Explicit uniform bounds for Brauer groups of singular K3 surfaces

TL;DR

This paper provides explicit bounds on the Brauer groups of certain singular K3 surfaces over number fields, enabling effective computation of the Brauer--Manin set and advancing the understanding of the distribution of singular K3 surfaces.

Contribution

It introduces explicit bounds for the Brauer groups of singular K3 surfaces related to CM elliptic curves, and proves an effective version of the strong Shafarevich conjecture for these surfaces.

Findings

Explicit bounds depending on degree and discriminant for Brauer groups.

Effective computability of the Brauer--Manin set for these surfaces.

Bound on the number of isomorphism classes of singular K3 surfaces over a number field.

Abstract

Let $k$ be a number field. We give an explicit bound, depending only on $[k:\mathbf{Q}]$ and the discriminant of the N\'{e}ron--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 $[k:\mathbf{Q}]$ and remove the condition that the elliptic curves be isogenous. In addition, we show how to obtain a bound, depending only on $[k:\mathbf{Q}]$, on the number of $\mathbf{C}$-isomorphism classes of singular K3 surfaces defined over $k$, thus proving an effective version of the strong Shafarevich conjecture for singular K3 surfaces.