## Algebra & Number Theory

### Nonemptiness of Newton strata of Shimura varieties of Hodge type

Dong Uk Lee

#### Abstract

For a Shimura variety of Hodge type with hyperspecial level at a prime $p$, the Newton stratification on its special fiber at $p$ is a stratification defined in terms of the isomorphism class of the rational Dieudonné module of parameterized abelian varieties endowed with a certain fixed set of Frobenius-invariant crystalline tensors (“$Gℚp$-isocrystal”). There has been a conjectural group-theoretic description of the $F$-isocrystals that are expected to show up in the special fiber. We confirm this conjecture. More precisely, for any $Gℚp$-isocrystal that is expected to appear (in a precise sense), we construct a special point whose reduction has associated $F$-isocrystal equal to the given one.

#### Article information

Source
Algebra Number Theory, Volume 12, Number 2 (2018), 259-283.

Dates
Revised: 7 May 2017
Accepted: 23 October 2017
First available in Project Euclid: 23 May 2018

https://projecteuclid.org/euclid.ant/1527040844

Digital Object Identifier
doi:10.2140/ant.2018.12.259

Mathematical Reviews number (MathSciNet)
MR3803703

Zentralblatt MATH identifier
06880888

#### Citation

Lee, Dong Uk. Nonemptiness of Newton strata of Shimura varieties of Hodge type. Algebra Number Theory 12 (2018), no. 2, 259--283. doi:10.2140/ant.2018.12.259. https://projecteuclid.org/euclid.ant/1527040844

