Abstract
The isomorphism number (resp. isogeny cutoff) of a $p$-divisible group $D$ over an algebraically closed field of characteristic $p$ is the least positive integer $m$ such that $D[p^m]$ determines $D$ up to isomorphism (resp. up to isogeny). We show that these invariants are lower semicontinuous in families of $p$-divisible groups of constant Newton polygon. Thus they allow refinements of Newton polygon strata. In each isogeny class of $p$-divisible groups, we determine the maximal value of isogeny cutoffs and give an upper bound for isomorphism numbers, which is shown to be optimal in the isoclinic case. In particular, the latter disproves a conjecture of Traverso. As an application, we answer a question of Zink on the liftability of an endomorphism of $D[p^m]$ to $D$.
-
[Be]
P. Berthelot, "Théorie de Dieudonné sur un anneau de valuation parfait," Ann. Sci. École Norm. Sup., vol. 13, iss. 2, pp. 225-268, 1980.@article {Be, MRKEY = {0584086},
AUTHOR = {Berthelot, Pierre},
TITLE = {Théorie de {D}ieudonné sur un anneau de valuation parfait},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {13},
YEAR = {1980},
NUMBER = {2},
PAGES = {225--268},
ISSN = {0012-9593},
CODEN = {ENAQAF},
MRCLASS = {14L05 (14F30 14G20)},
MRNUMBER = {0584086},
MRREVIEWER = {William E. Lang},
URL = {http://www.numdam.org/item?id=ASENS_1980_4_13_2_225_0},
ZBLNUMBER = {0441.14012},
} -
[dJO] A. J. de Jong and F. Oort, "Purity of the stratification by Newton polygons," J. Amer. Math. Soc., vol. 13, iss. 1, pp. 209-241, 2000.
@article {dJO, KEY={dJO00},
MRKEY = {1703336},
AUTHOR = {{de Jong},
A. J. and Oort, F.},
TITLE = {Purity of the stratification by {N}ewton polygons},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {13},
YEAR = {2000},
NUMBER = {1},
PAGES = {209--241},
ISSN = {0894-0347},
MRCLASS = {14L05 (14B05)},
MRNUMBER = {1703336},
MRREVIEWER = {Nobuo Tsuzuki},
DOI = {10.1090/S0894-0347-99-00322-7},
ZBLNUMBER = {0954.14007},
} -
[De] M. Demazure, Lectures on $p$-Divisible Groups, New York: Springer-Verlag, 1972, vol. 302.
@book {De, MRKEY = {0344261},
AUTHOR = {Demazure, Michel},
TITLE = {Lectures on {$p$}-Divisible Groups},
SERIES = {Lecture Notes in Math.},
VOLUME={302},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1972},
PAGES = {v+98},
MRCLASS = {14L05},
MRNUMBER = {0344261},
MRREVIEWER = {Loren D. Olson},
ZBLNUMBER = {0247.14010},
DOI = {10.1007/BFb0060741},
} -
[GV] O. Gabber and A. Vasiu, "Dimensions of group schemes of automorphisms of truncated Barsotti–Tate groups," Int. Math. Res. Notices, 2013.
@article{GV,
author={Gabber, O. and Vasiu, A.},
TITLE={Dimensions of group schemes of automorphisms of truncated {B}arsotti--{T}ate groups},
JOURNAL={Int. Math. Res. Notices},
YEAR={2013},
NOTE={published online July 26, 2012, 49 pp.},
DOI = {10.1093/imrn/rns165},
} -
[Ill] L. Illusie, "Déformations de groupes de Barsotti-Tate (d’après A. Grothendieck)," in Seminar on Arithmetic Bundles: the Mordell Conjecture, Paris: Soc. Math. France, 1985, vol. 127, pp. 151-198.
@incollection {Ill, MRKEY = {0801922},
AUTHOR = {Illusie, Luc},
TITLE = {Déformations de groupes de {B}arsotti-{T}ate (d'après {A}. {G}rothendieck)},
BOOKTITLE = {Seminar on Arithmetic Bundles: the Mordell Conjecture},
VENUE={Paris, 1983/84},
SERIES = {Astérisque},
FJOURNAL = {Astérisque},
VOLUME = {127},
PUBLISHER={Soc. Math. France},
ADDRESS={Paris},
YEAR = {1985},
PAGES = {151--198},
ISSN = {0303-1179},
MRCLASS = {14L05 (11G07 11G10 11S31)},
MRNUMBER = {0801922},
ZBLNUMBER = {1182.14050},
} -
[Ka] N. M. Katz, "Slope filtration of $F$-crystals," in Journées de Géométrie Algébrique de Rennes, Vol. I, Paris: Soc. Math. France, 1979, vol. 63, pp. 113-163.
