On reduction of Hilbert-Blumenthal varieties

Soit $O_{𝐅}$ l’anneau des entiers d’un corps de nombres totalement réel $𝐅$ de degré $g$. Nous étudions un nombre premier $p$ fixé, la réduction modulo $p$ de l’espace de modules classifiant les $O_{𝐅}$-variétés abéliennes séparablement polarisées de dimension $g$. Nous construisons une stratification schématique par les types $a$ 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 $p$ n’est pas ramifié dans $O_{𝐅}$. Nous démontrons également la conjecture de Grothendieck forte pour les espaces de modules dans certains cas, notamment lorsque $p$ est totalement ramifié dans $O_{𝐅}$.

Let $O_{𝐅}$ be the ring of integers of a totally real field $𝐅$ of degree $g$. We study the reduction of the moduli space of separably polarized abelian $O_{𝐅}$-varieties of dimension $g$ modulo $p$ for a fixed prime $p$. The invariants and related conditions for the objects in the moduli space are discussed. We construct a scheme-theoretic stratification by $a$-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 $p$ is unramified in $O_{𝐅}$. We also prove the strong Grothendieck conjecture for the moduli space in some restricted cases, particularly when $p$ is totally ramified in $O_{𝐅}$.

[BG] E. Bachmat; E. Z. Goren On the non-ordinary locus in Hilbert-Blumenthal surfaces, Math. Ann., Volume 313 (1999), pp. 475-506 | Article | MR 1678541 | Zbl 0919.14014

[C] C.-L. Chai Newton polygons as lattice points, Amer. J. Math., Volume 122 (2000), pp. 967-990 | MR 1781927 | Zbl 1057.11506

[CN] C.-L. Chai; P. Norman Bad reduction of the Siegel moduli scheme of genus two with ${\Gamma }_0\left(p\right)$-level structure, Amer. J. Math., Volume 112 (1990), pp. 1003-1071 | Article | MR 1081813 | Zbl 0734.14010

[dJO] A.J. de Jong; F. Oort Purity of the stratification by Newton polygons, J. Amer. Math. Soc., Volume 13 (2000), pp. 209-241 | Article | MR 1703336 | Zbl 0954.14007

[DP] P. Deligne; G. Pappas Singularités des espaces de modules de Hilbert en caractéristiques divisant le discriminant, Compositio Math., Volume 90 (1994), pp. 59-79 | Numdam | MR 1266495 | Zbl 0826.14027

[GO] E.Z. Goren; F. Oort Stratifications of Hilbert modular varieties in positive characteristic, J. Algebraic Geom., Volume 9 (2000), pp. 111-154 | MR 1713522 | Zbl 0973.14010

[Gr] A. Grothendieck Groupes de Barsotti-Tate et Cristaux de Dieudonné, Les Presses de l'Université de Montréal, 1974 | MR 417192 | Zbl 0331.14021

[K1] R. E. Kottwitz Isocrystal with additional structure, Compositio Math., Volume 56 (1985), pp. 201-220 | Numdam | MR 809866 | Zbl 0597.20038

[K2] R. E. Kottwitz Points on some Shimura varieties over finite fields, J. Amer. Math. Soc., Volume 5 (1992), pp. 373-444 | Article | MR 1124982 | Zbl 0796.14014

[LO] K.-Z. Li; F. Oort Moduli of Supersingular Abelian Varieties, Lecture Notes in Math, vol. 1680, Springer-Verlag, 1998 | MR 1611305 | Zbl 0920.14021

[Me] W. Messing The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Math, 264, Springer-Verlag, 1972 | MR 347836 | Zbl 0243.14013

[Mu] D. Mumford Abelian Varieties, Oxford University Press, 1974 | Zbl 0223.14022

[N] P. Norman An algorithm for computing moduli of abelian varieties, Ann. Math., Volume 101 (1975), pp. 499-509 | Article | MR 389928 | Zbl 0309.14031

[NO] P. Norman; F. Oort Moduli of abelian varieties, Ann. Math., Volume 112 (1980), pp. 413-439 | Article | MR 595202 | Zbl 0483.14010

[O1] F. Oort Moduli of abelian varieties and Newton polygons, C. R. Acad. Sci. Paris, Sér. I Math., Volume 312 (1991), pp. 385-389 | MR 1096617 | Zbl 0734.14016

[O2] F. Oort Newton polygons and formal groups: conjectures by Manin and Grothendieck, Ann. Math., Volume 152 (2000), pp. 183-206 | Article | MR 1792294 | Zbl 0991.14016

[O3] F. Oort; (ed. by C. Faber, G. van der Geer and F. Oort) Newton Polygon strata in the moduli space of abelian varieties, Moduli of Abelian Varieties (Progress in Mathematics), Volume 195 (2001), pp. 417-440 | Zbl 01644006

[O4] F. Oort Endomorphism algebras of abelian varieties, Algebraic geometry and commutative algebra, in honor of M. Nagata (1988), pp. 469-502 | MR 977774 | Zbl 0697.14029

[P] G. Pappas Arithmetic models for Hilbert modular varieties, Compositio Math., Volume 98 (1995), pp. 43-76 | Numdam | MR 1353285 | Zbl 0890.14010

[R] M. Rapoport Compactifications de l'espaces de modules de Hilbert-Blumenthal, Compositio Math., Volume 36 (1978), pp. 255-335 | Numdam | MR 515050 | Zbl 0386.14006

[RR] M. Rapoport; M. Richartz On the classification and specialization of F-isocrystals with additional structure, Compositio Math., Volume 103 (1996), pp. 153-181 | Numdam | MR 1411570 | Zbl 0874.14008

[RZ] M. Rapoport; Th. Zink Period Spaces for $p$-divisible groups, Ann. Math. Studies, 141, Princeton Univ. Press, 1996 | MR 1393439 | Zbl 0873.14039

[S] J.-P. Serre Local fields, GTM, 67, Springer-Verlag, 1979 | MR 554237 | Zbl 0423.12016

[Y1] C.-F. Yu On the supersingular locus of Hilbert-Blumenthal 4-folds, J. Alg. Geom., Volume 12 (2003), pp. 653-698 | Article | MR 1993760 | Zbl 1059.14031

[Y2] C.-F. Yu Lifting abelian varieties with additional structures, Math. Z., Volume 242 (2002), pp. 427-441 | Article | MR 1985459 | Zbl 1051.14052

[Z1] T. Zink Cartiertheorie kommutativer formaler Gruppen, Teubner-Texte Math., Teubner, Leipzig, 1984 | MR 767090 | Zbl 0578.14039

[Z2] T. Zink The display of a formal $p$-divisible groups, Cohomologies $p$-adiques et applications arithmétiques (Astérisque), Volume 278 (2002), pp. 127-248 | Zbl 1008.14008

[Z3] T. Zink Isogenieklassen von Punkten von Shimuramannigfaltigkeiten mit Werten in einem endlichen Körper, Math. Nachr., Volume 112 (1983), pp. 103-124 | Article | MR 726854 | Zbl 0604.14029