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Grothendieck's duality theory for coherent cohomology is a
fundamental tool in algebraic geometry and number theory, in areas
ranging from the moduli of curves to the arithmetic theory of
modular forms. Presented is a systematic overview of the entire
theory, including many basic definitions and a detailed study of
duality on curves, dualizing sheaves, and Grothendieck's residue
symbol. Along the way proofs are given of some widely used
foundational results which are not proven in existing treatments of
the subject, such as the general base change compatibility of the
trace map for proper Cohen-Macaulay morphisms (e.g., semistable
curves). This should be of interest to mathematicians who have some
familiarity with Grothendieck's work and wish to understand the
details of this theory.
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.
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.
Pseudo-reductive groups arise naturally in the study of general
smooth linear algebraic groups over non-perfect fields and have
many important applications. This monograph provides a
comprehensive treatment of the theory of pseudo-reductive groups
and gives their classification in a usable form. In this second
edition there is new material on relative root systems and Tits
systems for general smooth affine groups, including the extension
to quasi-reductive groups of famous simplicity results of Tits in
the semisimple case. Chapter 9 has been completely rewritten to
describe and classify pseudo-split absolutely pseudo-simple groups
with a non-reduced root system over arbitrary fields of
characteristic 2 via the useful new notion of 'minimal type' for
pseudo-reductive groups. Researchers and graduate students working
in related areas, 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.
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 in the arithmetic of abelian varieties, such as the Main
Theorem of Complex Multiplication and its generalisations, the
finer aspects of Tate's work on abelian varieties over finite
fields, and deformation theory. This book provides an ideal
illustration of how modern techniques in arithmetic geometry (such
as descent theory, crystalline methods, and group schemes) can be
fruitfully combined with class field theory to answer concrete
questions about abelian varieties. It will be a useful reference
for researchers and advanced graduate students at the interface of
number theory and algebraic geometry.
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