By studying the degeneration of abelian varieties with PEL structures,
this book explains the compactifications of smooth integral models of
all PEL-type Shimura varieties, providing the logical foundation for
several exciting recent developments. The book is designed to be
accessible to graduate students who have an understanding of schemes and
abelian varieties. PEL-type Shimura varieties, which are natural
generalizations of modular curves, are useful for studying the
arithmetic properties of automorphic forms and automorphic
representations, and they have played important roles in the development
of the Langlands program. As with modular curves, it is desirable to
have integral models of compactifications of PEL-type Shimura varieties
that can be described in sufficient detail near the boundary. This book
explains in detail the following topics about PEL-type Shimura varieties
and their compactifications: A construction of smooth integral models of
PEL-type Shimura varieties
by defining and representing moduli problems of abelian schemes with PEL
structures An analysis of the degeneration of abelian varieties with PEL
structures into semiabelian schemes, over noetherian normal complete
adic base rings A construction of toroidal and minimal compactifications
of smooth integral models of PEL-type Shimura varieties, with detailed
descriptions of their structure near the boundaryThrough these topics,
the book generalizes the theory of degenerations of polarized abelian
varieties and the application of that theory to the construction of
toroidal and minimal compactifications of Siegel moduli schemes over the
integers (as developed by Mumford, Faltings, and Chai).
