Abelian in/i-division fields of elliptic curves and Brauer groups of product Kummer surfaces

In this talk we will discuss uniform bounds for the size of the transcendental Brauer groups of certain one-parameter families of Kummer surfaces with fixed geometric Néron-Severi lattice. We will show, among other things, that over a number field of fixed degree and for a fixed prime $p$, the $p$-primary torsion of these Brauer groups is uniformly bounded. For $n$ odd, we will show how to relate the existence of an $n$-torsion transcendental element on these Kummer surfaces to the existence of certain abelian division fields for associated non-CM elliptic curves. This is joint work with Bianca Viray.