Over complex numbers, the famous global Torelli theorem for K3 surfaces says that two integral Hodge isometric K3 surfaces are isomorphic. In this talk, I will first give an overview of the Huybrechts’twisted Torelli theorem for K3 surfaces, which proves the Hodge conjecture for rational Hodge isometries. I will talk about how to extend Huybrechts’ result to abelian surfaces over all fields and its relation to Tate conjectures.