2016/06/29 by Anthony Várilly‐Alvarado, Bianca Viray, Várilly-Alvarado, Anthony +1 · 1 citation
Computer Science · Mathematics · #11F80 #11G05 #11G18 #14F22 (Primary) #14J28 (Secondary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1606.09240
openalex publication_date 2016/06/29 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
Let Y be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of ℚ. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes Br Y/ Br1 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 Néron-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 Br Y / Br1 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 Néron-Severi lattice, (Br Y / Br1 Y)[p^∞] is bounded by a constant that depends only on p, r, and the discriminant.