Geodesic
Canonical integral models for Shimura varieties of abelian type
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e69

Voir la notice de l'article provenant de la source Cambridge University Press

We prove a conjecture of Pappas and Rapoport for all Shimura varieties of abelian type with parahoric level structure when $p>2$ by showing that the Kisin–Pappas–Zhou integral models of Shimura varieties of abelian type are canonical. In particular, this shows that these models are independent of the choices made during their construction, and that they satisfy functoriality with respect to morphisms of Shimura data. 
Daniels, Patrick; Youcis, Alexander. Canonical integral models for Shimura varieties of abelian type. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e69. doi: 10.1017/fms.2025.27
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[ALY21] Achinger, P., Lara, M. and Youcis, A., ‘Variants of the de Jong fundamental group’, to appear in Amer. J. Math. Google Scholar

[BLR90] Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models, (Ergebnisse der Mathematik und ihrer Grenzgebiete), (Springer-Verlag, Berlin, 1990). Google Scholar | DOI

[Bor91] Borel, A., Linear algebraic groups, second ed. (Graduate Texts in Mathematics, Vol. 126) (Springer-Verlag, New York, 1991). MR 1102012 Google Scholar | DOI

[Bor98] Borovoi, M., ‘Abelian Galois cohomology of reductive groups’, Mem. Amer. Math. Soc. 132 (626)(1998), viii+50. MR 1401491 Google Scholar

[BS15] Bhatt, B. and Scholze, P., ‘The pro-étale topology for schemes’, Astérisque (369)(2015), 99–201. MR 3379634. Google Scholar

[BT84] Bruhat, F. and Tits, J., ‘Groupes réductifs sur un corps local. II. Schémas en groups. existence d’une donnée radicielle valuée’, Inst. Hautes Études Sci. Publ. Math. (60)(1984), 197–376. Google Scholar

[Dan25] Daniels, P., ‘Canonical integral models for Shimura varieties of toral type’, Algebra Number Theory 19 (2)(2025), 247–286. MR 4859066. Google Scholar | DOI

[Del79] Deligne, P., ‘Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques’, In Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977, Part 2, Proc. Sympos. Pure Math. XXXIII) (Amer. Math. Soc., Providence, R.I., 1979), 247–289. MR 546620. Google Scholar | DOI

[DvHKZ24a] Daniels, P., Van Hoften, P., Kim, D. and Zhang, M., ‘Igusa stacks for Shimura varieties of Hodge type’, Preprint (2024), . Google Scholar | arXiv

[DvHKZ24b] Daniels, P., Van Hoften, P., Kim, D. and Zhang, M., ‘On a conjecture of Pappas and Rapoport’, Preprint (2024), . Google Scholar | arXiv

[FK18] Fujiwara, K. and Kato, F., Foundations of rigid geometry. I, (EMS Monographs in Mathematics), (European Mathematical Society (EMS), Zürich, 2018). MR 3752648. Google Scholar | DOI

[Gir71] Giraud, J., Cohomologie non abélienne (Die Grundlehren der mathematischen Wissenschaften), Band 179 (Springer-Verlag, Berlin-New York, 1971). MR 0344253. Google Scholar | DOI

[Gle22] Gleason, I., ‘On the geometric connected components of moduli spaces of -adic shtukas and local Shimura varieties’, Preprint (2022), 33. . Google Scholar | arXiv

[Gro66] Grothendieck, A., ‘Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III’, Inst. Hautes Études Sci. Publ. Math. (28)(1966), 255. MR 217086. Google Scholar

[Gro67] Grothendieck, A., ‘Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV’, Inst. Hautes Études Sci. Publ. Math. (32)(1967), 361. MR 238860. Google Scholar

[GW20] Görtz, U. and Wedhorn, T., Algebraic geometry I. Schemes—with examples and exercises, (Springer Studium Mathematik—Master), (Springer Spektrum, Wiesbaden, 2020). MR 4225278. Google Scholar | DOI

[He16] He, X., ‘Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties’, Ann. Sci. Éc. Norm. Supér. (4) 49 (5)(2016), 1125–1141. Google Scholar | DOI

[HL23] Hamann, L. and Lee, S.Y., ‘Torsion vanishing for some Shimura varieties’ Preprint, (2023), . Google Scholar | arXiv

[HR08] Haines, T. and Rapoport, M., ‘On parahoric subgroups’, Adv. in Math. 219 (1)(2008), 188–198. Google Scholar | DOI

[IKY24a] Imai, N., Kato, H. and Youcis, A., ‘A Tannakian framework for prismatic F-crystals’, Preprint, (2024), . Google Scholar | arXiv

[IKY24b] Imai, N., Kato, H. and Youcis, A., ‘The prismatic realization functor for Shimura varieties of abelian type’, Preprint, (2024), . Google Scholar | arXiv

[Kis10] Kisin, M., ‘Integral models for Shimura varieties of abelian type’, J. Amer. Math. Soc. 23 (4)(2010), 967–1012. Google Scholar | DOI

[Kot84] Kottwitz, R. E., ‘Shimura varieties and twisted orbital integrals’, Math. Ann. 269 (3)(1984), 287–300. MR 761308. Google Scholar | DOI

[Kot90] Kottwitz, R. E., ‘Shimura varieties and -adic representations’, Automorphic forms, Shimura varieties, and -functions, Vol. I (Ann Arbor, MI, 1988) (Perspect. Math., Vol. 10), (Academic Press, Boston, MA, 1990), 161–209. MR 1044820. Google Scholar

