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Abelian varieties and their moduli are a central topic of
increasing importance in todays mathematics. Applications range
from algebraic geometry and number theory to mathematical
physics.
The present collection of 17 refereed articles originates from the
third "Texel Conference" held in 1999. Leading experts discuss and
study the structure of the moduli spaces of abelian varieties and
related spaces, giving an excellent view of the state of the art in
this field.
The book will appeal to pure mathematicians, especially algebraic
geometers and number theorists, but will also be relevant for
researchers in mathematical physics.
Abelian varieties and their moduli are a central topic of
increasing importance in todays mathematics. Applications range
from algebraic geometry and number theory to mathematical
physics.
The present collection of 17 refereed articles originates from the
third "Texel Conference" held in 1999. Leading experts discuss and
study the structure of the moduli spaces of abelian varieties and
related spaces, giving an excellent view of the state of the art in
this field.
The book will appeal to pure mathematicians, especially algebraic
geometers and number theorists, but will also be relevant for
researchers in mathematical physics.
In September 1997, the Working Week on Resolution of Singularities
was held at Obergurgl in the Tyrolean Alps. Its objective was to
manifest the state of the art in the field and to formulate major
questions for future research. The four courses given during this
week were written up by the speakers and make up part I of this
volume. They are complemented in part II by fifteen selected
contributions on specific topics and resolution theories. The
volume is intended to provide a broad and accessible introduction
to resolution of singularities leading the reader directly to
concrete research problems.
In September 1997, the Working Week on Resolution of Singularities
was held at Obergurgl in the Tyrolean Alps. Its objective was to
manifest the state of the art in the field and to formulate major
questions for future research. The four courses given during this
week were written up by the speakers and make up part I of this
volume. They are complemented in part II by fifteen selected
contributions on specific topics and resolution theories. The
volume is intended to provide a broad and accessible introduction
to resolution of singularities leading the reader directly to
concrete research problems.
Abelian varieties can be classified via their moduli. In positive
characteristic the structure of the p-torsion-structure is an
additional, useful tool. For that structure supersingular abelian
varieties can be considered the most special ones. They provide a
starting point for the fine description of various structures. For
low dimensions the moduli of supersingular abelian varieties is by
now well understood. In this book we provide a description of the
supersingular locus in all dimensions, in particular we compute the
dimension of it: it turns out to be equal to AEg.g/4UE, and we
express the number of components as a class number, thus completing
a long historical line where special cases were studied and general
results were conjectured (Deuring, Hasse, Igusa, Oda-Oort,
Katsura-Oort).
This book originated in the idea that open problems act as
crystallization points in mathematical research. Mathematical books
usually deal with fully developed theories. But here we present
work at an earlier stage-when challenging questions can give new
directions to mathematical research. In mathematics, significant
progress is often made by looking at the underlying structures of
open problems and discovering new directions that are developed to
find solutions. In that process, the search for finding the "true"
nature of the problem at hand is the impetus for our thoughts. It
is only much later, in retrospect, that we see the "flow of
mathematics"-from problem to theory and new insight. This is the
gist of the present volume. The origin of this volume lies in a
collection of nineteen problems presented in 1995 to the
participants of the conference Arithmetic and Geometry of Abelian
Varieties.
This is the first of two volumes constituting The Legacy of
Bernhard Riemann After One Hundred and Fifty Years. Bernhard
Riemann (1826-1866) possessed an original and broad vision of
mathematics together with powerful skill. His work continues to
influence almost all major branches of mathematics. The
twenty-three papers in the two-volume set examine Riemann, his
work, and his significance in the context of modern mathematical
developments. Contributing authors (to the two-volume set): Michael
F. Atiyah, M. V. Berry, Ching-Li Chai, Brian Conrey, Jean-Pierre
Demailly, F. T. Farrell, James Glimm, David Harbater, Lizhen Ji,
Jurgen Jost, Wolfgang Luck, Dan Marchesin, William H. Meeks III,
Peter W. Michor, James S. Milne, Frans Oort, Joaquin Perez, Bradley
Plohr, Herman J. J. te Riele, Michael Rubinstein, Norbert
Schappacher, Chi-Wang Shu, Dennis Sullivan, Claire Voisin,
Shing-Tung Yau.
This is the second of two volumes constituting The Legacy of
Bernhard Riemann After One Hundred and Fifty Years. Bernhard
Riemann (1826-1866) possessed an original and broad vision of
mathematics together with powerful skill. His work continues to
influence almost all major branches of mathematics. The
twenty-three papers in the two-volume set examine Riemann, his
work, and his significance in the context of modern mathematical
developments. Contributing authors (to the two-volume set): Michael
F. Atiyah, M. V. Berry, Ching-Li Chai, Brian Conrey, Jean-Pierre
Demailly, F. T. Farrell, James Glimm, David Harbater, Lizhen Ji,
Jurgen Jost, Wolfgang Luck, Dan Marchesin, William H. Meeks III,
Peter W. Michor, James S. Milne, Frans Oort, Joaquin Perez, Bradley
Plohr, Herman J. J. te Riele, Michael Rubinstein, Norbert
Schappacher, Chi-Wang Shu, Dennis Sullivan, Claire Voisin,
Shing-Tung Yau.
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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