Compactifications of Pel-Type Shimura Varieties in Ramified Characteristics

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.

l-Adic étale cohomology of Shimura varieties of Hodge type with non-trivial coefficients

“…Following the reasoning of the proof of Mantovan's formula for PEL Shimura varieties in their paper, we see that it suffices to prove the isomorphy (respectively, the existence of good compatifications) for the more standard choice of the integral model given by the relative normalisation of the integral model of the Shimura variety with parahoric level structure (instead of the integral models constructed via Drinfeld level structure as in Mantovan's papers [Man04,Man05]). This was done in the PEL case in [Lan16] (see also [LS18a,Prop. 2…”

Compactifications of Subschemes of Integral Models of Shimura Varieties

“…For a long time, it was mainly the good reduction integral models or some parahoric variants of them which had been considered at all (see the introductions of [36,78], and [79]). Nevertheless, in recent works by Madapusi Pera (see [50]) and us (with more elementary arguments; see [38,41], and [43], and also [1] and [70]), a general principle has emerged-the difficulties in the construction of compactifications and in the construction of normal integral models with nice local properties are essentially disjoint from each other.…”

“…of a moduli M H → S 0 = Spec(F 0 ) at a neat level H ⊂ G(Ẑ) (essentially the same as above, but with P = ∅) defined by taking normalizations over certain auxiliary good reduction models as in [38,Section 6] (which allow bad reductions due to arbitrarily high levels, ramifications, polarization degrees, and collections of isogenies). (In this case, we also allow F 0 to be a finite extension of the reflex field, with M H and others replaced with their pullbacks.)…”

“…(In this case, we also allow F 0 to be a finite extension of the reflex field, with M H and others replaced with their pullbacks.) For simplicity, we shall assume that, in the choice of the collection {(g j , L j , • , • j )} j∈J in [38,Section 2], we have g j = 1 for all j ∈ J and (L j 0 , • , • j 0 ) = ( p r 0 L , p −2r 0 • , • ) for some j 0 ∈ J and some r 0 ∈ Z. (This simplifying assumption imposes no restriction on the integral models we consider.)…”

Level structures on abelian varieties and Vojta’s conjecture

“…The purpose of this appendix is to lay out certain facts about toroidal compactifications of the moduli of principally polarized abelian varieties with full level structure at 'bad' primes. This is a straight-forward extension of the theory of [3], and could possibly also be extracted from the work of K.-W. Lan [10]. The stack A g,m has no obvious moduli interpretation over Z, and we know little about the singularities of its fibers over primes dividing m. However, this is not an obstruction to studying its general structure at the boundary.…”