[Kot97] Kottwitz, R.E., ‘Isocrystals with additional structure. II’, Compositio Math. 109 (3)(1997), 255–339. MR 1485921. Google Scholar | DOI

[KP18] Kisin, M. and Pappas, G., ‘Integral models of Shimura varieties with parahoric level structure’, Publ. Math. Inst. Hautes Études Sci. (128)(2018), 121–218. Google Scholar | DOI

[KP23] Kaletha, T. and Prasad, G., Bruhat-Tits Theory—A New Approach, (New Mathematical Monographs, Vol. 44), (Cambridge University Press, Cambridge, 2023). MR 4520154. Google Scholar | DOI

[KPZ24] Kisin, M., Pappas, G. and Zhou, R., ‘Integral models of Shimura varieties with parahoric level structure, II’, (2024), . Google Scholar | arXiv

[KSZ21] Kisin, M., Shin, S.W. and Zhu, Y., ‘The stable trace formula for Shimura varieties of abelian type’, Preprint, (2021), . Google Scholar | arXiv

[KZ21] Kisin, M. and Zhou, R., Independence of for Frobenius conjugacy classes attached to abelian varieties, arxiv preprint (2021), 66. Google Scholar | arXiv

[Lan00] Landvogt, E., ‘Some functorial properties of the Bruhat-Tits building’, J. Reine Angew. Math. 518 (2000), 213 – 241. Google Scholar

[Lov17a] Lovering, T., ‘Integral canonical models for automorphic vector bundles of abelian type’, Algebra Number Theory 11 (8)(2017), 1837–1890. Google Scholar | DOI

[Lov17b] Lovering, T., ‘Filtered F-crystals on Shimura varieties of abelian type’, (2017), . Google Scholar | arXiv

[LR87] Langlands, R.P. and Rapoport, M., ‘Shimuravarietäten und Gerben’, J. Reine Angew. Math. 378 (1987), 113–220. MR 895287. Google Scholar

[MFK94] Mumford, D., Fogarty, J. and Kirwan, F., Geometric Invariant Theory, third ed., (Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], Vol. 34), (Springer-Verlag, Berlin, 1994). MR 1304906. Google Scholar | DOI

[Mil05] Milne, J. S., ‘Introduction to Shimura varieties’, In Harmonic Analysis, the Trace Formula, and Shimura Varieties, (Clay Math. Proc., Vol. 4), (Amer. Math. Soc., 2005), 265–378. Google Scholar

[Mil13] Milne, J. S., ‘Shimura varieties and moduli’, In Handbook of Moduli (Vol. II, Adv. Lect. Math. (ALM), Vol. 25) (Int. Press, Somerville, MA, 2013), 467–548. MR 3184183. Google Scholar

[Mil17] Milne, J.S., Algebraic Groups: The Theory of Group Schemes of Finite Type Over a Field, Vol. 170, (Cambridge University Press, Cambridge, 2017). Google Scholar | DOI

[Moo98] Moonen, B., ‘Models of Shimura varieties in mixed characteristics’, Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996) (London Math. Soc. Lecture Note Ser. Vol. 254) (Cambridge Univ. Press, 1998), 267–350. Google Scholar

[MP16] Pera, K.M., ‘Integral canonical models for spin Shimura varieties’, Compos. Math. 152 (4)(2016), 769–824. MR 3484114. Google Scholar | DOI

[MS81] Milne, J. S. and Shih, K-Y, ‘The action of complex conjugation on a Shimura variety’, Ann. of Math. (2) 113 (3)(1981), 569–599. MR 621017. Google Scholar | DOI

[Pap22] Pappas, G., ‘On integral models of Shimura varieties’, Math. Ann. 385 (3–4)(2022), 2037–2097. Google Scholar

[PR22] Pappas, G. and Rapoport, M., ‘On integral local Shimura varieties’, Preprint (2022), , 56. Google Scholar | arXiv

[PR24] Pappas, G. and Rapoport, M., ‘ -adic shtukas and the theory of global and local Shimura varieties’, Camb. J. Math. 12 (1)(2024), 1–164. MR 4701491. Google Scholar | DOI

[RV14] Rapoport, M. and Viehmann, E., ‘Towards a theory of local Shimura varieties’, Münster Journal of Mathematics 7(1) (2014), 273–326. Google Scholar

[Sch13] Scholze, P., ‘ -adic Hodge theory for rigid-analytic varieties’, Forum Math. Pi. 1 (2013), 77. Google Scholar | DOI

[Sch17] Scholze, P., ‘Étale cohomology of diamonds’, Preprint (2017), . Google Scholar | arXiv

[SP17] The Stacks Project Authors, Stacks Project (2017), http://stacks.math.columbia.edu. Google Scholar

[Ste65] Steinberg, R., ‘Regular elements of semisimple algebraic groups’, Inst. Hautes Études Sci. Publ. Math. (25)(1965), 49–80. MR 180554. Google Scholar | DOI

[SW20] Scholze, P. and Weinstein, J., Berkeley Lectures on -adic Geometry: (AMS-207), (Annals of Mathematical Studies, Princeton University Press, Princeton, NJ, 2020). Google Scholar

[Tak24] Takaya, Y., ‘Moduli spaces of level structures on mixed characteristic local shtukas’, (2024), . Google Scholar | arXiv

[Zha23] Zhang, M., ‘A PEL-type Igusa stack and the -adic geometry of Shimura varieties’, Preprint (2023), . Google Scholar | arXiv

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