Complex multiplication and Brauer groups of \(K3\) surfaces
type: Item
instance of: scholarly article
title: Complex multiplication and Brauer groups of \(K3\) surfaces
zbMATH Open document ID: 1469.14082
author: Domenico Valloni
published in: Advances in Mathematics
publication date: 2021-06-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1804.08763
review text: In this very interesting and well-written paper, the author presents an algorithm which takes as input a CM-field \(E\) and a number field \(K\), and produces as output a finite list of groups with the following property: If \(X / K\) is a \(K3\) surface with CM by \(\mathcal{O}_E\) then the group of Galois-invariant elements of the geometric Brauer group of \(X\) is isomorphic to a group in the list. The result is quite relevant to the theory of the Brauer-Manin obstruction for \(K3\) surfaces both for theoretical and for practical considerations; this is well explained in the introduction of the paper. The algorithm is illustrated by two examples; one with \(K=E=\mathbb{Q}(i)\) and the other with \(K=E=\mathbb{Q}(\sqrt{-3})\). In order to establish the algorithm, the author further develops the theory of complex multiplication for \(K3\) surfaces. Recall that a complex \(K3\) surface \(X\) has CM if the Mumford-Tate group of the natural Hodge structure on \(H^2(X(\mathbb{C}),\mathbb{Q})\) is abelian. This implies, by results of \textit{Yu. G. Zarkhin} [J. Reine Angew. Math. 341, 193--220 (1983; Zbl 0506.14034)], that the endomorphism ring of \(\mathbb{Q}\)-Hodge structures of the transcedental lattice is a CM-field, call it \(E\). By definition \(X\) has CM by \(\mathcal{O}_E\) if the endomorphism ring of \(\mathbb{Z}\)-Hodge structures of the transcendental lattice is \(\mathcal{O}_E\), the full ring of integers of \(E\). The author builds on the main theorem of CM for \(K3\) surfaces, proved by \textit{J. Rizov} [``Complex multiplication for \(K3\) surfaces'', Preprint, \url{arXiv:math/0508018}]. He adapts Shimura's ideas on CM abelian varieties to \(K3\) surfaces with CM and then translates the main theorem of CM for \(K3\) surfaces in the language of ideal lattices. Using the abstract results he establishes, the author then employs a strategy analogous to the one employed by Silverberg for torsion points of CM abelian varieties [\textit{A. Silverberg}, Compos.…
reviewed by: Evis Ieronymou
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14G12
Mathematics Subject Classification ID: 14F22
zbMATH DE Number: 7358509
zbMATH Keywords: complex multiplication
zbMATH Keywords: \(K3\) surfaces
zbMATH Keywords: Brauer groups
Wikidata QID: Q114211609
MaRDI profile type: Publication
OpenAlex ID: W3163398941
arXiv ID: 1804.08763
cites work: Q3376483
cites work: Q4434669
cites work: Q4428499
cites work: Every rational Hodge isometry between two K3 surfaces is algebraic
cites work: Q1252893
cites work: The elementary obstruction and homogeneous spaces
cites work: La conjecture de Weil pour les surfaces K3
cites work: Hodge cycles, motives, and Shimura varieties
cites work: Modular forms and \(K3\) surfaces
cites work: Examples of \(K3\) surfaces with real multiplication
cites work: Motives of derived equivalent \(K3\) surfaces
cites work: Lectures on K3 Surfaces
cites work: Odd order Brauer-Manin obstruction on diagonal quartic surfaces
cites work: On the Brauer group of diagonal quartic surfaces
cites work: Effectivity of Brauer-Manin obstructions on surfaces
cites work: Q3320521
cites work: Q3995821
cites work: Q4298518
cites work: The modularity of \(K3\) surfaces with non-symplectic group actions
cites work: The Tate conjecture for \(K3\) surfaces in odd characteristic
cites work: Q5650797
cites work: A survey of Torelli and monodromy results for holomorphic-symplectic varieties
cites work: Transcendental Brauer groups of products of CM elliptic curves
cites work: Q4195021
cites work: ON CORRESPONDENCES BETWEEN K3 SURFACES
cites work: Finiteness theorems for K3 surfaces and abelian varieties of CM type
cites work: Q5623829
cites work: Fields of definition of singular \(K3\) surfaces
cites work: \(K3\) surfaces with Picard rank 20
cites work: Q4371936
cites work: Q4152652
cites work: Q4732552
cites work: Explicit methods in number theory. Abstracts from the workshop held July 12th -- July 18th, 2009
cites work: A finiteness theorem for the Brauer group of abelian varieties and 𝐾3 surfaces
cites work: The Brauer group of Kummer surfaces and torsion of elliptic curves
cites work: \(K3\) surfaces over finite fields with given \(L\)-function
cites work: ABELIAN -DIVISION FIELDS OF ELLIPTIC CURVES AND BRAUER GROUPS OF PRODUCT KUMMER & ABELIAN SURFACES
cites work: Hodge groups of K3 surfaces.
DOI: 10.1016/J.AIM.2021.107772
description: scientific article