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-Thélène - Krishna Dasaratha - Brendan Hassett - Colin Ingalls - Martí Lahoz - Emanuele Macrì - 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 Várilly-Alvarado - Bianca Viray - Rong Zhou