I am not sure that all automorphism groups of algebraic varieties have natrual algebraic group structure. But if the automorphism group of a variety has algebraic group structure, how do I know the automorphism group is an algebraic group. For example, the automorphism group of an elliptic curve $A$ is an extension of the group $G$ of automorphisms which preserve the structure of the elliptic curve, by the group $A(k)$ of translations in the points of $A$, i.e. the sequence of groups $0\to A(k)\to \text{Aut}(A)\to G \to 0$ is exact, see Springer online ref - automorphism group of algebraic variaties. In this example, how do I know $\text{Aut}(A)$ is an algebraic group.