no. 2
On reduction of moduli schemes of abelian varieties with definite quaternion multiplications
[Sur la réduction des schémas de modules de variétés abéliennes à multiplication quaternionique définie]
Yu, Chia-Fu1
1 Institute of Mathematics, Academia Sinica and NCTS 6th Floor, Astronomy Mathematics Building No. 1, Roosevelt Rd. Sec. 4 Taipei, Taiwan, 10617
Annales de l'Institut Fourier, Tome 71 (2021) no. 2, pp. 539-613.

Dans cet article, nous faisons une étude initiale sur les espaces de modules de type D en caractéristique positive p≠2, où le nombre premier p peut être ramifié dans la donnée définissant l’espace de modules. Nous classifions explicitement les classes d’isogénie des groupes p-divisibles avec structures supplémentaires en question. Nous étudions également la réduction des espaces de modules de type D de rang minimal.

In this paper we make an initial study on type D moduli spaces in positive characteristic p≠2, where we allow the prime p to ramify in the defining datum. We classify explicitly the isogeny classes of p-divisible groups with additional structures in question. We also study the reduction of the type D moduli spaces of minimal rank.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3349
Classification : 14G35, 11G18
Keywords: Shimura varieties of type D, bad reduction, isocrystals
Mot clés : variétés de Shimura de type D, mauvaise réduction, isocristaux

Yu, Chia-Fu 1

1 Institute of Mathematics, Academia Sinica and NCTS 6th Floor, Astronomy Mathematics Building No. 1, Roosevelt Rd. Sec. 4 Taipei, Taiwan, 10617
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2021__71_2_539_0,
     author = {Yu, Chia-Fu},
     title = {On reduction of moduli schemes of abelian varieties with definite quaternion multiplications},
     journal = {Annales de l'Institut Fourier},
     pages = {539--613},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {71},
     number = {2},
     year = {2021},
     doi = {10.5802/aif.3349},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3349/}
}
TY  - JOUR
AU  - Yu, Chia-Fu
TI  - On reduction of moduli schemes of abelian varieties with definite quaternion multiplications
JO  - Annales de l'Institut Fourier
PY  - 2021
SP  - 539
EP  - 613
VL  - 71
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3349/
DO  - 10.5802/aif.3349
LA  - en
ID  - AIF_2021__71_2_539_0
ER  - 
%0 Journal Article
%A Yu, Chia-Fu
%T On reduction of moduli schemes of abelian varieties with definite quaternion multiplications
%J Annales de l'Institut Fourier
%D 2021
%P 539-613
%V 71
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3349/
%R 10.5802/aif.3349
%G en
%F AIF_2021__71_2_539_0
Yu, Chia-Fu. On reduction of moduli schemes of abelian varieties with definite quaternion multiplications. Annales de l'Institut Fourier, Tome 71 (2021) no. 2, pp. 539-613. doi : 10.5802/aif.3349. https://aif.centre-mersenne.org/articles/10.5802/aif.3349/

[1] Artin, Michael; Winters, Gayn B. Degenerate fibres and stable reduction of curves, Topology, Volume 10 (1971), pp. 373-383 | DOI | MR | Zbl

[2] Faltings, Gerd Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., Volume 73 (1983) no. 3, pp. 349-366 | DOI | Zbl

[3] Görtz, Ulrich Topological flatness of local models in the ramified case, Math. Z., Volume 250 (2005) no. 4, pp. 775-790 | DOI | MR | Zbl

[4] Grothendieck, Alexander Groupes de Barsotti–Tate et Cristaux de Dieudonné, Séminaire de Mathématiques Supérieures, 45, Les Presses de l’Université de Montréal, 1974 | Zbl

[5] Grothendieck, Alexander; Raynaud, Michèle; Rim, Dock Sang Groupes de monodromie en géometrie algébrique I, Lecture Notes in Mathematics, 288, Springer, 1972, pp. 1967-1969 | Zbl

[6] Hartwig, Philippe Kottwitz–Rapoport and p-rank strata in the reduction of Shimura varieties of PEL type, Ann. Inst. Fourier, Volume 65 (2015) no. 3, pp. 1031-1103 | DOI | Numdam | MR | Zbl

