I have the following definition of what an Abelian Variety $A$ over an arbitrary field $k$ is:
it is a geometrically integral and proper group scheme over $Spec(k)$.
By a group scheme I understand a scheme over $k$ which has $k-$ morphisms
$m: A\times A \rightarrow A$,
$i: A \rightarrow A$,
$e: Spec(k) \rightarrow A$,
which formally fulfill the axioms of a group, i.e. one has the corresponding commutative diagrams.
By Yoneda this is equivalent to the datum of: for all $k-schemes$ $T$ the structure of a group on $A(T)$ which is functorial in $T$, i.e. for a $k-$morphism $S \rightarrow T$ one has a homomorphism of groups $A(T) \rightarrow A(S)$.
In particular this holds for the algebraic closure $\bar k$ of $k$.
Now my question: is it already enough to have the structure of a group on $A(\bar k)$ in order to regain the whole data described above?
And similarly for morphisms of abelian varieties: does a homomorphism between $A(\bar k)$ and $B(\bar k)$ already induce a scheme morphism $A \rightarrow B$?