Let be the ring of integers of a totally real field of degree . We study the reduction of the moduli space of separably polarized abelian -varieties of dimension modulo for a fixed prime . The invariants and related conditions for the objects in the moduli space are discussed. We construct a scheme-theoretic stratification by -types on the Rapoport locus and study the relation with the slope stratification. In particular, we recover the main results of Goren and Oort [J. Alg. Geom., 2000] on the stratifications when is unramified in . We also prove the strong Grothendieck conjecture for the moduli space in some restricted cases, particularly when is totally ramified in .
Soit l’anneau des entiers d’un corps de nombres totalement réel de degré . Nous étudions un nombre premier fixé, la réduction modulo de l’espace de modules classifiant les -variétés abéliennes séparablement polarisées de dimension . Nous construisons une stratification schématique par les types du lieu de Rapoport et étudions sa relation avec la stratification par les pentes. En particulier, nous retrouvons les résultats principaux de Goren et Oort [J. Alg. Geom., 2000] sur les stratifications lorsque n’est pas ramifié dans . Nous démontrons également la conjecture de Grothendieck forte pour les espaces de modules dans certains cas, notamment lorsque est totalement ramifié dans .
Keywords: Hilbert-Blumenthal varieties, Dieudonné modules, stratifications, deformations
Mot clés : variétés de Hilbert-Blumenthal, modules de Dieudonné, stratifications, déformations
@article{AIF_2003__53_7_2105_0, author = {Yu, Chia-Fu}, title = {On reduction of {Hilbert-Blumenthal} varieties}, journal = {Annales de l'Institut Fourier}, pages = {2105--2154}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {53}, number = {7}, year = {2003}, doi = {10.5802/aif.2002}, mrnumber = {2044169}, zbl = {02093468}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2002/} }
TY - JOUR AU - Yu, Chia-Fu TI - On reduction of Hilbert-Blumenthal varieties JO - Annales de l'Institut Fourier PY - 2003 SP - 2105 EP - 2154 VL - 53 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2002/ DO - 10.5802/aif.2002 LA - en ID - AIF_2003__53_7_2105_0 ER -
Yu, Chia-Fu. On reduction of Hilbert-Blumenthal varieties. Annales de l'Institut Fourier, Volume 53 (2003) no. 7, pp. 2105-2154. doi : 10.5802/aif.2002. http://www.numdam.org/articles/10.5802/aif.2002/
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