Notes on Geometry and Arithmetic (Universitext)

December 4, 2020
Notes on Geometry and Arithmetic (Universitext)

This English translation of Daniel Coray’s original Frenchtextbook Notes de géométrie et d’arithmétique introduces studentsto Diophantine geometry. It engages the reader with concrete andinteresting problems using the language of classical geometry,setting aside all but the most essential ideas from algebraicgeometry and commutative algebra. Readers are invited to discoverrational points on varieties through an appealing ‘hands on’approach that offers a pathway toward active research in arithmeticgeometry. Along the way, the reader encounters the state of the arton solving certain classes of polynomial equations with beautifulgeometric realizations, and travels a unique ascent towardsvariations on the Hasse Principle.Highlighting the importance of Diophantus of Alexandria as aprecursor to the study of arithmetic over the rational numbers,this textbook introduces basic notions with an emphasis onHilbert’s Nullstellensatz over an arbitrary field. A digression onEuclidian rings is followed by a thorough study of the arithmetictheory of cubic surfaces. Subsequent chapters are devoted to p-adicfields, the Hasse principle, and the subtle notion of Diophantinedimension of fields. All chapters contain exercises, with hints orcomplete solutions.Notes on Geometry and Arithmetic will appeal to a widereadership, ranging from graduate students through to researchers.Assuming only a basic background in abstract algebra and numbertheory, the text uses Diophantine questions to motivate readersseeking an accessible pathway into arithmetic geometry.