The Northcott property in the double struck upper Q overbar Q ¯ Q̄-isogeny class of an abelian variety

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 ¯ K̄ . Assuming the Mumford–Tate conjecture for A , we show that the isogeny class of A over sans serif upper K overbar K ¯ K̄ 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.