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.