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Published: 9 March 2023
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1

Scholze, Peter. "Perfectoid Shimura varieties." Japanese Journal of Mathematics 11, no. 1 (November 17, 2015): 15–32. http://dx.doi.org/10.1007/s11537-016-1484-6.

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2

Giacomini, Michele. "Holomorphic curves in Shimura varieties." Archiv der Mathematik 111, no. 4 (August 17, 2018): 379–88. http://dx.doi.org/10.1007/s00013-018-1227-4.

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Abstract We prove a hyperbolic analogue of the Bloch–Ochiai theorem about the Zariski closure of holomorphic curves in abelian varieties. We consider the case of non compact Shimura varieties completing the proof of the result for all Shimura varieties. The statement which we consider here was first formulated and proven by Ullmo and Yafaev for compact Shimura varieties.
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3

Gao, Ziyang. "Towards the Andre–Oort conjecture for mixed Shimura varieties: The Ax–Lindemann theorem and lower bounds for Galois orbits of special points." Journal für die reine und angewandte Mathematik (Crelles Journal) 2017, no. 732 (November 1, 2017): 85–146. http://dx.doi.org/10.1515/crelle-2014-0127.

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Abstract We prove in this paper the Ax–Lindemann–Weierstraß theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular, we reprove a result of Silverberg [57] in a different approach. Then combining these results we prove the André–Oort conjecture unconditionally for any mixed Shimura variety whose pure part is a subvariety of {\mathcal{A}_{6}^{n}} and under the Generalized Riemann Hypothesis for all mixed Shimura varieties of abelian type.
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4

Fargues, Laurent, Ulrich Görtz, Eva Viehmann, and Torsten Wedhorn. "Reductions of Shimura Varieties." Oberwolfach Reports 9, no. 3 (2012): 1961–2011. http://dx.doi.org/10.4171/owr/2012/32.

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5

Fargues, Laurent, Ulrich Görtz, Eva Viehmann, and Torsten Wedhorn. "Reductions of Shimura Varieties." Oberwolfach Reports 12, no. 3 (2015): 2265–328. http://dx.doi.org/10.4171/owr/2015/39.

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6

Fargues, Laurent, Ulrich Görtz, Eva Viehmann, and Torsten Wedhorn. "Arithmetic of Shimura Varieties." Oberwolfach Reports 16, no. 1 (February 26, 2020): 65–131. http://dx.doi.org/10.4171/owr/2019/2.

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7

Edixhoven, Bas, and Andrei Yafaev. "Subvarieties of Shimura varieties." Annals of Mathematics 157, no. 2 (March 1, 2003): 621–45. http://dx.doi.org/10.4007/annals.2003.157.621.

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8

Milne, J. S. "Descent for Shimura varieties." Michigan Mathematical Journal 46, no. 1 (May 1999): 203–8. http://dx.doi.org/10.1307/mmj/1030132370.

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9

Boyer, Pascal. "Conjecture de monodromie-poids pour quelques variétés de Shimura unitaires." Compositio Mathematica 146, no. 2 (January 26, 2010): 367–403. http://dx.doi.org/10.1112/s0010437x09004588.

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AbstractIn Boyer [Monodromy of perverse sheaves on vanishing cycles on some Shimura varieties, Invent. Math. 177 (2009), 239–280 (in French)], a sheaf version of the monodromy-weight conjecture for some unitary Shimura varieties was proved by giving explicitly the monodromy filtration of the complex of vanishing cycles in terms of local systems introduced in Harris and Taylor [The geometry and cohomology of some simple Shimura varieties (Princeton University Press, Princeton, NJ, 2001)]. The main result of this paper is the cohomological version of the monodromy-weight conjecture for these Shimura varieties, which we prove by means of an explicit description of the groups of cohomology in terms of automorphic representations and the local Langlands correspondence.
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10

Shin, Sug Woo. "A stable trace formula for Igusa varieties." Journal of the Institute of Mathematics of Jussieu 9, no. 4 (March 23, 2010): 847–95. http://dx.doi.org/10.1017/s1474748010000046.

