ABELIAN n-DIVISION FIELDS OF ELLIPTIC CURVES AND BRAUER GROUPS OF PRODUCT KUMMER & ABELIAN SURFACES

Let Y be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of Q. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes Br Y/Br-1 Y is finite. We study this quotient for the family of surfaces that are geometrically isomorphic to a product of isogenous non-CM elliptic curves, as well as the related family of geometrically Kummer surfaces; both families can be characterized by their geometric Neron-Severi lattices. Over a field of characteristic 0, we prove that the existence of a strong uniform bound on the size of the odd torsion of BrY/Br1Y is equivalent to the existence of a strong uniform bound on integers n for which there exist non-CM elliptic curves with abelian n-division fields. Using the same methods we show that, for a fixed prime l, a number field k of fixed degree r, and a fixed discriminant of the geometric Neron-Severi lattice, #(BrY/Br1Y)[l(infinity)] is bounded by a constant that depends only on l, r, and the discriminant.