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Let $Y$ be a principal homogeneous space of an abelian surface, or a K3surface, over a finitely generated extension of $\mathbb{Q}$. In 2008,Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes$\text{Br}\, Y/ \text{Br}_1\, Y$ is finite. We study this quotient for thefamily of surfaces that are geometrically isomorphic to a product of isogenousnon-CM elliptic curves, as well as the related family of geometrically Kummersurfaces; both families can be characterized by their geometric N\'eron-Severilattices. Over a field of characteristic $0$, we prove that the existence of astrong uniform bound on the size of the odd-torsion of $\text{Br}\, Y /\text{Br}_1\, Y$ is equivalent to the existence of a strong uniform bound onintegers $n$ for which there exist non-CM elliptic curves with abelian$n$-division fields. Using the same methods we show that, for a fixed prime$p$, a number field $k$ of fixed degree $r$, and a fixed discriminant of thegeometric N\'eron-Severi lattice, $(\text{Br}\, Y / \text{Br}_1\, Y)[p^\infty]$is bounded by a constant that depends only on $p$, $r$, and the discriminant.

Abelian $n$-division fields of elliptic curves and Brauer groups of product Kummer & abelian surfaces