Alcôves et p-rang des variétés abéliennes  [ Alcoves and p-rank of abelian varieties ]
Annales de l'Institut Fourier, Volume 52 (2002) no. 6, p. 1665-1680
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.
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.
DOI : https://doi.org/10.5802/aif.1930
Classification:  14K10,  20G05
Keywords: abelian varieties, p-rank, local models, alcoves
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {6},
     year = {2002},
     pages = {1665-1680},
     doi = {10.5802/aif.1930},
     mrnumber = {1952527},
     zbl = {1046.14023},
     language = {fr},
     url = {https://aif.centre-mersenne.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, Volume 52 (2002) no. 6, pp. 1665-1680. doi : 10.5802/aif.1930. https://aif.centre-mersenne.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, Tome 320 (1995) no. 3, pp. 335-340 | MR 1320381 | Zbl 0852.13005

[2] G. Faltings; C.-L. Chai Degeneration of abelian varieties, Springer-Verlag, Erb. Math (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, Tome 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, math.RT/0103048, e-print. À paraître dans Amer. M. Journal

[8] J. Humphreys Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, Tome 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, Tome 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, Tome 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, Tome 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), Tome 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, Princeton University Press, Princeton, NJ, Annals of Mathematics Studies, Tome 141 (1996) | MR 1393439 | Zbl 0873.14039