Complex multiplication and Brauer groups of \(K3\) surfaces
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. Math. 68, No. 3, 241--249 (1988; Zbl 0683.14002)] in order to describe the algorithm mentioned in the first paragraph.
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