On reduction of Hilbert-Blumenthal varieties

Chia-Fu Yu[1]

  • [1] National Tsing-Hua University, National Center for Theoretical Sciences, 3rd General Bldg, 101 Sec. Kuang-Fu road, Tsinchu 30043 (Taiwan)

Annales de l'Institut Fourier (2003)

  • Volume: 53, Issue: 7, page 2105-2154
  • ISSN: 0373-0956

Abstract

top
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 .

How to cite

top

Yu, Chia-Fu. "On reduction of Hilbert-Blumenthal varieties." Annales de l'Institut Fourier 53.7 (2003): 2105-2154. <http://eudml.org/doc/116095>.

@article{Yu2003,
abstract = {Let $O_\{\{\bf F\}\}$ be the ring of integers of a totally real field $\{\bf F\}$ of degree $g$. We study the reduction of the moduli space of separably polarized abelian $O_\{\{\bf F\}\}$-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 $O_\{\{\bf F\}\}$. We also prove the strong Grothendieck conjecture for the moduli space in some restricted cases, particularly when $p$ is totally ramified in $O_\{\{\bf F\}\}$.},
affiliation = {National Tsing-Hua University, National Center for Theoretical Sciences, 3rd General Bldg, 101 Sec. Kuang-Fu road, Tsinchu 30043 (Taiwan)},
author = {Yu, Chia-Fu},
journal = {Annales de l'Institut Fourier},
keywords = {Hilbert-Blumenthal varieties; Dieudonné modules; stratifications; deformations},
language = {eng},
number = {7},
pages = {2105-2154},
publisher = {Association des Annales de l'Institut Fourier},
title = {On reduction of Hilbert-Blumenthal varieties},
url = {http://eudml.org/doc/116095},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Yu, Chia-Fu
TI - On reduction of Hilbert-Blumenthal varieties
JO - Annales de l'Institut Fourier
PY - 2003
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 7
SP - 2105
EP - 2154
AB - Let $O_{{\bf F}}$ be the ring of integers of a totally real field ${\bf F}$ of degree $g$. We study the reduction of the moduli space of separably polarized abelian $O_{{\bf F}}$-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 $O_{{\bf F}}$. We also prove the strong Grothendieck conjecture for the moduli space in some restricted cases, particularly when $p$ is totally ramified in $O_{{\bf F}}$.
LA - eng
KW - Hilbert-Blumenthal varieties; Dieudonné modules; stratifications; deformations
UR - http://eudml.org/doc/116095
ER -

References

top
  1. E. Bachmat, E. Z. Goren, On the non-ordinary locus in Hilbert-Blumenthal surfaces, Math. Ann. 313 (1999), 475-506 Zbl0919.14014MR1678541
  2. C.-L. Chai, Newton polygons as lattice points, Amer. J. Math. 122 (2000), 967-990 Zbl1057.11506MR1781927
  3. C.-L. Chai, P. Norman, Bad reduction of the Siegel moduli scheme of genus two with Γ 0 ( p ) -level structure, Amer. J. Math. 112 (1990), 1003-1071 Zbl0734.14010MR1081813
  4. A.J. de Jong, F. Oort, Purity of the stratification by Newton polygons, J. Amer. Math. Soc. 13 (2000), 209-241 Zbl0954.14007MR1703336
  5. P. Deligne, G. Pappas, Singularités des espaces de modules de Hilbert en caractéristiques divisant le discriminant, Compositio Math. 90 (1994), 59-79 Zbl0826.14027MR1266495
  6. E.Z. Goren, F. Oort, Stratifications of Hilbert modular varieties in positive characteristic, J. Algebraic Geom. 9 (2000), 111-154 Zbl0973.14010MR1713522
  7. A. Grothendieck, Groupes de Barsotti-Tate et Cristaux de Dieudonné, (1974), Les Presses de l'Université de Montréal Zbl0331.14021MR417192
  8. R. E. Kottwitz, Isocrystal with additional structure, Compositio Math. 56 (1985), 201-220 Zbl0597.20038MR809866
  9. R. E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), 373-444 Zbl0796.14014MR1124982
  10. K.-Z. Li, F. Oort, Moduli of Supersingular Abelian Varieties, vol. 1680 (1998), Springer-Verlag Zbl0920.14021MR1611305
  11. W. Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, 264 (1972), Springer-Verlag Zbl0243.14013MR347836
  12. D. Mumford, Abelian Varieties, (1974), Oxford University Press Zbl0223.14022
  13. P. Norman, An algorithm for computing moduli of abelian varieties, Ann. Math. 101 (1975), 499-509 Zbl0309.14031MR389928
  14. P. Norman, F. Oort, Moduli of abelian varieties, Ann. Math. 112 (1980), 413-439 Zbl0483.14010MR595202
  15. F. Oort, Moduli of abelian varieties and Newton polygons, C. R. Acad. Sci. Paris, Sér. I Math. 312 (1991), 385-389 Zbl0734.14016MR1096617
  16. F. Oort, Newton polygons and formal groups: conjectures by Manin and Grothendieck, Ann. Math. 152 (2000), 183-206 Zbl0991.14016MR1792294
  17. F. Oort, Newton Polygon strata in the moduli space of abelian varieties, Moduli of Abelian Varieties 195 (2001), 417-440, Birkhäuser Zbl1086.14037
  18. F. Oort, Endomorphism algebras of abelian varieties, Algebraic geometry and commutative algebra, in honor of M. Nagata (1988), 469-502 Zbl0697.14029MR977774
  19. G. Pappas, Arithmetic models for Hilbert modular varieties, Compositio Math. 98 (1995), 43-76 Zbl0890.14010MR1353285
  20. M. Rapoport, Compactifications de l'espaces de modules de Hilbert-Blumenthal, Compositio Math. 36 (1978), 255-335 Zbl0386.14006MR515050
  21. M. Rapoport, M. Richartz, On the classification and specialization of F-isocrystals with additional structure, Compositio Math. 103 (1996), 153-181 Zbl0874.14008MR1411570
  22. M. Rapoport, Th. Zink, Period Spaces for p -divisible groups, 141 (1996), Princeton Univ. Press Zbl0873.14039MR1393439
  23. J.-P. Serre, Local fields, 67 (1979), Springer-Verlag Zbl0423.12016MR554237
  24. C.-F. Yu, On the supersingular locus of Hilbert-Blumenthal 4-folds, J. Alg. Geom. 12 (2003), 653-698 Zbl1059.14031MR1993760
  25. C.-F. Yu, Lifting abelian varieties with additional structures, Math. Z. 242 (2002), 427-441 Zbl1051.14052MR1985459
  26. T. Zink, Cartiertheorie kommutativer formaler Gruppen, (1984), Teubner, Leipzig Zbl0578.14039MR767090
  27. T. Zink, The display of a formal p -divisible groups, Cohomologies -adiques et applications arithmétiques 278 (2002), 127-248, SMF Zbl1008.14008
  28. T. Zink, Isogenieklassen von Punkten von Shimuramannigfaltigkeiten mit Werten in einem endlichen Körper, Math. Nachr. 112 (1983), 103-124 Zbl0604.14029MR726854

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.