Kai-Wen Lan

Arithmetic compactifications of PEL-type Shimura varieties

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 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).

COMPARISON BETWEEN ANALYTIC AND ALGEBRAIC CONSTRUCTIONS OF TOROIDAL COMPACTIFICATIONS OF PEL-TYPE SHIMURA VARIETIES

Abstract Using explicit identifications between algebraic and analytic theta functions, we compare the algebraic constructions of toroidal compactifications by Faltings–Chai and the author, with the analytic constructions of toroidal compactifications following Ash–Mumford–Rapoport–Tai. As one of the applications, we obtain the corresponding comparison for Fourier–Jacobi expansions of holomorphic automorphic forms.

Vanishing theorems for torsion automorphic sheaves on compact PEL-type Shimura varieties

(1) In Def. 1.1, the parenthetical remark “(If L 6= {0}, then the value of r is uniquely determined by g.)” is incorrect and should be “(This is an abuse of notation, because r is not always determined by g.)” (2) In Lem. 1.20, “an object in W ∈ RepR(M1)” should be “an object W ∈ RepR(M1)”. (3) In the second paragraph of Section 2.1, “simple” should be “indecomposable”, and “every projective O1-module” should be more precisely “every finitely generated projective O1-module”. (4) In the third paragraph in Section 2.4, “reductive group scheme G over Spec(R1)” should be “reductive group scheme G1 over Spec(R1)”. (5) In the paragraphs preceding Def. 2.26 and 2.27, the action of the distribution algebras are redundant. (6) In the two paragraphes after Def. 2.29, we should only define the split objects over Z(p), but not Z, to avoid saying that split (even) orthogonal groups are reductive at 2. And, again, the action of distribution algebras are redundant. Moreover, “minimal among admissible lattices” should be more precisely only “minimal among the admissible lattices containing the same highest weight vector”. (7) In Section 4.1, when defining and studying the Hodge filtration on HdR(A /MH,1), all instances of “Ω • A/S1 ” should be “ΩAn/MH,1”. (8) The assertion in Prop. 7.10 that “W ν has trivial tensor square as a line bundle over MH,1 if its coefficients (kτ )τ∈Υ of ν satisfy kτ + kτ◦c = 0” is too strong. The correct assertion is that “W ν defines a torsion element in the Picard group of MH,1 if its coefficients (kτ )τ∈Υ of ν satisfy the condition that kτ + kτ◦c = 0”. The simplest proof is to use the complex fiber of MH, which we spell out as follows, for the convenience of the reader:

Compactifications of PEL-type Shimura varieties and Kuga families with ordinary loci

With applications towards the constructions of overconvergent cusp forms and Galois representations in mind, we construct projective normal flat p-integral models of various algebraic compactifications of PEL-type Shimura varieties and Kuga families, allowing both ramification and levels at p, such that, along the ordinary loci where certain canonical subgroups can be defined, the partial compactifications behave almost exactly as in the good reduction case.

Nearby Cycles of Automorphic Étale Sheaves, II

We review some recent results of ours on the nearby cycles of automorphic etale sheaves, and record some improvements of the arguments.