Compactifications of PEL-Type Shimura Varieties and Kuga Families with Ordinary Loci

The following sections are included:

• Preface
• Contents

The following sections are included:

• Background and Aim
• Overview
• Outline of the Constructions
• What is Known, What is New, and What Can Be Studied Next
• What to Note and to Skip in Special Cases
• Notation and Conventions
• Acknowledgements

In this chapter, we review the main definitions and results in [62] and [61] (based on earlier results of others) and specialize them to the case of characteristic zero bases (over the reflex fields). (Also, we take this opportunity to correct or improve certain assertions in [62] and [61].) Readers who are already familiar with these results should feel free to skip this chapter (and return to here only for references).

From now on, let us fix a choice of a rational prime number p > 0.

In this chapter, we introduce the notions of ordinary (semi-)abelian schemes and ordinary level structures, define the moduli problems parameterizing them, and construct the ordinary loci by normalizing these moduli problems after suitable base changes. The terminology of ordinary loci is, admittedly, an abuse of language. Nevertheless, these ordinary loci can be embedded into (normalizations of suitable base changes) of the at integral models constructed in Chapter 2 (which we view as the total models). The main point is that, while we cannot describe the local structures of the total models in detail, we can describe the local structures of these ordinary loci rather precisely, because these ordinary loci are constructed as normalizations of moduli problems with explicit and mild singularities.

In this chapter, we explain how to incorporate the considerations of ordinary level structures into the theory of degeneration data and the boundary construction in [62]. (We no longer assume as in Section 3.4.5 that p is a good prime.) This is the technical heart of the whole work. Readers are encouraged to read this chapter only after mastering the earlier results in [82], [28, Ch. II–IV], and [62, Ch. 4–6]. Although the notation is quite heavy, it is designed to be as close as possible to the one in [62, Ch. 4–6], so that readers who are already familiar with the arguments there can easily see what the new considerations here are.

The goal of this chapter is to construct the partial toroidal compactifications for the ordinary loci defined in Chapter 3, based on the theory of degeneration and the construction of boundary charts given in Chapter 4.

The first goal of this chapter is to construct the partial minimal compactifications for the ordinary loci defined in Chapter 3, based on the partial toroidal compactifications constructed in Chapter 5 and on the total minimal compactifications constructed in Chapter 2 (which is in turn based on the projective minimal compactifications constructed in [62] in the good reduction case, for the auxiliary models). The second goal is to show that the partial toroidal compactifications are quasi-projective when the levels are neat away from p and when the compatible choices of smooth admissible cone decompositions are projective. The third goal is to show that the reductions of the partial minimal compactifications modulo powers of p are affine. These are all indispensable for the application of our work to the construction of p-adic modular forms as in, for example, [39].

In this chapter, we continue to assume the same settings as in Section 5.2. Our main goal is to generalize the results in Section 1.3.3 to the context of ordinary loci. We will first introduce some analogues of the definitions and results in Section 1.3.3, and explain how the proofs in [61] can be translated into this context.

In this chapter, we explain the construction of automorphic bundles and their canonical extensions over the partial and total toroidal compactifications defined in Chapters 5 and 2, respectively. These generalize the constructions in characteristic zero in Section 1.4.

The following sections are included:

• Bibliography
• Index