Arithmetic & moduli of K3 surfaces

On Tuesday 31 January 2017, we will hold a mini-workshop on arithmetic and moduli of K3 surfaces at the University of Amsterdam. Lectures take place in room F3.20 at the Korteweg-de Vries Institute for Mathematics. Since the lecture room has limited capacity, registration is compulsory.


Martin Orr (Imperial College)
Introduction to K3 surfaces
Alexei Skorobogatov (Imperial College)
Finiteness theorems for abelian varieties and K3 surfaces with complex multiplication
Abstract. We study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. Building on a result of Jacob Tsimerman we show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational numbers. As an application we confirm finiteness conjectures of Shafarevich, Coleman and Várilly-Alvarado in the CM case. This is a joint work with Martin Orr.
Anthony Várilly-Alvarado (Rice University)
Abelian n-division fields of elliptic curves and Brauer groups of product Kummer surfaces
Abstract. In this talk we will discuss uniform bounds for the size of the transcendental Brauer groups of certain one-parameter families of Kummer surfaces with fixed geometric Néron-Severi lattice. We will show, among other things, that over a number field of fixed degree and for a fixed prime p, the p-primary torsion of these Brauer groups is uniformly bounded. For n odd, we will show how to relate the existence of an n-torsion transcendental element on these Kummer surfaces to the existence of certain abelian division fields for associated non-CM elliptic curves. This is joint work with Bianca Viray.
Bianca Viray (University of Washington)
On the dependence of the Brauer-Manin obstruction on the degree of a variety
Abstract. Let X be a smooth projective variety of degree d over a number field k. In 1970 Manin observed that elements of the Brauer group of X can obstruct the existence of a k-point, even when X is everywhere locally soluble. In joint work with Brendan Creutz, we prove that if X is a geometrically abelian, Kummer, or bielliptic surface then this Brauer-Manin obstruction to the existence of a k-point can be detected from only the d-primary torsion Brauer classes.