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Finiteness theorems for K3 surfaces and abelian varieties of CM type

Part of: Arithmetic algebraic geometry Surfaces and higher-dimensional varieties (Co)homology theory

Published online by Cambridge University Press:  18 July 2018

Martin Orr  and
Alexei N. Skorobogatov
Martin Orr
Affiliation:
Department of Mathematics, South Kensington Campus, Imperial College London, London SW7 2BZ, UK email m.orr@imperial.ac.uk
Alexei N. Skorobogatov
Affiliation:
Department of Mathematics, South Kensington Campus, Imperial College London, London SW7 2BZ, UK email a.skorobogatov@imperial.ac.uk Institute for the Information Transmission Problems, Russian Academy of Sciences, 19 Bolshoi Karetnyi, Moscow 127994, Russia
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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.

Keywords

abelian varietiesK3 surfacescomplex multiplicationBrauer group

MSC classification

Primary: 11G15: Complex multiplication and moduli of abelian varieties 14J28: $K3$ surfaces and Enriques surfaces 14F22: Brauer groups of schemes

Information

Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 8 , August 2018 , pp. 1571 - 1592
Copyright
© The Authors 2018 

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