The subject of this book is arithmetic algebraic geometry, an area between number theory and algebraic geometry. It is about applying geometric methods to the study of polynomial equations in rational numbers (Diophantine equations). This book represents the first complete and coherent exposition in a single volume, of both the theory and applications of torsors to rational points. Some very recent material is included. It is demonstrated that torsors provide a unified approach to several branches of the theory which were hitherto developing in parallel.

Sir Peter Swinnerton-Dyer's mathematical career encompasses more than 60 years' work of amazing creativity. This volume provides contemporary insight into several subjects in which Sir Peter's influence has been notable, and is dedicated to his 75th birthday. The opening section reviews some of his many remarkable contributions to mathematics and other fields. The remaining contributions come from leading researchers in analytic and arithmetic number theory, and algebraic geometry. The topics treated include: rational points on algebraic varieties, the Hasse principle, Shafarevich-Tate groups of elliptic curves and motives, Zagier's conjectures, descent and zero-cycles, Diophantine approximation, and Abelian and Fano varieties.