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AbstractIgusa varieties are smooth varieties in positive characteristic p which are closely related to Shimura varieties and Rapoport–Zink spaces. One motivation for studying Igusa varieties is to analyse the representations in the cohomology of Shimura varieties which may be ramified at p. The main purpose of this work is to stabilize the trace formula for the cohomology of Igusa varieties arising from a PEL datum of type (A) or (C). Our proof is unconditional thanks to the recent proof of the fundamental lemma by Ngô, Waldspurger and many others.An earlier work of Kottwitz, which inspired our work and proves the stable trace formula for the special fibres of PEL Shimura varieties with good reduction, provides an explicit way to stabilize terms at ∞. Stabilization away from p and ∞ is carried out by the usual Langlands–Shelstad transfer as in work of Kottwitz. The key point of our work is to develop an explicit method to handle the orbital integrals at p. Our approach has the technical advantage that we do not need to deal with twisted orbital integrals or the twisted fundamental lemma.One application of our formula, among others, is the computation of the arithmetic cohomology of some compact PEL-type Shimura varieties of type (A) with non-trivial endoscopy. This is worked out in a preprint of the author's entitled ‘Galois representations arising from some compact Shimura varieties’.
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11

Pappas, G., and M. Rapoport. "Local models in the ramified case. III Unitary groups." Journal of the Institute of Mathematics of Jussieu 8, no. 3 (March 26, 2009): 507–64. http://dx.doi.org/10.1017/s1474748009000139.

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AbstractWe continue our study of the reduction of PEL Shimura varieties with parahoric level structure at primespat which the group defining the Shimura variety ramifies. We describe ‘good’p-adic integral models of these Shimura varieties and study their étale local structure. In the present paper we mainly concentrate on the case of unitary groups for a ramified quadratic extension. Some of our results are applications of the theory of twisted affine flag varieties that we developed in a previous paper.
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12

Tian, Yichao, and Liang Xiao. "On Goren–Oort stratification for quaternionic Shimura varieties." Compositio Mathematica 152, no. 10 (September 21, 2016): 2134–220. http://dx.doi.org/10.1112/s0010437x16007326.

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Let $F$ be a totally real field in which a prime $p$ is unramified. We define the Goren–Oort stratification of the characteristic-$p$ fiber of a quaternionic Shimura variety of maximal level at $p$. We show that each stratum is a $(\mathbb{P}^{1})^{r}$-bundle over other quaternionic Shimura varieties (for an appropriate integer $r$). As an application, we give a necessary condition for the ampleness of a modular line bundle on a quaternionic Shimura variety in characteristic $p$.
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13

Andreatta, Fabrizio, Eyal Z. Goren, Benjamin Howard, and Keerthi Madapusi Pera. "Height pairings on orthogonal Shimura varieties." Compositio Mathematica 153, no. 3 (March 2017): 474–534. http://dx.doi.org/10.1112/s0010437x1600779x.

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Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of special divisors and complex multiplication points on $M$ to the central derivatives of certain $L$-functions. Each such $L$-function is the Rankin–Selberg convolution associated with a cusp form of half-integral weight $n/2+1$, and the weight $n/2$ theta series of a positive definite quadratic space of rank $n$. When $n=1$ the Shimura variety $M$ is a classical quaternionic Shimura curve, and our result is a variant of the Gross–Zagier theorem on heights of Heegner points.
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14

Virdol, Cristian. "On L-Functions of Twisted 3-Dimensional Quaternionic Shimura Varieties." Nagoya Mathematical Journal 190 (2008): 87–104. http://dx.doi.org/10.1017/s0027763000009570.

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In this paper we compute and continue meromorphically to the entire complex plane the zeta functions of twisted quaternionic Shimura varieties of dimension 3. The twist of the quaternionic Shimura varieties is done by a mod ℘ representation of the absolute Galois group.
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15

Blasius, Don, and Lucio Guerberoff. "Complex conjugation and Shimura varieties." Algebra & Number Theory 11, no. 10 (December 31, 2017): 2289–321. http://dx.doi.org/10.2140/ant.2017.11.2289.

