Let $A$ and $B$ be abelian schemes over a base scheme $S$. There is the $\underline{\mathrm{Hom}}(A,B)$ functor $T \mapsto \mathrm{Hom}(A \times T, B \times T)$, where $\mathrm{Hom}$ means homomorphisms of group schemes.



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> Is it true that $\underline{\mathrm{Hom}}(A,B)$ is representable by a scheme of locally finite presentation over $S$?
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(By an abelian scheme over $S$, I mean a smooth proper group scheme over $S$, with geometrically integral fibers.)


I am ok with assuming that $S$ is locally Noetherian if necessary.


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If the abelian schemes $A$ and $B$ are moreover projective, one can make use of representability of the Hilbert scheme. I am not sure what to do if $A$ and $B$ are just proper, and not necessarily projective.


In the answer to this question


[Representability of Hom of two finite flat group schemes](https://mathoverflow.net/questions/314723/representability-of-hom-of-two-finite-flat-group-schemes)


R. van Dobben de Bruyn mentions there is still an argument to show the representability of $\underline{\mathrm{Hom}}(A,B)$ by a scheme, but I have not yet found the argument.



Representability of Hom of two abelian schemes
Thanks to Will Sawin for the suggestion. That works, here are the details:


Write $\underline{\mathrm{Mor}}(A,B)$ for the functor describing morphisms of schemes (as opposed to $\underline{\mathrm{Hom}}(A,B)$ for morphisms of group schemes).


By [[Tag 0D1C]](https://stacks.math.columbia.edu/tag/0D1C) we know that $\underline{\mathrm{Mor}}(A,B)$ is representable by an algebraic space, locally of finite presentation over $S$, and admits an open immersion into the Hilbert functor $\mathrm{Hilb}\_{A \times\_S B/S}$ [[Tag 0D1B]](https://stacks.math.columbia.edu/tag/0D1B) (which is also an algebraic space). This Hilbert functor is separated over $S$ [[Tag 0DM7]](https://stacks.math.columbia.edu/tag/0DM7), so $\underline{\mathrm{Mor}}(A,B)$ is also separated over $S$. Taking a fiber product as in the prevously linked post [https://mathoverflow.net/questions/314723/representability-of-hom-of-two-finite-flat-group-schemes] shows that $\underline{\mathrm{Hom}}(A,B)$ is representable by an algebraic space, which is separated and locally of finite presentation over $S$.


By the criterion in [https://mathoverflow.net/questions/4573/when-is-an-algebraic-space-a-scheme] (more precisely, the reference to Théorème A.2 in Champs Algébriques by Laumon and Moret-Bailly which says that a separated locally quasi-finite morphism of algebraic spaces is represented by schemes), it is enough to show that $\underline{\mathrm{Hom}}(A,B)$ is locally quasi-finite over $S$. For this, we reduce to the case where $S = \mathrm{Spec}~ k$ for a field $k$. In this case, one knows that $\underline{\mathrm{Hom}}(A,B)$ is in fact étale over $\mathrm{Spec}~k$ (rigidity for morphisms of abelian schemes), hence locally quasi-finite.


