zbMATH — the first resource for mathematics

Compactification of Siegel modular varieties with bad reduction. (Compactification de variétés de Siegel aux places de mauvaise réduction.) (English. French summary) Zbl 1203.14048
Summary: 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 G. Faltings and C.-L. Chai [Degeneration of abelian varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 22. Berlin etc.: Springer-Verlag. xii, 316 p. (1990; Zbl 0744.14031)] 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.

14K10 Algebraic moduli of abelian varieties, classification
14G35 Modular and Shimura varieties
Full Text: DOI Link