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16

Mok, Pila, and Tsimerman. "Ax-Schanuel for Shimura varieties." Annals of Mathematics 189, no. 3 (2019): 945. http://dx.doi.org/10.4007/annals.2019.189.3.7.

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17

Harris, Michael, and Steven Zucker. "Boundary cohomology of Shimura varieties." Inventiones Mathematicae 116, no. 1 (December 1994): 243–308. http://dx.doi.org/10.1007/bf01231562.

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18

Ullmo, Emmanuel, and Andrei Yafaev. "Algebraic flows on Shimura varieties." manuscripta mathematica 155, no. 3-4 (July 3, 2017): 355–67. http://dx.doi.org/10.1007/s00229-017-0949-0.

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19

Hida, Haruzo. "Automorphism groups of Shimura varieties." Documenta Mathematica 11 (2006): 25–56. http://dx.doi.org/10.4171/dm/203.

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20

Madapusi Pera, Keerthi. "Integral canonical models for Spin Shimura varieties." Compositio Mathematica 152, no. 4 (December 7, 2015): 769–824. http://dx.doi.org/10.1112/s0010437x1500740x.

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We construct regular integral canonical models for Shimura varieties attached to Spin and orthogonal groups at (possibly ramified) primes$p>2$where the level is not divisible by$p$. We exhibit these models as schemes of ‘relative PEL type’ over integral canonical models of larger Spin Shimura varieties with good reduction at$p$. Work of Vasiu–Zink then shows that the classical Kuga–Satake construction extends over the integral models and that the integral models we construct are canonical in a very precise sense. Our results have applications to the Tate conjecture for K3 surfaces, as well as to Kudla’s program of relating intersection numbers of special cycles on orthogonal Shimura varieties to Fourier coefficients of modular forms.
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21

Shen, Xu. "ON THE -ADIC COHOMOLOGY OF SOME -ADICALLY UNIFORMIZED SHIMURA VARIETIES." Journal of the Institute of Mathematics of Jussieu 17, no. 5 (December 1, 2016): 1197–226. http://dx.doi.org/10.1017/s1474748016000360.

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We determine the Galois representations inside the$\ell$-adic cohomology of some unitary Shimura varieties at split places where they admit uniformization by finite products of Drinfeld upper half spaces. Our main results confirm Langlands–Kottwitz’s description of the cohomology of Shimura varieties in new cases.
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22

Sankaran, Siddarth. "Unitary cycles on Shimura curves and the Shimura lift II." Compositio Mathematica 150, no. 12 (September 15, 2014): 1963–2002. http://dx.doi.org/10.1112/s0010437x14007507.

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AbstractWe consider two families of arithmetic divisors defined on integral models of Shimura curves. The first was studied by Kudla, Rapoport and Yang, who proved that if one assembles these divisors in a formal generating series, one obtains the$q$-expansion of a modular form of weight 3/2. The present work concerns the Shimura lift of this modular form: we identify the Shimura lift with a generating series comprising divisors arising in the recent work of Kudla and Rapoport regarding cycles on Shimura varieties of unitary type. In the prequel to this paper, the author considered the geometry of the two families of cycles. These results are combined with the Archimedean calculations found in this work in order to establish the theorem. In particular, we obtain new examples of modular generating series whose coefficients lie in arithmetic Chow groups of Shimura varieties.
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23

Rapoport, M., B. Smithling, and W. Zhang. "Arithmetic diagonal cycles on unitary Shimura varieties." Compositio Mathematica 156, no. 9 (September 2020): 1745–824. http://dx.doi.org/10.1112/s0010437x20007289.

