On the supersingular locus in Hilbert-Blumenthal $4$-folds
Author:
Chia-Fu Yu
Journal:
J. Algebraic Geom. 12 (2003), 653-698
DOI:
https://doi.org/10.1090/S1056-3911-03-00352-7
Published electronically:
July 9, 2003
MathSciNet review:
1993760
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References |
Additional Information
Abstract: We study the supersingular locus of Hilbert-Blumenthal four-folds modulo $p$ when $p$ is inert in the totally real field. The dimension, local moduli spaces, number of the irreducible components, and a description of intersections of these components are given. We also show that each irreducible component is a smooth algebraic stack which is a quotient of a ruled surface over ${\mathbf P}^1$ by a finite group.
beauville A. Beauville, Complex algebraic surfaces. Translated from the French by R. Barlow, N.I. Shepherd-Barron and M. Reid. London Mathematical Society Lecture Note Series 68, Cambridge University Press, Cambridge-New York, 1983.
breen:dieu L. Breen, Rapport sur la théorie de Dieudonné. Journées de Géométrie Algébrique de Rennes (1978), Vol. I, pp. 39–66, Astérisque, no. 63. Soc. Math. France, Paris, 1979.
chai:np C.-L. Chai, Newton polygons as lattice points. Amer. J. Math. 122 (2000), 967–990.
chai-norman:gamma C.-L. Chai and P. Norman, Bad reduction of the Siegel moduli scheme of genus two with $\Gamma _0(p)$-level structure. Amer. J. Math. 112 (1990), 1003–1071.
chai-norman:gamma2 C.-L. Chai and P. Norman, Singularities of the $\Gamma _0(p)$-level structure. J. Algebraic Geom. 1 (1992), 251–178.
dejong-oort:purity A.J. de Jong and F. Oort, Purity of the stratification by Newton polygons. J. Amer. Math. Soc. 13 (2000), 209–241.
deligne-pappas P. Deligne and G. Pappas, Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant. Compositio Math. 90 (1994), 59–79.
ekedahl:ss T. Ekedahl, On supersingular curves and supersingular abelian varieties. Math. Scand. 60 (1987), 151–178.
ekedahl-oort T. Ekedahl and F. Oort, Connected subsets of a moduli space of abelian varieties. (to appear).
goren-oort E.Z. Goren and F. Oort, Stratifications of Hilbert modular varieties in positive characteristic. J. Algebraic Geom. 9 (2000), 111–154.
grothendieck:bt A. Grothendieck, Groupes de Barsotti-Tate et Cristaux de Dieudonné. Les Presses de l’Université de Montréal, 1974.
ibukiyama-katsura-oort T. Ibukiyama, T. Katsura and F. Oort, Supersingular curves of genus two and class numbers. Compositio Math. 57 (1986), 127–152.
katz:slope N. M. Katz, Slope filtration of $F$-crystals. Journées de Géométrie Algébrique de Rennes (1978), Vol. I, pp. 113–163, Astérisque, no. 63. Soc. Math. France, Paris, 1979.
koblitz:thesis N. Koblitz, $p$-adic variant of the zeta-function of families of varieties defined over finite fields. Compositio Math. 31 (1975), 119–218.
katsura-oort:surface T. Katsura and F. Oort, Families of supersingular abelian surfaces, Compositio Math. 62 (1987), 107–167.
kstsura-oort:classnumber T. Katsura and F. Oort, Supersingular abelian varieties of dimension two or three and class numbers. Adv. St. Pure Math. (Algebraic Geometry, Sendai, 1985, Ed. T. Oda) (1987), 253–281.
kottwitz:isocrystals R. E. Kottwitz, Isocrystal with additional structure. Compositio Math. 56 (1985), 201–220.
li K.-Z. Li, Classification of supersingular abelian varieties. Math. Ann. 283 (1989), 333–351.
li-oort K.-Z. Li and F. Oort, Moduli of Supersingular Abelian Varieties. Lecture Notes in Math., vol. 1680, Springer-Verlag, 1998.
manin:thesis Yu. Manin, Theory of commutative formal groups over fields of finite characteristic. Russian Math. Surveys 18 (1963), no. 6, 1–80.
messing:bt W. Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes. Lecture Notes in Math. 264, Springer-Verlag, 1972.
mumford:biextension D. Mumford, Bi-extensions of formal groups. Algebraic Geometry. 307–322 (Proceedings of Internat. Coll. Bombay, 1968), Oxford Univ. Press, 1969.
mumford:av D. Mumford, Abelian Varieties. Oxford University Press, 1974.
norman:algo P. Norman, An algorithm for computing moduli of abelian varieties. Ann. Math. 101 (1975), 499–509.
norman-oort P. Norman and F. Oort, Moduli of abelian varieties, Ann. Math. 112 (1980), 413–439.
ogus:ss A. Ogus, Supersingular K3 crystals. Journées de Géométrie Algébrique de Rennes (1978), Vol. II, pp. 3–86, Astérisque, no. 64. Soc. Math. France, Paris, 1979.
