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On uniformity conjectures for abelian varieties and K3 surfaces over number fields Skorobogatov, Alexei


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.

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