$k$ is a field of characteristic p, $C_k$ is the category of all artinian local rings with residue field an extension of $k$. $A$ is a dim-$g$ abelian variety over $k$, $L$ is a CM field with $[L:\mathbb Q]=2g$, $\alpha: O_L\rightarrow End(A)$ is an embedding of $L$ into $A$'s endomorphism algebra. Consider the deformation functor $Def(A,\alpha)$ on the category of $C_k$. Can one recommend any reference for the representability of such deformation functors that allow a possible extension of the base field? Thank you very much!

$k$ is a field of characteristic p, $C_k$ is the category of all artinian local rings with residue field an extension of $k$. $A$ is a dim-$g$ abelian variety over $k$, $L$ is a CM field with $[L:\mathbb Q]=2g$, $\alpha: O_L\rightarrow End(A)$ is an embedding of $O_L$ into $A$'s endomorphism ring. Consider the deformation functor $Def(A,\alpha)$ on the category of $C_k$. Can one recommend any reference for the representability of such deformation functors that allow a possible extension of the base field? Thank you very much!