@incollection {Ka, MRKEY = {0563463},
AUTHOR = {Katz, Nicholas M.},
TITLE = {Slope filtration of {$F$}-crystals},
BOOKTITLE = {Journées de {G}éométrie {A}lgébrique de {R}ennes, {V}ol. {I}},
VENUE={{R}ennes, 1978},
SERIES = {Astérisque},
VOLUME = {63},
PAGES = {113--163},
PUBLISHER = {Soc. Math. France},
ADDRESS = {Paris},
YEAR = {1979},
MRCLASS = {14F30 (14G20 14L05)},
MRNUMBER = {0563463},
MRREVIEWER = {Noriko Yui},
ZBLNUMBER = {0426.14007},
} -
[La] E. Lau, "Displays and formal $p$-divisible groups," Invent. Math., vol. 171, iss. 3, pp. 617-628, 2008.
@article {La, MRKEY = {2372808},
AUTHOR = {Lau, Eike},
TITLE = {Displays and formal {$p$}-divisible groups},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {171},
YEAR = {2008},
NUMBER = {3},
PAGES = {617--628},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {14L05},
MRNUMBER = {2372808},
DOI = {10.1007/s00222-007-0090-x},
ZBLNUMBER = {1186.14048},
} -
[Ma] Y. I. Manin, "The theory of formal commutative groups over fields of finite characteristic," Russian Math. Surv., vol. 18, p. no. 6, 1-83, 1963.
@article{Ma,
author={Manin, Y. I.},
TITLE={The theory of formal commutative groups over fields of finite characteristic},
JOURNAL={Russian Math. Surv.},
VOLUME={18},
YEAR={1963},
PAGES={no.~6, 1--83},
ZBLNUMBER = {128.15603},
DOI = {10.1070/RM1963v018n06ABEH001142},
} -
[NV1] M. Nicole and A. Vasiu, "Minimal truncations of supersingular $p$-divisible groups," Indiana Univ. Math. J., vol. 56, iss. 6, pp. 2887-2897, 2007.
@article {NV1, MRKEY = {2375706},
AUTHOR = {Nicole, Marc-Hubert and Vasiu, Adrian},
TITLE = {Minimal truncations of supersingular {$p$}-divisible groups},
JOURNAL = {Indiana Univ. Math. J.},
FJOURNAL = {Indiana University Mathematics Journal},
VOLUME = {56},
YEAR = {2007},
NUMBER = {6},
PAGES = {2887--2897},
ISSN = {0022-2518},
CODEN = {IUMJAB},
MRCLASS = {14L05},
MRNUMBER = {2375706},
MRREVIEWER = {Takao Yamazaki},
DOI = {10.1512/iumj.2007.56.3110},
ZBLNUMBER = {1163.11046},
} -
[NV2] M. Nicole and A. Vasiu, "Traverso’s isogeny conjecture for $p$-divisible groups," Rend. Semin. Mat. Univ. Padova, vol. 118, pp. 73-83, 2007.
@article {NV2, MRKEY = {2378390},
AUTHOR = {Nicole, Marc-Hubert and Vasiu, Adrian},
TITLE = {Traverso's isogeny conjecture for {$p$}-divisible groups},
JOURNAL = {Rend. Semin. Mat. Univ. Padova},
FJOURNAL = {Rendiconti del Seminario Matematico della Università di Padova. Mathematical Journal of the University of Padua},
VOLUME = {118},
YEAR = {2007},
PAGES = {73--83},
ISSN = {0041-8994},
ISBN = {978-88-77842-84-8},
MRCLASS = {14L05},
MRNUMBER = {2378390},
MRREVIEWER = {Eva Viehmann},
ZBLNUMBER = {1167.14322},
URL = {http://www.numdam.org/item?id=ASNSP_1969_3_23_3_481_0},
} -
[NVW] M. Nicole, A. Vasiu, and T. Wedhorn, "Purity of level $m$ stratifications," Ann. Sci. Éc. Norm. Supér., vol. 43, iss. 6, pp. 925-955, 2010.