[7] He, Xuhua; Rapoport, Michael Stratifications in the reduction of Shimura varieties, Manuscr. Math., Volume 152 (2017) no. 3-4, pp. 317-343 | MR | Zbl

[8] Helgason, Siguldur Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, 34, American Mathematical Society, 2001 | Zbl

[9] Katsura, Toshiyuki; Oort, Frans Families of supersingular abelian surfaces, Compos. Math., Volume 62 (1987) no. 2, pp. 107-167 | Numdam | MR | Zbl

[10] Kottwitz, Robert E. Isocrystals with additional structure, Compos. Math., Volume 56 (1985) no. 3, pp. 201-220 | Numdam | MR | Zbl

[11] Kottwitz, Robert E. Points on some Shimura varieties over finite fields, J. Am. Math. Soc., Volume 5 (1992) no. 2, pp. 373-444 | DOI | MR | Zbl

[12] Kottwitz, Robert E. Isocrystals with additional structure. II, Compos. Math., Volume 109 (1997) no. 3, pp. 255-339 | DOI | MR | Zbl

[13] Manin, Yu I. The theory of commutative formal groups over fields of finite characteristic, Russ. Math. Surv., Volume 18 (1963) no. 6, pp. 1-83 | DOI | MR | Zbl

[14] Messing, William The crystals associated to Barsotti–Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, 264, Springer, 1972 | Zbl

[15] Moonen, Ben Group schemes with additional structures and Weyl group cosets, Moduli of Abelian Varieties (Faber, Carel; van der Geer, Gerard; Oort, Frans, eds.) (Progress in Mathematics), Volume 195, Birkhäuser, 2001, pp. 255-298 | DOI | MR | Zbl

[16] Moonen, Ben A dimension formula for Ekedahl-Oort strata, Ann. Inst. Fourier, Volume 54 (2004) no. 3, pp. 666-698 | DOI | Numdam | MR | Zbl

[17] Moonen, Ben Serre-Tate Theory for Moduli Spaces of PEL Type, Ann. Sci. Éc. Norm. Supér., Volume 37 (2004) no. 2, pp. 223-269 | DOI | Numdam | MR | Zbl

[18] Moret-Bailly, Laurent Familles de courbes et de variétés abéliennes sur P 1 . I. Descente des polarisations. II. Exemples, Séminaire sur les pinceaux de courbes de genre au moins deux (Szpiro, L., ed.) (Astérisque), Volume 86, Société Mathématique de France, 1981, pp. 109-140

[19] Norman, Peter; Oort, Frans Moduli of abelian varieties, Ann. Math., Volume 112 (1980) no. 3, pp. 413-439 | DOI | MR | Zbl

[20] O’Meara, O. Timothy Introduction to quadratic forms, Classics in Mathematics, Springer, 2000 | MR | Zbl

[21] Oort, Frans Finite group schemes, local moduli for abelian varieties, and lifting problems, Compos. Math., Volume 23 (1971), pp. 265-296 | Numdam | MR | Zbl

[22] Oort, Frans The isogeny class of a CM-type abelian variety is defined over a finite extension of the prime field, J. Pure Appl. Algebra, Volume 3 (1973), pp. 399-408 | DOI | MR | Zbl

[23] Pappas, Georgios On the arithmetic moduli schemes of PEL Shimura varieties, J. Algebr. Geom., Volume 9 (2000) no. 3, pp. 577-605 | MR | Zbl

[24] Pappas, Georgios; Rapoport, Michael Local models in the ramified case I. The EL-case, J. Algebr. Geom., Volume 12 (2003) no. 1, pp. 107-145 | DOI | MR | Zbl

[25] Pappas, Georgios; Rapoport, Michael Local models in the ramified case. III. Unitary groups, J. Inst. Math. Jussieu, Volume 8 (2009) no. 3, pp. 507-564 | DOI | MR | Zbl

[26] Pappas, Georgios; Zhu, Xinwen Local models of Shimura varieties and a conjecture of Kottwitz, Invent. Math., Volume 194 (2013) no. 1, pp. 147-254 | DOI | MR | Zbl

[27] Rapoport, Michael; Richartz, Melanie On the classification and specialization of F-isocrystals with additional structure, Compos. Math., Volume 103 (1996) no. 2, pp. 153-181 | Numdam | MR | Zbl

