Finiteness theorems for K3 surfaces and abelian varieties of CM type

Abstract

We study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. 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 and Coleman in the CM case. In addition we prove the uniform boundedness of the Galois invariant subgroup of the geometric Brauer group for forms of a smooth projective variety satisfying the integral Mumford--Tate conjecture. When applied to K3 surfaces, this affirms a conjecture of Várilly-Alvarado in the CM case.

25 pages, to appear in Compositio Math

References in corpus (1)

• The unpolarized Shafarevich Conjecture for K3 Surfaces

Cited by in corpus (8)

• On uniformity conjectures for abelian varieties and K3 surfaces
• Arithmetic of rational points and zero-cycles on products of Kummer varieties and K3 surfaces
• Local to global principle for the moduli space of K3 surfaces
• A uniform bound on the Brauer groups of certain log K3 surfaces
• The Brauer-Manin obstruction for zero-cycles on K3 surfaces
• Endomorphism algebras of geometrically split abelian surfaces over $\mathbb{Q}$
• On the irrationality of moduli spaces of projective hyperkähler manifolds
• Arithmetic of rational points and zero-cycles on Kummer varieties