Let $K$ be a field of characteristic $0$ (number fields is a sufficient generality), $A/K$ an abelian variety, and $X\subseteq A$ a closed reduced subscheme.

I am looking for a reference for the statement that the stablizer of $X$ in $A$ is a $K$-abelian subvariety of $A$.

I hope that the reference will also include a discussion on the notion of stabilizers. I have two naive options in mind: The first is the group stabilizer of $X(\bar{K})$ in $A(\bar{K})$ together with the Galois action of $G_K$; maybe taking Zariski closure is needed? The second is just $\{a\in A: a+X \subseteq X\}$.