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

Abstract

Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, 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 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 N\'eron-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 $\text{Br}\, Y / \text{Br}_1\, Y$ 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 $p$, a number field $k$ of fixed degree $r$, and a fixed discriminant of the geometric 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.