Let $S$ be a locally noetherian scheme, $C$ the category of proper smooth $S$-schemes with geometrical connected fibres and $C_*$ the category of pointed objects of $S$, i.e. objects of $C$ together with a morphism $S \to C$. Also denote $A$ the category of abelian schemes over $S$. There is a well-known rigidity result stating that a pointed morphism between $X,Y \in A$ is already a group morphism. In other words, the inclusion functor

$A \to C_*$

is fully faithful. Is there a nice description for the image? In other words, which purely scheme-theoretic properties do abelian schemes have and are there enough to characterize them? For example, $X \in A$ is "homogeneous".