Alcôves et $p$-rang des variétés abéliennes

Ngô, Bao Chau; Genestier, Alain

Annales de l'Institut Fourier, Tome 52 (2002) no. 6, pp. 1665-1680.

On étudie la relation entre le $p$-rang des variétés abéliennes en caractéristique $p$ et la stratification de Kottwitz-Rapoport de la fibre spéciale en $p$ de l’espace de module des variétés abéliennes principalement polarisées avec structure de niveau de type Iwahori en $p$. En particulier, on démontre la densité du lieu ordinaire dans cette fibre spéciale.

We study the relation between the $p$-rank of abelian varieties in characteristic $p$ and the Kottwitz-Rapoport’s stratification of the special fiber modulo $p$ of the moduli space of principally polarized abelian varieties with Iwahori type level structure on $p$. In particular, the density of the ordinary locus in that special fiber is proved.

[1] A. Beauville; Y. Laszlo Un lemme de descente, C. R. Acad. Sci. Paris, Sér. I Math, Volume 320 (1995) no. 3, pp. 335-340 | MR | Zbl

[2] G. Faltings; C.-L. Chai Degeneration of abelian varieties, Erb. Math, Springer-Verlag, 1990 | MR | Zbl

[3] A. Genestier Un modèle semi-stable de la variété de Siegel de genre 3 avec structures de niveau de type ${\Gamma }_0\left(p\right)$, Compositio Math, Volume 123 (2000) no. 3, pp. 303-328 | DOI | MR | Zbl

[4] U. Goertz On the flatness of local models for the symplectic group (e-print, math.AG/0011202)

[5] T. Haines Test functions for Shimura varieties: The Drinfeld case, Duke Math. J (2001), pp. 19-40 | DOI | MR | Zbl

[6] T. Haines; B.C. Ngô Nearby cycles on local models of some Shimura varieties (E-print. À paraître dans Compo. Math., math.AG/0103047) | Zbl

[7] T. Haines; B.C. Ngô Alcoves associated to special fibers of local models, e-print. À paraître dans Amer. M. Journal, math.RT/0103048

[8] J. Humphreys Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, Volume 29 | MR | Zbl

[9] N. Iwahori; H. Matsumoto On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups, Inst. Hautes études Sci. Publ. Math, Volume 25 (1965), pp. 5-48 | DOI | Numdam | MR | Zbl

[10] A. de Jong The moduli spaces of principally polarized abelian varieties with ${\Gamma }_0\left(p\right)$-level structure, J. Algebraic Geom, Volume 2 (1993) no. 4, pp. 667-688 | MR | Zbl

[11] R. Kottwitz; M. Rapoport Minuscule alcoves for ${\mathrm{GL}}_n$ and $G{\mathrm{Sp}}_{2n}$, Manuscripta Math, Volume 102 (2000) no. 4, pp. 403-428 | MR | Zbl

[12] P. Norman; F. Oort Moduli of abelian varieties, Ann. of Math (2), Volume 112 (1980), pp. 419-439 | MR | Zbl

[13] M. Rapoport Communication privée (mars 2001)

[14] M. Rapoport; T. Zink Period spaces for $p$-divisible groups, Annals of Mathematics Studies, 141, Princeton University Press, Princeton, NJ, 1996 | MR | Zbl