oda-oort T. Oda and F. Oort, Supersingular abelian varieties, Internat. Symp. on Algebraic Geometry, Kyoto (1977), 595–621, Kinokuniya Book Store, Tokyo, 1978.
oort:1975 F. Oort, Isogenies of formal groups. Indag. Math. 37 (1975), 391–400.
oort:product F. Oort, Which abelian surfaces are products of elliptic curves. Math. Ann. 214 (1975), 35–47.
oort:np91 F. Oort, Moduli of abelian varieties and Newton polygons. C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 385–289.
oort:cayley F. Oort, Newton polygons and formal groups: conjectures by Manin and Grothendieck. Ann. Math. 152 (2000), 183–206.
oort:98 F. Oort, Moduli of abelian varieties, finite group schemes and formal groups. Preprint Univ. Utrecht (1998), 26 pp.
pappas:thesis G. Pappas, Arithmetic models for Hilbert modular varieties. Compositio Math. 98 (1995), 43–76.
rapoport:thesis M. Rapoport, Compactifications de l’espaces de modules de Hilbert-Blumenthal. Compositio Math. 36 (1978), 255–335.
rapoport-richartz M. Rapoport and M. Richartz, On the classification and specialization of $F$-isocrystals with additional structure. Compositio Math. 103 (1996), 153–181.
rapoport-zink M. Rapoport and Th. Zink, Period spaces for $p$-divisible groups. Ann. of Math. Studies, vol. 141, Princeton Univ. Press, 1996.
stamm H. Stamm, On the reduction of the Hilbert-Blumenthal-moduli scheme with $\Gamma _0(p)$-level structure. Forum Math. 4 (1997), 405–455.
yu:lift C.-F. Yu, Lifting abelian varieties with additional structure. To appear Math. Z. 242 (2002), 427–441.
yu:thesis C.-F. Yu, On the supersingular locus of Hilbert-Blumenthal 4-folds. Thesis, Univ. of Penn. 1999, 76 pp.
zink:cartier T. Zink, Cartiertheorie kommutativer formaler Gruppen. Teubner-Texte Math., Teubner, Leipzig, 1984.
zink:display T. Zink, The display of a formal $p$-divisible group. Cohomologies $p$-adiques et applications arithmétiques, pp. 127–248. Astérisque, no. 278, Soc. Math. France, Paris, 2002.
zink:isogeny T. Zink, Isogenieklassen von Punkten von Shimuramannigfaltigkeiten mit Werten in einem endlichen Körper. Math. Nachr. 112 (1983), 103–124.
beauville A. Beauville, Complex algebraic surfaces. Translated from the French by R. Barlow, N.I. Shepherd-Barron and M. Reid. London Mathematical Society Lecture Note Series 68, Cambridge University Press, Cambridge-New York, 1983.
breen:dieu L. Breen, Rapport sur la théorie de Dieudonné. Journées de Géométrie Algébrique de Rennes (1978), Vol. I, pp. 39–66, Astérisque, no. 63. Soc. Math. France, Paris, 1979.
chai:np C.-L. Chai, Newton polygons as lattice points. Amer. J. Math. 122 (2000), 967–990.
chai-norman:gamma C.-L. Chai and P. Norman, Bad reduction of the Siegel moduli scheme of genus two with $\Gamma _0(p)$-level structure. Amer. J. Math. 112 (1990), 1003–1071.
chai-norman:gamma2 C.-L. Chai and P. Norman, Singularities of the $\Gamma _0(p)$-level structure. J. Algebraic Geom. 1 (1992), 251–178.
dejong-oort:purity A.J. de Jong and F. Oort, Purity of the stratification by Newton polygons. J. Amer. Math. Soc. 13 (2000), 209–241.
deligne-pappas P. Deligne and G. Pappas, Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant. Compositio Math. 90 (1994), 59–79.
ekedahl:ss T. Ekedahl, On supersingular curves and supersingular abelian varieties. Math. Scand. 60 (1987), 151–178.
ekedahl-oort T. Ekedahl and F. Oort, Connected subsets of a moduli space of abelian varieties. (to appear).
goren-oort E.Z. Goren and F. Oort, Stratifications of Hilbert modular varieties in positive characteristic. J. Algebraic Geom. 9 (2000), 111–154.
grothendieck:bt A. Grothendieck, Groupes de Barsotti-Tate et Cristaux de Dieudonné. Les Presses de l’Université de Montréal, 1974.
ibukiyama-katsura-oort T. Ibukiyama, T. Katsura and F. Oort, Supersingular curves of genus two and class numbers. Compositio Math. 57 (1986), 127–152.
katz:slope N. M. Katz, Slope filtration of $F$-crystals. Journées de Géométrie Algébrique de Rennes (1978), Vol. I, pp. 113–163, Astérisque, no. 63. Soc. Math. France, Paris, 1979.
