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Brauer Groups of K3 Surfaces

The algebraic Brauer group of a K3 surface is fairly well-understood, as long as a description of Pic as a Galois module is available, e.g. M. Bright has classified the algebraic Brauer group of diagonal quartic surfaces. This talk will explain how to determine the full Brauer group for several classes of K3 surfaces over number fields. One first gets hold of the Galois action on the transcendental part of the second l-adic cohomology via the main theorem of complex multiplication for K3 surfaces. Then one determines the full Brauer group using an argument in derived categories which builds on work by Colliot-Thélène and Skorobogatov. While for diagonal quartics over Q the odd order transcendental part of the Brauer group was previously known by different methods, the 2-primary torsion of their Brauer groups is more subtle. Surprisingly, it turns out to be algebraic with two explicit exceptions. This is partly joint work with A. Skorobogatov.