# Finiteness theorems for K3 surfaces and abelian varieties of CM type

## TL;DR

This paper proves finiteness results for CM abelian varieties and K3 surfaces over number fields, confirming several conjectures and establishing uniform bounds for related Galois invariants.

## Contribution

It establishes finiteness of isomorphism classes of CM abelian varieties and K3 surfaces over fixed-degree fields, confirming conjectures of Shafarevich, Coleman, and Várilly-Alvarado.

## Key findings

- Finiteness of CM abelian varieties over fixed-degree fields.
- Finiteness of CM K3 surfaces over fixed-degree fields.
- Uniform boundedness of Galois-invariant Brauer group subgroups.

## 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 V\'arilly-Alvarado in the CM case.