Bjorn Poonen, Massachusetts Institute of Technology, Cambridge, MA.

“The reviewer cannot emphasize enough how brilliant and necessary this book is. It will be a great reference text for researchers and essential reading for graduate students in arithmetic geometry for many years to come. I will certainly be recommending that all my Ph.D. students study it in great detail.” - Daniel Loughran, Mathematical Reviews“A monograph/textbook whose main goals are to introduce the interested reader to the methods and problems of arithmetic geometry and at the same time discuss open problems of interest for further research is therefore a most welcome addition to a classical subject..The choice of topics and the decisions on what to spell out and what to just barely sketch, with adequate pointers to the existing literature, make the book under review an excellent quick introduction and reference on this subject..The book is well structured, balancing explicit constructions, terse arguments, and precise references to the literature when needed.” - Felipe Zaldivar, MAA Reviews“The origins of arithmetic (or Diophantine) geometry can be traced back to antiquity, and it remains a lively and wide research domain up to our days. The book by Bjorn Poonen, a leading expert in the field, opens doors to this vast field for many readers with different experiences and backgrounds. It leads through various algebraic geometric constructions towards its central subject: obstructions to existence of rational points.” - Yuri Manin, Max-Planck-Institute, Bonn“It is clear that my mathematical life would have been very different if a book like this had been around at the time I was a student.” - Hendrik Lenstra, University Leiden“Understanding rational points on arbitrary algebraic varieties is the ultimate challenge. We have conjectures but few results. Poonen's book, with its mixture of basic constructions and openings into current research, will attract new generations to the Queen of Mathematics.” - Jean-Louis Colliot-Thelene, Universite Paris-Sud“A beautiful subject, handled by a master.” - Joseph Silverman, Brown University

• FieldsVarieties over arbitrary fieldsProperties of morphismsFaithfully flat descentAlgebraic groupsEtale and fppf cohomologyThe Weil conjectureCohomological obstructions to rational pointsSurfacesUniversesOther kinds of fieldsProperties under base extensionBibliographyIndex

Bjorn Poonen, Massachusetts Institute of Technology, Cambridge, MA.

“The reviewer cannot emphasize enough how brilliant and necessary this book is. It will be a great reference text for researchers and essential reading for graduate students in arithmetic geometry for many years to come. I will certainly be recommending that all my Ph.D. students study it in great detail.” - Daniel Loughran, Mathematical Reviews“A monograph/textbook whose main goals are to introduce the interested reader to the methods and problems of arithmetic geometry and at the same time discuss open problems of interest for further research is therefore a most welcome addition to a classical subject..The choice of topics and the decisions on what to spell out and what to just barely sketch, with adequate pointers to the existing literature, make the book under review an excellent quick introduction and reference on this subject..The book is well structured, balancing explicit constructions, terse arguments, and precise references to the literature when needed.” - Felipe Zaldivar, MAA Reviews“The origins of arithmetic (or Diophantine) geometry can be traced back to antiquity, and it remains a lively and wide research domain up to our days. The book by Bjorn Poonen, a leading expert in the field, opens doors to this vast field for many readers with different experiences and backgrounds. It leads through various algebraic geometric constructions towards its central subject: obstructions to existence of rational points.” - Yuri Manin, Max-Planck-Institute, Bonn“It is clear that my mathematical life would have been very different if a book like this had been around at the time I was a student.” - Hendrik Lenstra, University Leiden“Understanding rational points on arbitrary algebraic varieties is the ultimate challenge. We have conjectures but few results. Poonen's book, with its mixture of basic constructions and openings into current research, will attract new generations to the Queen of Mathematics.” - Jean-Louis Colliot-Thelene, Universite Paris-Sud“A beautiful subject, handled by a master.” - Joseph Silverman, Brown University

• FieldsVarieties over arbitrary fieldsProperties of morphismsFaithfully flat descentAlgebraic groupsEtale and fppf cohomologyThe Weil conjectureCohomological obstructions to rational pointsSurfacesUniversesOther kinds of fieldsProperties under base extensionBibliographyIndex

Bjorn Poonen, Massachusetts Institute of Technology, Cambridge, MA.

“The reviewer cannot emphasize enough how brilliant and necessary this book is. It will be a great reference text for researchers and essential reading for graduate students in arithmetic geometry for many years to come. I will certainly be recommending that all my Ph.D. students study it in great detail.” - Daniel Loughran, Mathematical Reviews“A monograph/textbook whose main goals are to introduce the interested reader to the methods and problems of arithmetic geometry and at the same time discuss open problems of interest for further research is therefore a most welcome addition to a classical subject..The choice of topics and the decisions on what to spell out and what to just barely sketch, with adequate pointers to the existing literature, make the book under review an excellent quick introduction and reference on this subject..The book is well structured, balancing explicit constructions, terse arguments, and precise references to the literature when needed.” - Felipe Zaldivar, MAA Reviews“The origins of arithmetic (or Diophantine) geometry can be traced back to antiquity, and it remains a lively and wide research domain up to our days. The book by Bjorn Poonen, a leading expert in the field, opens doors to this vast field for many readers with different experiences and backgrounds. It leads through various algebraic geometric constructions towards its central subject: obstructions to existence of rational points.” - Yuri Manin, Max-Planck-Institute, Bonn“It is clear that my mathematical life would have been very different if a book like this had been around at the time I was a student.” - Hendrik Lenstra, University Leiden“Understanding rational points on arbitrary algebraic varieties is the ultimate challenge. We have conjectures but few results. Poonen's book, with its mixture of basic constructions and openings into current research, will attract new generations to the Queen of Mathematics.” - Jean-Louis Colliot-Thelene, Universite Paris-Sud“A beautiful subject, handled by a master.” - Joseph Silverman, Brown University

• FieldsVarieties over arbitrary fieldsProperties of morphismsFaithfully flat descentAlgebraic groupsEtale and fppf cohomologyThe Weil conjectureCohomological obstructions to rational pointsSurfacesUniversesOther kinds of fieldsProperties under base extensionBibliographyIndex