We show that the uniform boundedness of the transcendental Brauer group of K3 surfaces and abelian varieties of bounded dimension defined over number fields of bounded degree is a consequence of a conjecture of Coleman about rings of endomorphisms of abelian varieties. We also show that this conjecture of Coleman implies the conjecture of Shafarevich about the N\'eron-Severi lattices of K3 surfaces. This is a joint work with Martin Orr and Yuri Zarhin.