By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PELtype 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.
PELtype 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 PELtype Shimura varieties that can be described in sufficient detail near the boundary. This book explains in detail the following topics about PELtype Shimura varieties and their compactifications:
A construction of smooth integral models of PELtype Shimura varieties by defining and representing moduli problems of abelian schemes with PEL structuresAn analysis of the degeneration of abelian varieties with PEL structures into semiabelian schemes, over noetherian normal complete adic base ringsA construction of toroidal and minimal compactifications of smooth integral models of PELtype Shimura varieties, with detailed descriptions of their structure near the boundary
Through 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).

Front Matter Front Matter (pp. iiv) 
Table of Contents Table of Contents (pp. vx) 
Acknowledgments Acknowledgments (pp. xixii) 
Introduction Introduction (pp. xiiixxvi)The main subject of this book is the construction of good compactifications of PELtype Shimura varieties in mixed characteristics. In this introduction, we will summarize the history of such a construction, clarify its aims, and describe what has been achieved (by others and by us). After this, we will also give an outline of the topics in each chapter of the book.
To avoid introducing a heavy load of notation and definitions (which is perhaps typical for this subject), we shall not attempt to give any precise mathematical statement here. (We shall try not to introduce general nonPELtype Shimura varieties...

Chapter One Definition of Moduli Problems Chapter One Definition of Moduli Problems (pp. 190)In this chapter, we give the definition of the moduli problems providing integral models of PELtype Shimura varieties that we will compactify.
Just to make sure that potential logical problems do not arise in our use of categories, we assume that a pertinent choice of auniversehas been made (see Section A.1.1 for more details). This is harmless for our study, and we shall not mention it again in our work.
The main objective in this chapter is to state Definition 1.4.1.4 with justifications. In order to explain the relation between our definition and those in the literature, we...

Chapter Two Representability of Moduli Problems Chapter Two Representability of Moduli Problems (pp. 91142)In this chapter, let us assume the same setting as in Section 1.4. Let us fix a choice of an open compact subgroup
$\mathcal{H} \subset G({\widehat\mathbb{Z}^\square })$ .Our main objective is to prove Theorem 1.4.1.11, with Proposition 2.3.5.2 as a byproduct. Technical results worth noting are Proposition 2.1.6.8 and Corollary 2.2.4.12. The proof of Theorem 1.4.1.11 is carried out by verifying Artin’s criterion in Section 2.3.4 (see, in particular, Theorems B.3.7, B.3.9, and B.3.11). For readers who might have wondered, let us make it clear that we will not need Condition 1.4.3.10 in this chapter.
Let us outline the strategy of...

Chapter Three Structures of SemiAbelian Schemes Chapter Three Structures of SemiAbelian Schemes (pp. 143174)In this chapter, we review notions that are of fundamental importance in the study of degeneration of abelian varieties. The main objective is to understand the statement and the proof of the theory of degeneration data, to be presented in Chapter 4. Our main references for these will be [43], [61], and in particular, [103].
Technical results worth noting are Theorem 3.1.3.3, Propositions 3.1.5.1, 3.3.1.5, 3.3.1.7, and 3.3.3.6, Theorems 3.4.2.4 and 3.4.3.2, Lemma 3.4.3.1, and Proposition 3.4.4.1.
Definition 3.1.1.1 ([43, IX, 1.1]).A group(scheme)of multiplicative type over a scheme
$S$ is a commutative group scheme over$S$ that... 
Chapter Four Theory of Degeneration for Polarized Abelian Schemes Chapter Four Theory of Degeneration for Polarized Abelian Schemes (pp. 175284)In this chapter we reproduce the theory of degeneration data for abelian varieties, following Mumford’s original paper [105] and the first three chapters of Faltings and Chai’s monograph [46]. Although there is essentially nothing new in this chapter, some modifications have been introduced to make the statements compatible with our understanding of the proofs. Moreover, since [105] and [46] have supplied full details only in the completely degenerate case, we will balance the literature by avoiding the special case and treating all cases equally.
The main objective in this chapter will be to state and prove Theorems 4.2.1.14, 4.4.16, and...

Chapter Five Degeneration Data for Additional Structures Chapter Five Degeneration Data for Additional Structures (pp. 285372)In this chapter, we supply a theory of degeneration for abelian varieties with additional structures of PELtype, based on the theory developed in Chapter 4. The running assumptions and notation in Chapter 4 (see 4.1) will be continued in this chapter without further remark. Moreover, we fix some choices of
$B$ ,$O$ , and$(L,\langle \cdot , \cdot \rangle ,h)$ as in Section 1.4.The main objective is to state and prove Theorem 5.3.1.19, with Theorem 5.3.3.1 and the notion of cusp labels in Section 5.4 as byproducts. Technical results worth noting are Propositions 5.1.2.4, 5.2.2.23, 5.2.3.3, and 5.2.3.9, Theorem 5.2.3.14, and Proposition 5.4.3.8. The preparation...

Chapter Six Algebraic Constructions of Toroidal Compactifications Chapter Six Algebraic Constructions of Toroidal Compactifications (pp. 373446)We will generalize the techniques in [46] and construct the toroidal compactifications of the moduli problems we considered in Chapter 1.
The main objective of this chapter is to state and prove Theorem 6.4.1, with byproducts concerning Hecke actions given in Sections 6.4.2 and 6.4.3. Technical results worth noting are Propositions 6.2.2.4, 6.2.3.2, 6.2.3.18, and 6.2.5.18, Lemma 6.3.1.11, and Propositions 6.3.2.6, 6.3.2.10, 6.3.3.5, 6.3.3.13, and 6.3.3.17.
Let
$H$ be a group of multiplicative type of finite type over$S$ , so that its character group$\underline X (H) = {\underline {{\rm{Hom}}} _S}(H,{G_{m,S}})$ is an étale sheaf of finitely generated commutative groups.DEFINITION 6.1.1.1.The cocharacter group
$\underline X {(H)^{\rm{v}}}$ ... 
Chapter Seven Algebraic Constructions of Minimal Compactifications Chapter Seven Algebraic Constructions of Minimal Compactifications (pp. 447486)In this chapter we explain the construction of arithmetic toroidal compactifications and several other useful results as a byproduct of the construction of arithmetic toroidal compactifications.
Although all results to be stated have their analytic analogues over the complex numbers, the analytic techniques in [18] and [17] do not carry over naively. We need the arithmetic toroidal compactifications and the positivity of Hodge invertible sheaves (to be reviewed in Section 7.2.1, based on the theory of theta constants) to establish the finite generation of certain natural sheaves of graded algebras. This should be considered as the main technical input of...

Appendix A Algebraic Spaces and Algebraic Stacks Appendix A Algebraic Spaces and Algebraic Stacks (pp. 487518) 
Appendix B Deformations and Artin’s Criterion Appendix B Deformations and Artin’s Criterion (pp. 519534) 
Bibliography Bibliography (pp. 535544) 
Index Index (pp. 545561)