[28] Rapoport, Michael; Viehmann, Eva Towards a theory of local Shimura varieties, Münster J. Math., Volume 7 (2014) no. 1, pp. 273-326 | MR | Zbl

[29] Rapoport, Michael; Zink, Thomas Period Spaces for p-divisible groups, Annals of Mathematics Studies, 141, Princeton University Press, 1996 | DOI | Zbl

[30] Smithling, Brian D. Admissibility and permissibility for minuscule cocharacters in orthogonal groups, Manuscr. Math., Volume 136 (2011) no. 3-4, pp. 295-314 | DOI | MR | Zbl

[31] Smithling, Brian D. Topological flatness of orthogonal local models in the split, even case. I, Math. Ann., Volume 350 (2011) no. 2, pp. 381-416 | DOI | MR | Zbl

[32] Tsukamoto, T. On the local theory of quaternionic anti-hermitian forms, J. Math. Soc. Japan, Volume 13 (1961), pp. 387-400 | DOI | MR | Zbl

[33] Vasiu, Adrian Integral models in unramified mixed characteristic (0,2) of Hermitian orthogonal Shimura varieties of PEL type. Part I, J. Ramanujan Math. Soc., Volume 27 (2012) no. 4, pp. 425-477 | MR | Zbl

[34] Vasiu, Adrian Integral models in unramified mixed characteristic (0,2) of hermitian orthogonal Shimura varieties of PEL type. Part II, Math. Nachr., Volume 287 (2014) no. 14-15, pp. 1756-1773 | DOI | MR | Zbl

[35] Vollaard, Inken On the Hilbert–Blumenthal moduli problem, J. Inst. Math. Jussieu, Volume 4 (2005) no. 4, pp. 653-683 | DOI | MR | Zbl

[36] Wedhorn, Torsten Ordinariness in good reductions of Shimura varieties of PEL-type, Ann. Sci. Éc. Norm. Supér., Volume 32 (1999) no. 5, pp. 575-618 | DOI | Numdam | MR | Zbl

[37] Wedhorn, Torsten The dimension of Oort strata of Shimura varieties of PEL-type, Moduli of Abelian Varieties (Faber, Carel; van der Geer, Gerard; Oort, Frans, eds.) (Progress in Mathematics), Volume 195, Birkhäuser, 2001, pp. 441-471 | DOI | MR | Zbl

[38] Yu, Chia-Fu Lifting abelian varieties with additional structures, Math. Z., Volume 242 (2002) no. 3, pp. 427-441 | MR | Zbl

[39] Yu, Chia-Fu On reduction of Hilbert-Blumenthal varieties, Ann. Inst. Fourier, Volume 53 (2003) no. 7, pp. 2105-2154 | Numdam | MR | Zbl

[40] Yu, Chia-Fu The isomorphism classes of abelian varieties of CM-type, J. Pure Appl. Algebra, Volume 187 (2004) no. 1-3, pp. 305-319 | MR | Zbl

[41] Yu, Chia-Fu On the mass formula of supersingular abelian varieties with real multiplications, J. Aust. Math. Soc., Volume 78 (2005) no. 3, pp. 373-392 | MR | Zbl

[42] Yu, Chia-Fu An exact geometric mass formula, Int. Math. Res. Not., Volume 2008 (2008) no. 11, rnn113, 11 pages | Zbl

[43] Yu, Chia-Fu On finiteness of endomorphism rings of abelian varieties, Math. Res. Lett., Volume 17 (2010) no. 2, pp. 357-370 | MR | Zbl

[44] Yu, Chia-Fu Simple mass formulas on Shimura varieties of PEL-type, Forum Math., Volume 22 (2010) no. 3, pp. 565-582 | MR | Zbl

[45] Yu, Chia-Fu Embeddings of fields into simple algebras: generalizations and applications, J. Algebra, Volume 368 (2012), pp. 1-20 | MR | Zbl

[46] Zarhin, Ju. G. Isogenies of abelian varieties over fields of finite characteristics, Math. USSR, Sb., Volume 24 (1974) no. 3, pp. 451-461 | DOI | MR

[47] Zink, Thomas Cartiertheorie kommutativer formaler Gruppen. Unter Mitarb. von Harry Reimann, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], 68, Teubner, 1984, 124 pages | Zbl

Cité par Sources :