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

TL;DR

This paper explores the relationship between the Brauer groups of certain algebraic surfaces and the properties of elliptic curves, establishing bounds on torsion elements linked to the existence of specific division fields.

Contribution

It proves the equivalence between uniform bounds on Brauer group torsion and abelian division fields of elliptic curves, and provides bounds on Brauer group torsion depending only on fixed invariants.

Findings

Existence of uniform bounds on odd-torsion of Brauer groups is equivalent to bounds on elliptic curve division fields.

For fixed prime, degree, and discriminant, Brauer group $p$-primary torsion is uniformly bounded.

Bounds depend only on prime, degree, and discriminant, not on individual surfaces or fields.

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.