Abstract Let A be an abelian variety over a number field sans serif upper K comma K, K, with algebraic closure sans serif upper K overbar K ¯ K. Assuming the Mumford–Tate conjecture for A, we show that the isogeny class of A over sans serif upper K overbar K ¯ K contains only finitely many isomorphism classes of bounded Faltings height. As the Mumford–Tate conjecture is known for many abelian varieties, our theorem is unconditional in those cases.
Kisin et al. (Sun,) studied this question.