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We define variants of PEL type of the Shimura varieties that appear in the context of the arithmetic Gan–Gross–Prasad (AGGP) conjecture. We formulate for them a version of the AGGP conjecture. We also construct (global and semi-global) integral models of these Shimura varieties and formulate for them conjectures on arithmetic intersection numbers. We prove some of these conjectures in low dimension.
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24

Chen, Ke, Xin Lu, and Kang Zuo. "On the Oort conjecture for Shimura varieties of unitary and orthogonal types." Compositio Mathematica 152, no. 5 (February 2, 2016): 889–917. http://dx.doi.org/10.1112/s0010437x15007794.

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In this paper we study the Oort conjecture concerning the non-existence of Shimura subvarieties contained generically in the Torelli locus in the Siegel modular variety${\mathcal{A}}_{g}$. Using the poly-stability of Higgs bundles on curves and the slope inequality of Xiao on fibered surfaces, we show that a Shimura curve$C$is not contained generically in the Torelli locus if its canonical Higgs bundle contains a unitary Higgs subbundle of rank at least$(4g+2)/5$. From this we prove that a Shimura subvariety of$\mathbf{SU}(n,1)$type is not contained generically in the Torelli locus when a numerical inequality holds, which involves the genus$g$, the dimension$n+1$, the degree$2d$of CM field of the Hermitian space, and the type of the symplectic representation defining the Shimura subdatum. A similar result holds for Shimura subvarieties of$\mathbf{SO}(n,2)$type, defined by spin groups associated to quadratic spaces over a totally real number field of degree at least$6$subject to some natural constraints of signatures.
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25

Cadoret, Anna, and Arno Kret. "Galois-generic points on Shimura varieties." Algebra & Number Theory 10, no. 9 (November 22, 2016): 1893–934. http://dx.doi.org/10.2140/ant.2016.10.1893.

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26

de Shalit, Ehud, and Eyal Z. Goren. "Theta operators on unitary Shimura varieties." Algebra & Number Theory 13, no. 8 (October 9, 2019): 1829–77. http://dx.doi.org/10.2140/ant.2019.13.1829.

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27

Rapoport, M., B. Smithling, and W. Zhang. "On Shimura varieties for unitary groups." Pure and Applied Mathematics Quarterly 17, no. 2 (2021): 773–837. http://dx.doi.org/10.4310/pamq.2021.v17.n2.a8.

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28

Kisin, Mark. "Integral canonical models of Shimura varieties." Journal de Théorie des Nombres de Bordeaux 21, no. 2 (2009): 301–12. http://dx.doi.org/10.5802/jtnb.672.

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29

James S. Milne and Junecue Suh. "Nonhomeomorphic conjugates of connected Shimura varieties." American Journal of Mathematics 132, no. 3 (2010): 731–50. http://dx.doi.org/10.1353/ajm.0.0112.

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30

Rotger, Victor. "Modular Shimura varieties and forgetful maps." Transactions of the American Mathematical Society 356, no. 4 (October 6, 2003): 1535–50. http://dx.doi.org/10.1090/s0002-9947-03-03408-1.

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31

Görtz, Ulrich, Xuhua He, and Sian Nie. "Fully Hodge–Newton Decomposable Shimura Varieties." Peking Mathematical Journal 2, no. 2 (June 2019): 99–154. http://dx.doi.org/10.1007/s42543-019-00013-2.

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32

Ullmo, Emmanuel, and Andrei Yafaev. "Holomorphic curves in compact Shimura varieties." Annales de l’institut Fourier 68, no. 2 (2018): 647–59. http://dx.doi.org/10.5802/aif.3174.

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33

Haines, Thomas J. "On Connected Components of Shimura Varieties." Canadian Journal of Mathematics 54, no. 2 (April 1, 2002): 352–95. http://dx.doi.org/10.4153/cjm-2002-012-x.

