<p>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.</p><p>Contributors:<br></p><p>· Nicolas Addington</p>

· Benjamin Antieau<p></p>

<p>· Kenneth Ascher </p>· Asher Auel<p></p>

<p>· Fedor Bogomolov</p>

<p>· Jean-Louis Colliot-Thélène</p>

<p>· Krishna Dasaratha</p>

<p>· Brendan Hassett</p>

<p>· Colin Ingalls</p>

· Martí Lahoz<p></p>

<p>· Emanuele&nbsp;Macrì</p>

<p>· Kelly McKinnie</p>

<p>· Andrew Obus</p>

<p>· Ekin Ozman</p>

<p>· Raman Parimala</p>

<p>· Alexander Perry</p>

<p>· Alena Pirutka</p>

<p>· Justin Sawon</p>

<p>· Alexei N. Skorobogatov</p>

<p>· Paolo Stellari</p>

· Sho Tanimoto<p></p>

<p>· Hugh Thomas</p>

<p>· Yuri Tschinkel</p>

<p>· Anthony Várilly-Alvarado</p>

<p>· Bianca Viray</p>

<p>· Rong Zhou</p><br><br>