@article {NVW, MRKEY = {2778452},
AUTHOR = {Nicole, Marc-Hubert and Vasiu, Adrian and Wedhorn, Torsten},
TITLE = {Purity of level {$m$} stratifications},
JOURNAL = {Ann. Sci. Éc. Norm. Supér.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {43},
YEAR = {2010},
NUMBER = {6},
PAGES = {925--955},
ISSN = {0012-9593},
MRCLASS = {14L05 (11G18 14L15)},
MRNUMBER = {2778452},
MRREVIEWER = {Alan Koch},
ZBLNUMBER = {1223.14052},
} -
[Oo1] F. Oort, "Newton polygon strata in the moduli space of abelian varieties," in Moduli of Abelian Varieties, Basel: Birkhäuser, 2001, vol. 195, pp. 417-440.
@incollection {Oo1, MRKEY = {1827028},
AUTHOR = {Oort, Frans},
TITLE = {Newton polygon strata in the moduli space of abelian varieties},
BOOKTITLE = {Moduli of Abelian Varieties},
VENUE={{T}exel {I}sland, 1999},
SERIES = {Progr. Math.},
VOLUME = {195},
PAGES = {417--440},
PUBLISHER = {Birkhäuser},
ADDRESS = {Basel},
YEAR = {2001},
MRCLASS = {14K10 (14L05)},
MRNUMBER = {1827028},
MRREVIEWER = {Sigrid Wortmann},
ZBLNUMBER = {1086.14037},
} -
[Oo2] F. Oort, "Foliations in moduli spaces of abelian varieties," J. Amer. Math. Soc., vol. 17, iss. 2, pp. 267-296, 2004.
@article {Oo2, MRKEY = {2051612},
AUTHOR = {Oort, Frans},
TITLE = {Foliations in moduli spaces of abelian varieties},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {17},
YEAR = {2004},
NUMBER = {2},
PAGES = {267--296},
ISSN = {0894-0347},
MRCLASS = {14K10 (14L05)},
MRNUMBER = {2051612},
MRREVIEWER = {G. K. Sankaran},
DOI = {10.1090/S0894-0347-04-00449-7},
ZBLNUMBER = {1041.14018},
} -
[Oo3] F. Oort, "Minimal $p$-divisible groups," Ann. of Math., vol. 161, iss. 2, pp. 1021-1036, 2005.
@article {Oo3, MRKEY = {2153405},
AUTHOR = {Oort, Frans},
TITLE = {Minimal {$p$}-divisible groups},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {161},
YEAR = {2005},
NUMBER = {2},
PAGES = {1021--1036},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {14L05},
MRNUMBER = {2153405},
MRREVIEWER = {R. T. Hoobler},
DOI = {10.4007/annals.2005.161.1021},
ZBLNUMBER = {1081.14065},
} -
[Oo4] F. Oort, "Simple $p$-kernels of $p$-divisible groups," Adv. Math., vol. 198, iss. 1, pp. 275-310, 2005.
@article {Oo4, MRKEY = {2183256},
AUTHOR = {Oort, Frans},
TITLE = {Simple {$p$}-kernels of {$p$}-divisible groups},
JOURNAL = {Adv. Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {198},
YEAR = {2005},
NUMBER = {1},
PAGES = {275--310},
ISSN = {0001-8708},
CODEN = {ADMTA4},
MRCLASS = {14L05 (14L15)},
MRNUMBER = {2183256},
MRREVIEWER = {Alan Koch},
DOI = {10.1016/j.aim.2005.03.015},
ZBLNUMBER = {1093.14064},
} -
[OZ] F. Oort and T. Zink, "Families of $p$-divisible groups with constant Newton polygon," Doc. Math., vol. 7, pp. 183-201, 2002.
@article {OZ, MRKEY = {1938119},
AUTHOR = {Oort, Frans and Zink, Thomas},
TITLE = {Families of {$p$}-divisible groups with constant {N}ewton polygon},
JOURNAL = {Doc. Math.},
FJOURNAL = {Documenta Mathematica},
VOLUME = {7},
YEAR = {2002},
PAGES = {183--201},
ISSN = {1431-0635},
MRCLASS = {14L05 (14F30)},
MRNUMBER = {1938119},
MRREVIEWER = {Oliver B{ü}ltel},
ZBLNUMBER = {1022.14013},
} -
[RZ] M. Rapoport and T. Zink, Period Spaces for $p$-divisible Groups, Princeton, NJ: Princeton Univ. Press, 1996, vol. 141.
