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The Brauer–Grothendieck Group
This monograph provides a systematic treatment of the Brauer group of schemes, from the foundational work of Grothendieck to recent applications in arithmetic and algebraic geometry. The importance of the cohomological Brauer group for applications to Diophantine equations and algebraic geometry was discovered soon after this group was introduced by Grothendieck. The Brauer–Manin obstruction plays a crucial role in the study of rational points on varieties over global fields. The birational invariance of the Brauer group was recently used in a novel way to establish the irrationality of many new classes of algebraic varieties. The book covers the vast theory underpinning these and other ap...
Number Theory and Algebraic Geometry
This volume honors Sir Peter Swinnerton-Dyer's mathematical career spanning more than 60 years' of amazing creativity in number theory and algebraic geometry.
Torsors and Rational Points
This book, first published in 2001, is a complete and coherent exposition of the theory and applications of torsors to rational points.
Torsors, Étale Homotopy and Applications to Rational Points
Lecture notes and research articles on the use of torsors and étale homotopy in algebraic and arithmetic geometry.
Torsors, Étale Homotopy and Applications to Rational Points
Torsors, also known as principal bundles or principal homogeneous spaces, are ubiquitous in mathematics. The purpose of this book is to present expository lecture notes and cutting-edge research papers on the theory and applications of torsors and étale homotopy, all written from different perspectives by leading experts. Part one of the book contains lecture notes on recent uses of torsors in geometric invariant theory and representation theory, plus an introduction to the étale homotopy theory of Artin and Mazur. Part two of the book features a milestone paper on the étale homotopy approach to the arithmetic of rational points. Furthermore, the reader will find a collection of research articles on algebraic groups and homogeneous spaces, rational and K3 surfaces, geometric invariant theory, rational points, descent and the Brauer–Manin obstruction. Together, these give a state-of-the-art view of a broad area at the crossroads of number theory and algebraic geometry.
Brauer Groups and Obstruction Problems
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...
Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
This book constitutes the refereed proceedings of the 15th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-15, held in Toulouse, France, in May 2003. The 25 revised full papers presented together with 2 invited papers were carefully reviewed and selected from 40 submissions. Among the subjects addressed are block codes; algebra and codes: rings, fields, and AG codes; cryptography; sequences; decoding algorithms; and algebra: constructions in algebra, Galois groups, differential algebra, and polynomials.
Unramified Brauer Group and Its Applications
This book is devoted to arithmetic geometry with special attention given to the unramified Brauer group of algebraic varieties and its most striking applications in birational and Diophantine geometry. The topics include Galois cohomology, Brauer groups, obstructions to stable rationality, Weil restriction of scalars, algebraic tori, the Hasse principle, Brauer-Manin obstruction, and étale cohomology. The book contains a detailed presentation of an example of a stably rational but not rational variety, which is presented as series of exercises with detailed hints. This approach is aimed to help the reader understand crucial ideas without being lost in technical details. The reader will end up with a good working knowledge of the Brauer group and its important geometric applications, including the construction of unirational but not stably rational algebraic varieties, a subject which has become fashionable again in connection with the recent breakthroughs by a number of mathematicians.
Abelian Varieties and Number Theory
This book is a collection of articles on Abelian varieties and number theory dedicated to Gerhard Frey's 75th birthday. It contains original articles by experts in the area of arithmetic and algebraic geometry. The articles cover topics on Abelian varieties and finitely generated Galois groups, ranks of Abelian varieties and Mordell-Lang conjecture, Tate-Shafarevich group and isogeny volcanoes, endomorphisms of superelliptic Jacobians, obstructions to local-global principles over semi-global fields, Drinfeld modular varieties, representations of etale fundamental groups and specialization of algebraic cycles, Deuring's theory of constant reductions, etc. The book will be a valuable resource to graduate students and experts working on Abelian varieties and related areas.
Arithmetic and Geometry
The world's leading authorities describe the state of the art in Serre's conjecture and rational points on algebraic varieties.
