Elevators for Degenerations of Pel Structures

Abstract

We show that the maximal rank of mixed characteristic degenerations of abelian varieties parameterized by a PEL-type Shimura variety are the same as the maximal rank of equicharacteristic zero degenerations (of abelian varieties parameterized by the same Shimura variety). As a byproduct, we obtain a simple proof of Yasuo Morita's conjecture in 1975 that an abelian variety with additional structures parameterized by a compact PEL-type Shimura variety has potential good reductions everywhere.

Compactifications of Pel-Type Shimura Varieties in Ramified Characteristics

“…REMARK 12.6. Corollary 12.5 can also be proved by constructing elevators as in [27], which can be viewed as a minimalistic analog of the boundary charts constructed in Section 8. See the proof of [31, Proposition 2.2.1.7] for the special case where J = {j 0 } and …”

A proof of a conjecture of Y. Morita

“…If G A has no Q-rational unipotent element other than 1, then A has potentially good reduction at every finite prime of E. This was first conjectured by Morita [12] in the PEL-type setting (that is, (G A , {h A }) defines a PEL-type Shimura datum in the sense of Kottwitz [9]), and proved by himself in certain cases; consequently, we refer to this conjecture as the Morita conjecture from now on. Recently, Noot [13], Paugam [15], Vasiu [24] and Lan [10] proved the conjecture in more general situations. Paugam, among other things, generalized the well-known criterion on good reduction of abelian varieties over local fields of residue characteristic p in terms of the Galois representation on the l( = p)-adic Tate module to the p-adic case.…”

“…54-57], in particular the relation between the absolute rank n and the relative rank r, or [24, Remark 2.3.2]). Therefore, the Shimura datum (H A , X A ) is of PEL-type [23,Corollary 4.10], and this case is covered by a recent work of Lan [10].…”

Nearby cycles of automorphic étale sheaves

“…The crucial test function introduced by Scholze in [Sch13, Definition 4.1] is defined using the cohomology of certain deformation spaces (of Barsotti–Tate groups with additional structures) constructed in [Sch13, §3], which depends only on the data at and is local in nature. On the contrary, the properness of depends on the nonexistence of proper rational parabolic subgroups of (see the discussions in [Lan13, §5.3.3], [Lan11, §4.2], and [Lan15a]), which is global in nature. This convinced us that it is reasonable to consider the generalization of [Sch13, Theorem 5.7] to the nonproper case.…”