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AbstractWe study the cohomology of connected components of Shimura varieties coming from the group GSp2g, by an approach modeled on the stabilization of the twisted trace formula, due to Kottwitz and Shelstad. More precisely, for each character ϖ on the group of connected components of we define an operator L(ω) on the cohomology groups with compact supports Hic(, ), and then we prove that the virtual trace of the composition of L(ω) with a Hecke operator f away from p and a sufficiently high power of a geometric Frobenius , can be expressed as a sum of ω-weighted (twisted) orbital integrals (where ω-weighted means that the orbital integrals and twisted orbital integrals occuring here each have a weighting factor coming from the character ϖ). As the crucial step, we define and study a new invariant α1(γ0; γ, δ) which is a refinement of the invariant α(γ0; γ, δ) defined by Kottwitz. This is done by using a theorem of Reimann and Zink.
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34

Gärtner, Jérôme. "Darmon’s Points and Quaternionic Shimura Varieties." Canadian Journal of Mathematics 64, no. 6 (December 1, 2012): 1248–88. http://dx.doi.org/10.4153/cjm-2011-086-5.

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Abstract In this paper, we generalize a conjecture due to Darmon and Logan in an adelic setting. We study the relation between our construction and Kudla's works on cycles on orthogonal Shimura varieties. This relation allows us to conjecture a Gross-Kohnen-Zagier theorem for Darmon's points.
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35

Harris, Michael, and Richard Taylor. "Regular models of certain Shimura varieties." Asian Journal of Mathematics 6, no. 1 (2002): 61–94. http://dx.doi.org/10.4310/ajm.2002.v6.n1.a4.

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36

Jan Nekovar. "Eichler-Shimura relations and semisimplicity of étale cohomology of quaternionic Shimura varieties." Annales scientifiques de l'École normale supérieure 51, no. 5 (2018): 1179–252. http://dx.doi.org/10.24033/asens.2374.

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37

CHOJECKI, PRZEMYSLAW. "DENSITY OF CRYSTALLINE POINTS ON UNITARY SHIMURA VARIETIES." International Journal of Number Theory 09, no. 03 (April 7, 2013): 729–44. http://dx.doi.org/10.1142/s1793042112501576.

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38

Howard, Benjamin, and Georgios Pappas. "Rapoport–Zink spaces for spinor groups." Compositio Mathematica 153, no. 5 (April 10, 2017): 1050–118. http://dx.doi.org/10.1112/s0010437x17007011.

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After the work of Kisin, there is a good theory of canonical integral models of Shimura varieties of Hodge type at primes of good reduction. The first part of this paper develops a theory of Hodge type Rapoport–Zink formal schemes, which uniformize certain formal completions of such integral models. In the second part, the general theory is applied to the special case of Shimura varieties associated with groups of spinor similitudes, and the reduced scheme underlying the Rapoport–Zink space is determined explicitly.
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39

Cauchi, Antonio. "Norm-compatible systems of cohomology classes for GU(2,2)." International Journal of Number Theory 16, no. 03 (September 24, 2019): 461–510. http://dx.doi.org/10.1142/s1793042120500244.

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We describe work of Faltings on the construction of étale cohomology classes associated to symplectic Shimura varieties and show that they satisfy certain trace compatibilities similar to the ones of Siegel units in the modular curve case. Starting from those, we construct a two-variable family of trace compatible classes in the cohomology of a unitary Shimura variety.
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40

Linowitz, Benjamin, and Matthew Stover. "Parametrizing Shimura subvarieties of $${\mathrm{A}_1}$$ Shimura varieties and related geometric problems." Archiv der Mathematik 107, no. 3 (July 23, 2016): 213–26. http://dx.doi.org/10.1007/s00013-016-0944-9.

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41

de Shalit, Ehud, and Eyal Z. Goren. "Foliations on unitary Shimura varieties in positive characteristic." Compositio Mathematica 154, no. 11 (October 11, 2018): 2267–304. http://dx.doi.org/10.1112/s0010437x18007406.

