Alcôves et p-rang des variétés abéliennes
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.

DOI : https://doi.org/10.5802/aif.1930
Classification : 14K10,  20G05
Mots clés : variétés abéliennes, p-rang, modèles locaux, alcôves
@article{AIF_2002__52_6_1665_0,
     author = {Ng\^o, B\h ao Ch\^au and Genestier, Alain},
     title = {Alc\^oves et $p$-rang des vari\'et\'es ab\'eliennes},
     journal = {Annales de l'Institut Fourier},
     pages = {1665--1680},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {6},
     year = {2002},
     doi = {10.5802/aif.1930},
     zbl = {1046.14023},
     mrnumber = {1952527},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2002__52_6_1665_0/}
}
Ngô, Bao Chau; Genestier, Alain. Alcôves et $p$-rang des variétés abéliennes. Annales de l'Institut Fourier, Tome 52 (2002) no. 6, pp. 1665-1680. doi : 10.5802/aif.1930. http://www.numdam.org/item/AIF_2002__52_6_1665_0/

[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 1320381 | Zbl 0852.13005

[2] G. Faltings; C.-L. Chai Degeneration of abelian varieties, Erb. Math, Springer-Verlag, 1990 | MR 1083353 | Zbl 0744.14031

[3] A. Genestier Un modèle semi-stable de la variété de Siegel de genre 3 avec structures de niveau de type Γ 0 (p), Compositio Math, Volume 123 (2000) no. 3, pp. 303-328 | Article | MR 1795293 | Zbl 0974.11029

[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 | Article | MR 1810365 | Zbl 1014.20002

[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 1009.11042

[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 1066460 | Zbl 0768.20016

[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 | Article | Numdam | MR 185016 | Zbl 0228.20015

[10] A. de Jong The moduli spaces of principally polarized abelian varieties with Γ 0 ( p ) -level structure, J. Algebraic Geom, Volume 2 (1993) no. 4, pp. 667-688 | MR 1227472 | Zbl 0816.14020

[11] R. Kottwitz; M. Rapoport Minuscule alcoves for GL n and G Sp 2 n , Manuscripta Math, Volume 102 (2000) no. 4, pp. 403-428 | MR 1785323 | Zbl 0981.17003

[12] P. Norman; F. Oort Moduli of abelian varieties, Ann. of Math (2), Volume 112 (1980), pp. 419-439 | MR 595202 | Zbl 0483.14010

[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 1393439 | Zbl 0873.14039