koblitz:thesis N. Koblitz, $p$-adic variant of the zeta-function of families of varieties defined over finite fields. Compositio Math. 31 (1975), 119–218.
katsura-oort:surface T. Katsura and F. Oort, Families of supersingular abelian surfaces, Compositio Math. 62 (1987), 107–167.
kstsura-oort:classnumber T. Katsura and F. Oort, Supersingular abelian varieties of dimension two or three and class numbers. Adv. St. Pure Math. (Algebraic Geometry, Sendai, 1985, Ed. T. Oda) (1987), 253–281.
kottwitz:isocrystals R. E. Kottwitz, Isocrystal with additional structure. Compositio Math. 56 (1985), 201–220.
li K.-Z. Li, Classification of supersingular abelian varieties. Math. Ann. 283 (1989), 333–351.
li-oort K.-Z. Li and F. Oort, Moduli of Supersingular Abelian Varieties. Lecture Notes in Math., vol. 1680, Springer-Verlag, 1998.
manin:thesis Yu. Manin, Theory of commutative formal groups over fields of finite characteristic. Russian Math. Surveys 18 (1963), no. 6, 1–80.
messing:bt W. Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes. Lecture Notes in Math. 264, Springer-Verlag, 1972.
mumford:biextension D. Mumford, Bi-extensions of formal groups. Algebraic Geometry. 307–322 (Proceedings of Internat. Coll. Bombay, 1968), Oxford Univ. Press, 1969.
mumford:av D. Mumford, Abelian Varieties. Oxford University Press, 1974.
norman:algo P. Norman, An algorithm for computing moduli of abelian varieties. Ann. Math. 101 (1975), 499–509.
norman-oort P. Norman and F. Oort, Moduli of abelian varieties, Ann. Math. 112 (1980), 413–439.
ogus:ss A. Ogus, Supersingular K3 crystals. Journées de Géométrie Algébrique de Rennes (1978), Vol. II, pp. 3–86, Astérisque, no. 64. Soc. Math. France, Paris, 1979.
oda-oort T. Oda and F. Oort, Supersingular abelian varieties, Internat. Symp. on Algebraic Geometry, Kyoto (1977), 595–621, Kinokuniya Book Store, Tokyo, 1978.
oort:1975 F. Oort, Isogenies of formal groups. Indag. Math. 37 (1975), 391–400.
oort:product F. Oort, Which abelian surfaces are products of elliptic curves. Math. Ann. 214 (1975), 35–47.
oort:np91 F. Oort, Moduli of abelian varieties and Newton polygons. C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 385–289.
oort:cayley F. Oort, Newton polygons and formal groups: conjectures by Manin and Grothendieck. Ann. Math. 152 (2000), 183–206.
oort:98 F. Oort, Moduli of abelian varieties, finite group schemes and formal groups. Preprint Univ. Utrecht (1998), 26 pp.
pappas:thesis G. Pappas, Arithmetic models for Hilbert modular varieties. Compositio Math. 98 (1995), 43–76.
rapoport:thesis M. Rapoport, Compactifications de l’espaces de modules de Hilbert-Blumenthal. Compositio Math. 36 (1978), 255–335.
rapoport-richartz M. Rapoport and M. Richartz, On the classification and specialization of $F$-isocrystals with additional structure. Compositio Math. 103 (1996), 153–181.
rapoport-zink M. Rapoport and Th. Zink, Period spaces for $p$-divisible groups. Ann. of Math. Studies, vol. 141, Princeton Univ. Press, 1996.
stamm H. Stamm, On the reduction of the Hilbert-Blumenthal-moduli scheme with $\Gamma _0(p)$-level structure. Forum Math. 4 (1997), 405–455.
yu:lift C.-F. Yu, Lifting abelian varieties with additional structure. To appear Math. Z. 242 (2002), 427–441.
yu:thesis C.-F. Yu, On the supersingular locus of Hilbert-Blumenthal 4-folds. Thesis, Univ. of Penn. 1999, 76 pp.
zink:cartier T. Zink, Cartiertheorie kommutativer formaler Gruppen. Teubner-Texte Math., Teubner, Leipzig, 1984.
zink:display T. Zink, The display of a formal $p$-divisible group. Cohomologies $p$-adiques et applications arithmétiques, pp. 127–248. Astérisque, no. 278, Soc. Math. France, Paris, 2002.
zink:isogeny T. Zink, Isogenieklassen von Punkten von Shimuramannigfaltigkeiten mit Werten in einem endlichen Körper. Math. Nachr. 112 (1983), 103–124.
Additional Information
Chia-Fu Yu
Affiliation:
Department of Mathematics, Columbia University, New York, New York 10027
MR Author ID:
716493
ORCID:
0000-0003-1634-672X
Email:
chiafu@math.columbia.edu
Received by editor(s):
December 22, 2000
Received by editor(s) in revised form:
January 3, 2002
Published electronically:
July 9, 2003
