Explicit uniform bounds for Brauer groups of singular K3 surfaces

2020/06/26 by Balestrieri, Francesca, Johnson, Alexis, Newton, Rachel
#14F22 (Primary) 14J28 #14G05 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2006.14907

Abstract

Let k be a number field. We give an explicit bound, depending only on [k:Q] and the discriminant of the Néron--Severi lattice, on the size of the Brauer group of a K3 surface X/k that is geometrically isomorphic to the Kummer surface attached to a product of isogenous CM elliptic curves. As an application, we show that the Brauer--Manin set for such a variety is effectively computable. Conditional on GRH, we can also make the explicit bound depend only on [k:Q] and remove the condition that the elliptic curves be isogenous. In addition, we show how to obtain a bound, depending only on [k:Q], on the number of C-isomorphism classes of singular K3 surfaces defined over k, thus proving an effective version of the strong Shafarevich conjecture for singular K3 surfaces.

Related

• Abelian n-division fields of elliptic curves and Brauer groups of product Kummer & abelian surfaces
• Explicit bounds on the transcendental Brauer group of K3 surfaces with principal complex multiplication
• Neighbors and arithmetic of isogenous K3 surfaces
• Brauer groups of conic bundles over elliptic curves
• Effective computation of Picard groups and Brauer-Manin obstruction of degree two K3 surfaces over number fields