#### References

• P. Berthelot and A. Ogus, “$F$-isocrystals and de Rham cohomology, I”, Invent. Math. 72:2 (1983), 159–199.
• D. Blasius, “A $p$-adic property of Hodge classes on abelian varieties”, pp. 293–308 in Motives (Seattle, 1991), edited by U. Jannsen et al., Proceedings of Symposia in Pure Math. 55, American Mathematical Society, Providence, RI, 1994.
• F. Bruhat and J. Tits, “Groupes réductifs sur un corps local, II: Schémas en groupes. Existence d'une donnée radicielle valuée”, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 5–184.
• O. Bültel, “Density of the ordinary locus”, Bull. London Math. Soc. 33:2 (2001), 149–156.
• C.-L. Chai, “Newton polygons as lattice points”, Amer. J. Math. 122:5 (2000), 967–990.
• C.-L. Chai, B. Conrad, and F. Oort, Complex multiplication and lifting problems, Mathematical Surveys and Monographs 195, American Mathematical Society, Providence, RI, 2014.
• B. Conrad, “Reductive group schemes”, pp. 93–444 in Autour des schémas en groupes (Luminy, 2011), vol. I, Panoramas et Synthèses 42/43, Société Mathématique de France, Paris, 2014.
• P. Deligne, “Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques”, pp. 247–289 in Automorphic forms, representations and $L$-functions (Corvallis, OR, 1977), vol. 2, edited by A. Borel and W. Casselman, Proceedings of Symposia in Pure Math. 33, American Mathematical Society, Providence, RI, 1979.
• P. Deligne, J. S. Milne, A. Ogus, and K.-Y. Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math. 900, Springer, 1982.
• G. Faltings, “Crystalline cohomology and $p$-adic Galois-representations”, pp. 25–80 in Algebraic analysis, geometry, and number theory (Baltimore, 1988), edited by J.-I. Igusa, Johns Hopkins University Press, 1989.
• L. Fargues, “Cohomologie des espaces de modules de groupes $p$-divisibles et correspondances de Langlands locales”, pp. 1–199 in Variétés de Shimura, espaces de Rapoport–Zink et correspondances de Langlands locales, Astérisque 291, Société Mathématique de France, Paris, 2004.
• J.-M. Fontaine and W. Messing, “$p$-adic periods and $p$-adic étale cohomology”, pp. 179–207 in Current trends in arithmetical algebraic geometry (Arcata, CA, 1985), edited by K. A. Ribet, Contemporary Math. 67, American Mathematical Society, Providence, RI, 1987.
• P. Hamacher, “The geometry of Newton strata in the reduction modulo $p$ of Shimura varieties of PEL type”, Duke Math. J. 164:15 (2015), 2809–2895.
• P. Hamacher, “The almost product structure of Newton strata in the Deformation space of a Barsotti–Tate group with crystalline Tate tensors”, Math. Z. 287:3-4 (2017), 1255–1277.
• M. Kisin, “Integral models for Shimura varieties of abelian type”, J. Amer. Math. Soc. 23:4 (2010), 967–1012.
• M. Kisin, “${\rm mod}\,p$ points on Shimura varieties of abelian type”, J. Amer. Math. Soc. 30:3 (2017), 819–914.
• J.-S. Koskivirta, “Canonical sections of the Hodge bundle over Ekedahl–Oort strata of Shimura varieties of Hodge type”, J. Algebra 449 (2016), 446–459.
• R. E. Kottwitz, “Stable trace formula: cuspidal tempered terms”, Duke Math. J. 51:3 (1984), 611–650.
• R. E. Kottwitz, “Isocrystals with additional structure”, Compositio Math. 56:2 (1985), 201–220.
• R. E. Kottwitz, “Stable trace formula: elliptic singular terms”, Math. Ann. 275:3 (1986), 365–399.
• R. E. Kottwitz, “Shimura varieties and $\lambda$-adic representations”, pp. 161–209 in Automorphic forms, Shimura varieties, and $L$-functions (Ann Arbor, MI, 1988), vol. I, edited by L. Clozel and J. S. Milne, Perspectives in Math. 10, Academic Press, Boston, 1990.
• R. E. Kottwitz, “Points on some Shimura varieties over finite fields”, J. Amer. Math. Soc. 5:2 (1992), 373–444.
• R. E. Kottwitz, “Isocrystals with additional structure, II”, Compositio Math. 109:3 (1997), 255–339.
• A. Kret, “The trace formula and the existence of PEL type Abelian varieties modulo $p$”, preprint, 2012.
• R. P. Langlands, Les débuts d'une formule des traces stable, Publ. Math. 13, Université de Paris VII, U.E.R. de Mathématiques, 1983.
• R. P. Langlands and M. Rapoport, “Shimuravarietäten und Gerben”, J. Reine Angew. Math. 378 (1987), 113–220.
• D. U. Lee, “Non-emptiness of Newton strata of Shimura varieties of Hodge type”, preprint, 2016.
• J. S. Milne, “Shimura varieties and motives”, pp. 447–523 in Motives (Seattle, 1991), edited by U. Jannsen et al., Proceedings of Symposia in Pure Math. 55, American Mathematical Society, Providence, RI, 1994.
• J. S. Milne, “Complex multiplication”, course notes, 2006, http://www.jmilne.org/math/CourseNotes/CM.pdf.
• B. Moonen, “Models of Shimura varieties in mixed characteristics”, pp. 267–350 in Galois representations in arithmetic algebraic geometry (Durham, 1996), edited by A. J. Scholl and R. L. Taylor, London Math. Society Lecture Note Series 254, Cambridge University Press, 1998.
• F. Oort, “Newton polygon strata in the moduli space of abelian varieties”, pp. 417–440 in Moduli of abelian varieties (Texel Island, Netherlands, 1999), edited by C. Faber et al., Progress in Math. 195, Birkhäuser, Basel, 2001.
• F. Oort, “Moduli of abelian varieties in mixed and in positive characteristic”, pp. 75–134 in Handbook of moduli, vol. III, edited by G. Farkas and I. Morrison, Advanced Lectures in Math. 26, International Press, Somerville, MA, 2013.
• V. Platonov and A. Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics 139, Academic Press, Boston, 1994.
• M. Rapoport, “On the Newton stratification”, pp. [exposé] 903, pp. 207–224 in Séminaire Bourbaki, 2001/2002, Astérisque 290, Société Mathématique de France, Paris, 2003.
• M. Rapoport, “A guide to the reduction modulo $p$ of Shimura varieties”, pp. 271–318 in Formes automorphes, I, Astérisque 298, Société Mathématique de France, Paris, 2005.
• M. Rapoport and M. Richartz, “On the classification and specialization of $F$-isocrystals with additional structure”, Compositio Math. 103:2 (1996), 153–181.
• P. Scholze and S. W. Shin, “On the cohomology of compact unitary group Shimura varieties at ramified split places”, J. Amer. Math. Soc. 26:1 (2013), 261–294.
• M. Demazure and A. Grothendieck, Schémas en groupes, Tome II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, Exposés VIII–XVIII (Séminaire de Géométrie Algébrique du Bois Marie 1962–1964), Lecture Notes in Math. 152, Springer, 1970.
• D. Shelstad, “Characters and inner forms of a quasi-split group over ${\mathbb R}$”, Compositio Math. 39:1 (1979), 11–45.
• J. Tits, “Reductive groups over local fields”, pp. 29–69 in Automorphic forms, representations and $L$-functions (Corvallis, OR, 1977), vol. 1, edited by A. Borel and W. Casselman, Proceedings of Symposia in Pure Math. 33, American Mathematical Society, Providence, RI, 1979.
• A. Vasiu, “Integral canonical models of Shimura varieties of preabelian type”, Asian J. Math. 3:2 (1999), 401–518.
• A. Vasiu, “Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic, I”, preprint, 2007.
• A. Vasiu, “Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic, II”, preprint, 2007.
• A. Vasiu, “Geometry of Shimura varieties of Hodge type over finite fields”, pp. 197–243 in Higher-dimensional geometry over finite fields, edited by D. Kaledin and Y. Tschinkel, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur. 16, IOS, Amsterdam, 2008.
• A. Vasiu, “Manin problems for Shimura varieties of Hodge type”, J. Ramanujan Math. Soc. 26:1 (2011), 31–84.
• E. Viehmann, “On the geometry of the Newton stratification”, preprint, 2015.
• E. Viehmann and T. Wedhorn, “Ekedahl–Oort and Newton strata for Shimura varieties of PEL type”, Math. Ann. 356:4 (2013), 1493–1550.
• V. E. Voskresenskiĭ, Algebraic groups and their birational invariants, Translations of Mathematical Monographs 179, American Mathematical Society, Providence, RI, 1998.
• T. Wedhorn, “Ordinariness in good reductions of Shimura varieties of PEL-type”, Ann. Sci. École Norm. Sup. $(4)$ 32:5 (1999), 575–618.
• J.-P. Wintenberger, “Existence de $F$-cristaux avec structures supplémentaires”, Adv. Math. 190:1 (2005), 196–224.
• D. Wortmann, “The $\mu$-ordinary locus for Shimura varieties of Hodge type”, preprint, 2013.
• C.-F. Yu, “On the slope stratification of certain Shimura varieties”, Math. Z. 251:4 (2005), 859–873.