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On Goren–Oort stratification for quaternionic Shimura varieties

  • Yichao Tian (a1) and Liang Xiao (a2)


Let $F$ be a totally real field in which a prime $p$ is unramified. We define the Goren–Oort stratification of the characteristic- $p$ fiber of a quaternionic Shimura variety of maximal level at $p$ . We show that each stratum is a $(\mathbb{P}^{1})^{r}$ -bundle over other quaternionic Shimura varieties (for an appropriate integer $r$ ). As an application, we give a necessary condition for the ampleness of a modular line bundle on a quaternionic Shimura variety in characteristic $p$ .



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[AG04] Andreatta, F. and Goren, E., Hilbert modular varieties of low dimension , in Geometric aspects of Dwork theory I (De Gruyter, 2004), 113175.
[BG99] Bachmat, E. and Goren, E. Z., On the non-ordinary locus in Hilbert–Blumenthal surfaces , Math. Ann. 313 (1999), 475506.
[BBM82] Berthelot, P., Breen, L. and Messing, W., Théorie de Dieudonné cristalline II, Lecture Notes in Mathematics, vol. 930 (Springer, 1982).
[Bor91] Borel, A., Linear algebraic groups, Graduate Texts in Mathematics, vol. 126 (Springer, New York, 1991).
[Car86] Carayol, H., Sur la mauvaise réduction des courbes de Shimura , Compositio Math. 59 (1986), 151230.
[Del71] Deligne, P., Travaux de Shimura , in Séminaire Bourbaki, 23ème année (1970/71), Exp. No. 389, Lecture Notes in Mathematics, vol. 244 (Springer, Berlin, 1971), 123165.
[Del79] Deligne, P., Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques , in Automorphic forms, representations and L-functions, Proceedings of Symposia in Pure Mathematics, vol. XXXIII (American Mathematical Society, Providence, RI, 1979), 247289.
[EGAIV] Grothendieck, A., Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné): IV. Étude locale des schémas et des morphismes de schémas, Troisième partie , Publ. Math. Inst. Hautes Études Sci. 28 (1966), 5255.
[GO00] Goren, E. and Oort, F., Stratifications of Hilbert modular varieties , J. Algebraic Geom. 9 (2000), 111154.
[Gro74] Grothendieck, A., Groupes de Barsotti–Tate et cristaux de Dieudonné, Séminaire de Mathématiques Supérior, vol. 45 (Presses de l’Université de Montréal, 1974).
[Hel10] Helm, D., Towards a geometric Jacquet–Langlands correspondence for unitary Shimura varieties , Duke Math. J. 155 (2010), 483518.
[Hel12] Helm, D., A geometric Jacquet–Langlands correspondence for U (2) Shimura varieties , Israel J. Math. 187 (2012), 3780.
[Hid04] Hida, H., p-adic automorphic forms on Shimura varieties (Springer, 2004).
[Kis10] Kisin, M., Integral models for Shimura varieties of abelian type , J. Amer. Math. Soc. 23 (2010), 9671012.
[Kot92] Kottwitz, R., Points on some Shimura varieties over finite fields , J. Amer. Math. Soc. 5 (1992), 373444.
[Lan13] Lan, K.-W., Arithmetic compactifications of PEL-type Shimura varieties, London Mathematical Society Monographs, vol. 36 (Princeton University Press, Princeton, NJ, 2013).
[Mat86] Matsumura, H., Commutative ring theory (Cambridge University Press, 1986), translated by M. Reid.
[MM74] Mazur, B. and Messing, W., Universal extensions and one-dimensional crystalline cohomology, Lecture Notes in Mathematics, vol. 370 (Springer, 1974).
[Mil90] Milne, J., Canonical models of (mixed) Shimura varieties and automorphic vector bundles , in Shimura varieties and L-functions, I (Academic Press, New York, 1990), 284414.
[Mil05] Milne, J., Introduction to Shimura varieties , in Harmonic analysis, the trace formula, and Shimura varieties, Clay Mathematics Proceedings, vol. 4 (American Mathematical Society, Providence, RI, 2005), 265378.
[Min11] Minguez, A., Unramified representations of unitary groups , in On the stabilization of the trace formula (International Press, 2011), 389410.
[Moo98] Moonen, B., Models of Shimura varieties in mixed characteristics , in Galois representations in arithmetic algebraic geometry (Durham, 1996), London Mathematical Society Lecture Note Series, vol. 254 (Cambridge University Press, 1998), 267350.
[Rap78] Rapoport, M., Compactification de l’espace de modules de Hilbert–Blumenthal , Compositio Math. 36 (1978), 255335.
[RZ96] Rapoport, M. and Zink, T., Period spaces for p-divisible groups, Annals of Mathematics Studies, vol. 141 (Princeton University Press), 1996.
[Sai09] Saito, T., Hilbert modular forms and p-adic Hodge theory , Compositio Math. 145 (2009), 10811113.
[Ser96] Serre, J. P., Two letters on quaternions and modular forms (mod p) , Israel J. Math. 95 (1996), 281299.
[SGA5] Grothendieck, A., Séminaire de géométrie algébrique du Bois Marie 1965–1966, SGA 5 , in Cohomologie -adic et fonctions L , Lecture Notes in Mathematics, vol. 589 (Springer, 1977) (avec la collaboration de I. Bucur, C. Houzel, L. Illusie, J. P. Jouanolou et J. P. Serre).
[TX13] Tian, Y. and Xiao, L., $p$ -adic cohomology and classicality of overconvergent Hilbert modular forms, Astérisque, to appear. Preprint (2013), arXiv:1308.0779.
[TX14] Tian, Y. and Xiao, L., Tate cycles on quaternionic Shimura varieties over finite fields, Preprint (2014), arXiv:1410.2321.
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On Goren–Oort stratification for quaternionic Shimura varieties

  • Yichao Tian (a1) and Liang Xiao (a2)


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