Abstract
We show that, by taking normalizations over certain auxiliary good reduction integral models, one obtains integral models of toroidal and minimal compactifications of PEL-type Shimura varieties which enjoy many features of the good reduction theory studied as in the earlier works of Faltings and Chai’s and the author’s. We treat all PEL-type cases uniformly, with no assumption on the level, ramifications, and residue characteristics involved.
Highlights
In recent years, we have witnessed a rapid development in the arithmetic applications of noncompact Shimura varieties, in which the integral models of toroidal and minimal compactifications have played important roles
Let us compare the results obtained in this article with the main results in [30] in the good reduction case
Compared with [30, Theorem 6.4.1.1], which is the main result on integral models of toroidal compactifications in the good reduction case, the results obtained in this article achieved the following
Summary
We have witnessed a rapid development in the arithmetic applications of noncompact Shimura varieties, in which the integral models of toroidal and minimal compactifications have played important roles. For the sake of completeness, we consider collections of lattices twisted by group actions, which define moduli problems for collections of abelian schemes with PEL structures related to each other by Q×-isogenies (that is, quasiisogenies; see [30, Definitions 1.3.1.16 and 1.3.1.17]) (our theory applies, in particular, to the parahoric setting in [51] and in later works built on it). A = Aj0 → Aj1 → · · · → Ajm → A (whose composition is the multiplication by p on A) satisfying certain additional conditions, and extends to a moduli problem over S0 := Spec(OF0,(p)) given by the moduli scheme of chains of isogenies between abelian schemes (with additional PEL structures) as in [51] and later works built on it In this case, we will study (mixed characteristics) degenerations of such chains of isogenies. The key points are already novel under these two simplifying assumptions
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