Twisted derived equivalences and the Tate conjecture for K3 squares

There is a long standing connection between the Tate conjecture in codimension 1 and finiteness properties, which first appeared in Tate’s seminal work on the endomorphisms of abelian varieties. I will explain how one can possibly extend this connection to codimension 2 cycles, using the theory of Brauer groups, moduli of twisted sheaves, and twisted derived equivalences, and prove the Tate conjecture for K3 squares. This recovers an earlier result of Ito-Ito-Kashikawa, which was established via a CM lifting theory, and moreover provides a recipe of constructing all the cycles on these varieties by purely geometric methods.

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Password: The order of the permutation group on 9 elements.