Compactification de variétés de Siegel aux places de mauvaise réduction

Abstract

Résumé. -Nous construisons des compactifications toroïdales arithmétiques du champ de modules des variétés abéliennes principalement polarisées munies d'une structure de niveau parahorique. Pour ce faire, nous étendons la méthode de Faltings et Chai [7] à un cas de mauvaise réduction. Le voisinage du bord des compactifications obtenues n'est pas lisse, mais a pour singularités celles des champs de modules de variétés abéliennes avec structure parahorique de genre plus petit. Nous sommes amenés à reprendre la construction des compactifications sans niveau de Faltings et Chai, en modifiant l'étape d'approximation pour préserver le groupe de p-torsion des variétés abéliennes. Nous donnons comme application une nouvelle preuve de l'existence du sous-groupe canonique pour des familles de variétés abéliennes. Abstract (Compactification of Siegel modular varieties with bad reduction)We construct arithmetic toroidal compactifications of the moduli stack of principally polarized abelian varieties with parahoric level structure. To this end, we extend the methods of Faltings and Chai [7] to a case of bad reduction. Our compactifications are not smooth near the boundary; the singularities are those of the moduli stacks of abelian varieties with parahoric level structure of lower genus. We modify Faltings and Chai's construction of compactifications without level structure. The key point is that our approximation preserves the p-torsion subgroup of the abelian varieties. As an application, we give a new proof of the existence of the canonical subgroup for some families of abelian varieties.

Compactifications of Pel-Type Shimura Varieties in Ramified Characteristics

“…In fact, these integral models with Iwahoric and pro-p-Iwahoric levels at p have been shown to be normal and Cohen-Macaulay. If we use the constructions in this article instead, then we obtain the same (projective normal) minimal compactifications as in [56] and [57], and sufficiently many (but not all) normal and Cohen-Macaulay toroidal compactifications as in [55] and [57], which admit stratifications and formal local descriptions compatible with those in [10] and [30] in characteristic zero. REMARK 16.5.…”

“…REMARK 16.4. The toroidal and minimal compactifications constructed in [55] and [56] are for the Siegel moduli with parahoric levels at p defined by linear algebraic data that are otherwise split, in which case the naive moduli problems as in Example 13.12 are not naive and define good integral models. The constructions rely crucially on the assertion that the integral models (before compactification) are normal, which is shown there using results of [44] and [12].…”

Complex multiplication cycles and Kudla-Rapoport divisors, II

“…In this section we prove Theorem A of the introduction. The proof uses the same machinery as [10,22,32], and we omit those details that are adequately documented in the literature. We use the following notation throughout Section 2.…”

Compactification minimale et mauvaise réduction

“…L'ingrédient principal de la démonstration du théorème est la construction de compactifications toroïdales de A g,0 [12].…”

“…Dans la première partie, nous expliquons comment déduire l'existence de compactifications toroïdales de A g,n,0 des résultats de [12].…”