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

Banff International Research Station for Mathematical Innovation and Discovery

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 $\mathrm{Br}{Y}/ \mathrm{Br}_1{Y}$ 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 $\mathrm{Br}{Y}/ \mathrm{Br}_1{Y}$ 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, $(\mathrm{Br}{Y} / \mathrm{Br}_1{Y})[p^\infty]$ is bounded by a constant that depends only on $p$, $r$, and the discriminant. This is joint work with Anthony Várilly-Alvarado.

Mathematics; Number theory; Algebraic geometry; Arithmetic number theory

video/mp4

Author affiliation: University of Washington

2017-11-26

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63731

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/