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When$p$is inert in the quadratic imaginary field$E$and$m<n$, unitary Shimura varieties of signature$(n,m)$and a hyperspecial level subgroup at$p$, carry a naturalfoliationof height 1 and rank$m^{2}$in the tangent bundle of their special fiber$S$. We study this foliation and show that it acquires singularities at deep Ekedahl–Oort strata, but that these singularities are resolved if we pass to a natural smooth moduli problem$S^{\sharp }$, a successive blow-up of$S$. Over the ($\unicode[STIX]{x1D707}$-)ordinary locus we relate the foliation to Moonen’s generalized Serre–Tate coordinates. We study the quotient of$S^{\sharp }$by the foliation, and identify it as the Zariski closure of the ordinary-étale locus in the special fiber$S_{0}(p)$of a certain Shimura variety with parahoric level structure at$p$. As a result, we get that this ‘horizontal component’ of$S_{0}(p)$, as well as its multiplicative counterpart, are non-singular (formerly they were only known to be normal and Cohen–Macaulay). We study two kinds of integral manifolds of the foliation: unitary Shimura subvarieties of signature$(m,m)$, and a certain Ekedahl–Oort stratum that we denote$S_{\text{fol}}$. We conjecture that these are the only integral submanifolds.
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42

Lee, Min. "Mixed automorphic vector bundles on Shimura varieties." Pacific Journal of Mathematics 173, no. 1 (March 1, 1996): 105–26. http://dx.doi.org/10.2140/pjm.1996.173.105.

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43

Caraiani, Ana, Daniel R. Gulotta, Chi-Yun Hsu, Christian Johansson, Lucia Mocz, Emanuel Reinecke, and Sheng-Chi Shih. "Shimura varieties at level and Galois representations." Compositio Mathematica 156, no. 6 (May 26, 2020): 1152–230. http://dx.doi.org/10.1112/s0010437x20007149.

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We show that the compactly supported cohomology of certain $\text{U}(n,n)$- or $\text{Sp}(2n)$-Shimura varieties with $\unicode[STIX]{x1D6E4}_{1}(p^{\infty })$-level vanishes above the middle degree. The only assumption is that we work over a CM field $F$ in which the prime $p$ splits completely. We also give an application to Galois representations for torsion in the cohomology of the locally symmetric spaces for $\text{GL}_{n}/F$. More precisely, we use the vanishing result for Shimura varieties to eliminate the nilpotent ideal in the construction of these Galois representations. This strengthens recent results of Scholze [On torsion in the cohomology of locally symmetric varieties, Ann. of Math. (2) 182 (2015), 945–1066; MR 3418533] and Newton–Thorne [Torsion Galois representations over CM fields and Hecke algebras in the derived category, Forum Math. Sigma 4 (2016), e21; MR 3528275].
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44

Varshavsky, Yakov. "p-adic uniformization of unitary Shimura varieties." Publications mathématiques de l'IHÉS 87, no. 1 (December 1998): 57–119. http://dx.doi.org/10.1007/bf02698861.

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45

Varshavsky, Y. "On the characterization of complex Shimura varieties." Selecta Mathematica 8, no. 2 (June 2002): 283–314. http://dx.doi.org/10.1007/s00029-002-8107-1.

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46

Milne, J. S. "Automorphic vector bundles on connected Shimura varieties." Inventiones Mathematicae 92, no. 1 (February 1988): 91–128. http://dx.doi.org/10.1007/bf01393994.

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47

Vasiu, Adrian. "Integral Canonical Models of Unitary Shimura Varieties." Asian Journal of Mathematics 12, no. 2 (2008): 151–76. http://dx.doi.org/10.4310/ajm.2008.v12.n2.a1.

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48

Imai, Naoki, and Yoichi Mieda. "Potentially good reduction loci of Shimura varieties." Tunisian Journal of Mathematics 2, no. 2 (January 1, 2020): 399–454. http://dx.doi.org/10.2140/tunis.2020.2.399.

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49

Giacomini, Michele. "Correction to: Holomorphic curves in Shimura varieties." Archiv der Mathematik 114, no. 1 (November 18, 2019): 119–21. http://dx.doi.org/10.1007/s00013-019-01400-y.

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50

He, X., and M. Rapoport. "Stratifications in the reduction of Shimura varieties." manuscripta mathematica 152, no. 3-4 (July 15, 2016): 317–43. http://dx.doi.org/10.1007/s00229-016-0863-x.

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