$ decomposes as a product of abelian schemes, so is representable by an algebraic space over $S$.

The arguments used to prove Proposition 7.6.5 (f) and (h) of

*Néron Models*show that $\mathfrak{R}_{S'/S}(A)$ is smooth and proper as an algebraic space. Moreover the formation of the Weil restriction commutes with base change and its fibers are connected by Proposition A.5.9 of the book*Pseudo-reductive Groups*by Conrad,Gabber and Prasad.By 1) and 2), $\mathfrak{R}_{S'/S}(A)$ is an abelian algebraic space in the sense of Section I.1 of the book

*Degeneration of abelian varieties*by Chai and Faltings. By Theorem 1.9 of loc. cit. (due to Raynaud) it is actually an abelian scheme.

The "problem" with this proof is that 1) and 3) are relatively delicate results (at least to an algebraic space novice such as myself; it is not clear to me for instance if 1) can be proved in a simpler way in this very special case)relatively delicate result. Of course, if $A$ is projective over $S$, the Weil restriction is automatically a scheme and we can do without 1) and 3). However, the question of when an abelian scheme is projective over the base is subtle in general: it is known to hold over a noetherian normal base but the argument is a key part of the proof of 3) anyway.

Does anyone know a simpler proof ?

- Does anyone know a simpler proof ?

- If one assumes only that $S'/S$ is finite locally free, is $\mathfrak{R}_{S'/S}(A)$ (which might not be proper anymore) still a scheme ?