@book {RZ, MRKEY = {1393439},
AUTHOR = {Rapoport, M. and Zink, Th.},
TITLE = {Period Spaces for {$p$}-divisible Groups},
SERIES = {Ann. of Math. Stud.},
VOLUME = {141},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, NJ},
YEAR = {1996},
PAGES = {xxii+324},
ISBN = {0-691-02782-X; 0-691-02781-1},
MRCLASS = {14G20 (11G18 14F30 14L05 14M15 20G05 20G25)},
MRNUMBER = {1393439},
MRREVIEWER = {Robert E. Kottwitz},
ZBLNUMBER = {0873.14039},
} -
[Tr1] C. Traverso, "Sulla classificazione dei gruppi analitici commutativi di caratteristica positiva," Ann. Scuola Norm. Sup. Pisa, vol. 23, pp. 481-507, 1969.
@article {Tr1, MRKEY = {0260757},
AUTHOR = {Traverso, Carlo},
TITLE = {Sulla classificazione dei gruppi analitici commutativi di caratteristica positiva},
JOURNAL = {Ann. Scuola Norm. Sup. Pisa},
VOLUME = {23},
YEAR = {1969},
PAGES = {481--507},
MRCLASS = {14.50},
MRNUMBER = {0260757},
MRREVIEWER = {D. B. Scott},
ZBLNUMBER = {0214.48301},
} -
[Tr2] C. Traverso, "$p$-divisible groups over fields," in Symposia Mathematica, Vol. XI, London: Academic Press, 1973, pp. 45-65.
@incollection {Tr2, MRKEY = {0344262},
AUTHOR = {Traverso, Carlo},
TITLE = {{$p$}-divisible groups over fields},
BOOKTITLE = {Symposia {M}athematica, {V}ol. {XI}},
VENUE={{C}onvegno di {A}lgebra {C}ommutativa, {INDAM},
{R}ome, 1971},
PAGES = {45--65},
PUBLISHER = {Academic Press},
ADDRESS = {London},
YEAR = {1973},
MRCLASS = {14L05},
MRNUMBER = {0344262},
MRREVIEWER = {Loren D. Olson},
ZBLNUMBER = {0293.14011},
} -
[Tr3] C. Traverso, "Specializations of Barsotti-Tate Groups," in Symposia Mathematica, Vol. XXIV, London: Academic Press, 1981, pp. 1-21.
@incollection {Tr3, MRKEY = {0619238},
AUTHOR = {Traverso, Carlo},
TITLE = {Specializations of {B}arsotti-{T}ate Groups},
BOOKTITLE = {Symposia {M}athematica, {V}ol. {XXIV}},
VENUE={{S}ympos., {INDAM},
{R}ome, 1979},
PAGES = {1--21},
PUBLISHER = {Academic Press},
ADDRESS = {London},
YEAR = {1981},
MRCLASS = {14L05},
MRNUMBER = {0619238},
MRREVIEWER = {William E. Lang},
ZBLNUMBER = {0466.14016},
} -
[Va1] A. Vasiu, "Crystalline boundedness principle," Ann. Sci. École Norm. Sup., vol. 39, iss. 2, pp. 245-300, 2006.
@article {Va1, MRKEY = {2245533},
AUTHOR = {Vasiu, Adrian},
TITLE = {Crystalline boundedness principle},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {39},
YEAR = {2006},
NUMBER = {2},
PAGES = {245--300},
ISSN = {0012-9593},
CODEN = {ASENAH},
MRCLASS = {14L05 (14F30)},
MRNUMBER = {2245533},
MRREVIEWER = {Martin C. Olsson},
DOI = {10.1016/j.ansens.2005.12.003},
ZBLNUMBER = {1143.14037},
} -
[Va2] A. Vasiu, "Level $m$ stratifications of versal deformations of $p$-divisible groups," J. Algebraic Geom., vol. 17, iss. 4, pp. 599-641, 2008.
