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Pseudo-reductive Groups
Pseudo-reductive groups arise naturally in the study of general smooth linear algebraic groups over non-perfect fields and have many important applications. This self-contained monograph provides a comprehensive treatment of the theory of pseudo-reductive groups and gives their classification in a usable form. The authors present numerous new results and also give a complete exposition of Tits' structure theory of unipotent groups. They prove the conjugacy results (conjugacy of maximal split tori, minimal pseudo-parabolic subgroups, maximal split unipotent subgroups) announced by Armand Borel and Jacques Tits, and also give the Bruhat decomposition, of general smooth connected algebraic groups. Researchers and graduate students working in any related area, such as algebraic geometry, algebraic group theory, or number theory, will value this book as it develops tools likely to be used in tackling other problems.
Arithmetic Algebraic Geometry
The articles in this volume are expanded versions of lectures delivered at the Graduate Summer School and at the Mentoring Program for Women in Mathematics held at the Institute for Advanced Study/Park City Mathematics Institute. The theme of the program was arithmetic algebraic geometry. The choice of lecture topics was heavily influenced by the recent spectacular work of Wiles on modular elliptic curves and Fermat's Last Theorem. The main emphasis of the articles in the volume is on elliptic curves, Galois representations, and modular forms. One lecture series offers an introduction to these objects. The others discuss selected recent results, current research, and open problems and conjectures. The book would be a suitable text for an advanced graduate topics course in arithmetic algebraic geometry.
Classification of Pseudo-reductive Groups (AM-191)
In the earlier monograph Pseudo-reductive Groups, Brian Conrad, Ofer Gabber, and Gopal Prasad explored the general structure of pseudo-reductive groups. In this new book, Classification of Pseudo-reductive Groups, Conrad and Prasad go further to study the classification over an arbitrary field. An isomorphism theorem proved here determines the automorphism schemes of these groups. The book also gives a Tits-Witt type classification of isotropic groups and displays a cohomological obstruction to the existence of pseudo-split forms. Constructions based on regular degenerate quadratic forms and new techniques with central extensions provide insight into new phenomena in characteristic 2, which also leads to simplifications of the earlier work. A generalized standard construction is shown to account for all possibilities up to mild central extensions. The results and methods developed in Classification of Pseudo-reductive Groups will interest mathematicians and graduate students who work with algebraic groups in number theory and algebraic geometry in positive characteristic.
Complex Multiplication and Lifting Problems
Abelian varieties with complex multiplication lie at the origins of class field theory, and they play a central role in the contemporary theory of Shimura varieties. They are special in characteristic 0 and ubiquitous over finite fields. This book explores the relationship between such abelian varieties over finite fields and over arithmetically interesting fields of characteristic 0 via the study of several natural CM lifting problems which had previously been solved only in special cases. In addition to giving complete solutions to such questions, the authors provide numerous examples to illustrate the general theory and present a detailed treatment of many fundamental results and concepts...
Modular Forms and the Ramanujan Conjecture
The connection between Galois representations and modular forms has been a dominant theme in number theory in recent decades. It lies at the foundation of Deligne's proof of the classical Ramanujan Conjecture, Wiles' proof of Fermat's Last Theorem, fundamental discoveries by a number of mathematicians working on the Langlands program, and much more. In this advanced graduate textbook, a self-contained development is provided for the construction of Galois representations attached to classical modular forms, and a distinguishing feature is a careful and extensive development of the algebro-geometric machinery and techniques that one needs in this construction and its generalizations beyond th...
Kaufman-Kauffman
Family history and genealogical information about the descendants of David Kaufman who was likely born ca. 1700 in an area subject to the "Emperor of Germany" according to his oath of allegiance on 9 January 1929/30. He was one of two sons of Anne Kaufman. David immigrated to America with his mother and brother (Jacob) sometime prior to the year 1716. He married Veronica Hoch sometime prior to the year 1727. David and Veronica settled in Oley, Pennsylvania and their son (Johannes), whome this book concerns, later established the township of Maidencreek, Pennsylvania. David was the father of six children. Descendants lived primarily in Pennsylvania.
