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The contributions in this book explore various contexts in which the derived category of coherent sheaves on a variety determines some of its arithmetic. This setting provides new geometric tools for interpreting elements of the Brauer group. With a view towards future arithmetic applications, the book extends a number of powerful tools for analyzing rational points on elliptic curves, e.g., isogenies among curves, torsion points, modular curves, and the resulting descent techniques, as well as higher-dimensional varieties like K3 surfaces. Inspired by the rapid recent advances in our understanding of K3 surfaces, the book is intended to foster cross-pollination between the fields of complex algebraic geometry and number theory. Contributors: ? Nicolas Addington? Benjamin Antieau ? Kenneth Ascher ? Asher Auel ? Fedor Bogomolov ? Jean-Louis Colliot-Thelene ? Krishna Dasaratha ? Brendan Hassett ? Colin Ingalls? Marti Lahoz ? Emanuele Macri ? Kelly McKinnie ? Andrew Obus ? Ekin Ozman ? Raman Parimala ? Alexander Perry ? Alena Pirutka ? Justin Sawon ? Alexei N. Skorobogatov ? Paolo Stellari? Sho Tanimoto ? Hugh Thomas ? Yuri Tschinkel ? Anthony Varilly-Alvarado ? Bianca Viray ? Rong Zhou