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