Let A be an abelian variety over an algebraic closure of Q. A conjecture of Mocz asserts that there are only finitely many isomorphism classes of abelian varieties isogenous to A, and of height less than some fixed constant c.

In this talk, I will sketch a proof of the conjecture when the Mumford-Tate conjecture - which is known in many cases - holds for A. In particular, I will discuss finiteness in the case of a sequence of isogenies of pairwise coprime order.

This result should be compared with Faltings' famous theorem, which is about finiteness for abelian varieties defined over a fixed number field. This is joint work with Lucia Mocz.

 While this talk will have some overlap with the one given at the introductory workshop it will *not* be the same.