@article {Va2, MRKEY = {2424922},
AUTHOR = {Vasiu, Adrian},
TITLE = {Level {$m$} stratifications of versal deformations of {$p$}-divisible groups},
JOURNAL = {J. Algebraic Geom.},
FJOURNAL = {Journal of Algebraic Geometry},
VOLUME = {17},
YEAR = {2008},
NUMBER = {4},
PAGES = {599--641},
ISSN = {1056-3911},
MRCLASS = {14L05 (14D15)},
MRNUMBER = {2424922},
MRREVIEWER = {Eva Viehmann},
DOI = {10.1090/S1056-3911-08-00495-5},
ZBLNUMBER = {1152.14022},
} -
[Va3] A. Vasiu, "Reconstructing $p$-divisible groups from their truncations of small level," Comment. Math. Helv., vol. 85, iss. 1, pp. 165-202, 2010.
@article {Va3, MRKEY = {2563685},
AUTHOR = {Vasiu, Adrian},
TITLE = {Reconstructing {$p$}-divisible groups from their truncations of small level},
JOURNAL = {Comment. Math. Helv.},
FJOURNAL = {Commentarii Mathematici Helvetici. A Journal of the Swiss Mathematical Society},
VOLUME = {85},
YEAR = {2010},
NUMBER = {1},
PAGES = {165--202},
ISSN = {0010-2571},
MRCLASS = {14L05 (11E57 11G10 11G25)},
MRNUMBER = {2563685},
MRREVIEWER = {Alan Koch},
DOI = {10.4171/CMH/192},
ZBLNUMBER = {05648233},
} -
[Yu] C. Yu, "On finiteness of endomorphism rings of abelian varieties," Math. Res. Lett., vol. 17, iss. 2, pp. 357-370, 2010.
@article {Yu, MRKEY = {2644382},
AUTHOR = {Yu, Chia-Fu},
TITLE = {On finiteness of endomorphism rings of abelian varieties},
JOURNAL = {Math. Res. Lett.},
FJOURNAL = {Mathematical Research Letters},
VOLUME = {17},
YEAR = {2010},
NUMBER = {2},
PAGES = {357--370},
ISSN = {1073-2780},
MRCLASS = {14K05 (11G10 11G15 14G17)},
MRNUMBER = {2644382},
MRREVIEWER = {Benjamin Smith},
URL = {http://intlpress.com/site/pub/pages/journals/items/mrl/content/vols/0017/0002/00020486/index.html},
} -
[Zi1] T. Zink, "The display of a formal $p$-divisible group," in Cohomologies $p$-Adiques et Applications Arithmétiques, I, Paris: Soc. Math. France, 2002, vol. 278, pp. 127-248.
@incollection {Zi1, MRKEY = {1922825},
AUTHOR = {Zink, Thomas},
TITLE = {The display of a formal {$p$}-divisible group},
BOOKTITLE = {Cohomologies $p$-Adiques et Applications Arithm{é}tiques, I},
SERIES = {Astérisque},
FJOURNAL = {Astérisque},
PUBLISHER={Soc. Math. France},
ADDRESS={Paris},
VOLUME= {278},
YEAR = {2002},
PAGES = {127--248},
ISSN = {0303-1179},
MRCLASS = {14L05 (14F30)},
MRNUMBER = {1922825},
MRREVIEWER = {Oliver B{ü}ltel},
ZBLNUMBER = {1008.14008},
} -
[Zi2] T. Zink, "De Jong-Oort purity for $p$-divisible groups," in Algebra, Arithmetic, and Geometry: in honor of Yu. I. Manin. Vol. II, Boston, MA: Birkhäuser, 2009, vol. 270, pp. 693-701.
@incollection {Zi2, MRKEY = {2641208},
AUTHOR = {Zink, Thomas},
TITLE = {De {J}ong-{O}ort purity for {$p$}-divisible groups},
BOOKTITLE = {Algebra, Arithmetic, and Geometry: in honor of {Y}u. {I}. {M}anin. {V}ol. {II}},
SERIES = {Progr. Math.},
VOLUME = {270},
PAGES = {693--701},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {2009},
MRCLASS = {14L05 (14F30)},
MRNUMBER = {2641208},
MRREVIEWER = {Yichao Tian},
DOI = {10.1007/978-0-8176-4747-6_24},
ZBLNUMBER = {1241.14018},
}
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