On the ordinary Hecke orbit conjecture

We prove the ordinary Hecke orbit conjecture for Shimura varieties of Hodge type at primes of good reduction. We make use of the global Serre-Tate coordinates of Chai as well as recent results of D'Addezio about the $p$-adic monodromy of isocrystals. The new ingredients in this paper are a general monodromy theorem for Hecke-stable subvarieties for Shimura varieties of Hodge type, and a rigidity result for the formal completions of ordinary Hecke orbits. Along the way we show that classical Serre--Tate coordinates can be described using unipotent formal groups, generalising results of Howe.


Introduction
Let A g,n be the moduli space of g-dimensional principally polarised abelian varieties (A, λ) with level n ≥ 3 structure over F p , for a prime number p coprime to n. Recall that there are finite étale prime-to-p Hecke correspondences from A g,n to itself, and that two points x, y ∈ A g,n (F p ) are said to be in the same prime-to-p Hecke orbit if they share a preimage under one of these correspondences.Recall the following result of Chai: Theorem (Chai [6]).Let x ∈ A g,n (F p ) be a point corresponding to an ordinary principally polarised abelian variety.Then the prime-to-p Hecke orbit of x is Zariski dense in A g,n,Fp .
Our main result is a generalisation of this theorem to Shimura varieties of Hodge type.To state it, we will first to introduce some notation.
1.1.Main results.Let (G, X) be a Shimura datum of Hodge type with reflex field E and let p be a prime number.Let K p ⊂ G(Q p ) be a hyperspecial subgroup and let K p ⊂ G(A p f ) be a sufficiently small compact open subgroup.Let Sh G be the special fiber of the canonical integral model of the Shimura variety of level K p K p at a prime v above p of E, constructed in [25,27].
Let E v be the v-adic completion of E, which is a finite extension of Q p .There is a closed immersion Sh G,Fp → A g,n for some n (see [52]), and the intersection Sh G,ord of the ordinary locus of A g,n with Sh G is nonempty if and only if E v = Q p (see [35,Cor. 1.0.2]).Recall that there are also prime-to-p Hecke correspondences from Sh G to itself, and thus we can define prime-to-p Hecke orbits.
Theorem I. Suppose that E v = Q p .Then the prime-to-p Hecke orbit of a point x ∈ Sh G,ord (F p ) is Zariski dense in Sh G .
Our result generalises results of Maulik-Shankar-Tang [38], who deal with GSpin Shimura varieties associated to a quadratic space over Q and GU (1, n − 1) Shimura varieties associated to imaginary quadratic fields E with p split in E; their methods are completely disjoint from ours.There is also work of Shankar [46] for Shimura varieties of type C, using a group-theoretic version of Chai's strategy of using hypersymmetric points and reducing to the case of Hilbert modular varieties.Shankar crucially proves that the Hodge map Sh G → A g,n is a closed immersion over the ordinary locus via canonical liftings, whereas we use work of Xu [52].
Last we mention work of Zhou [53], who proves the Hecke orbit conjecture for the µ-ordinary locus of certain quaternionic Shimura varieties.Our results do not imply his, but there is some overlap between the cases that we cover.
A fairly direct consequence of Theorem I is a density result for prime-to-p Hecke orbits of an F p -point in the µ-ordinary locus of a Shimura variety of abelian type, at primes v above p of the reflex field E where E v = Q p , see Corollary 6.4.1.
1.2.Monodromy theorems.An important ingredient in our proof is an ℓ-adic monodromy theorem for prime-to-p Hecke-stable subvarieties of special fibers of Shimura varieties, in the style of [7,Cor. 3.5].To state it, let (G, X) be as above and assume for simplicity that G ad is simple over Q.Let V ℓ be the rational ℓ-adic Tate module of the abelian variety A over Sh G coming from the map Sh G,Fp → A g,n ; it is an ℓ-adic local system of rank 2g.
Theorem II.Let Z ⊂ Sh G be a smooth locally closed subvariety that is stable under the primeto-p Hecke operators.Suppose that Z is not contained in the smallest Newton stratum of Sh G .Let z ∈ Z(F p ) and let Z • ⊂ Z Fp be the connected component of Z containing z. Then the neutral component M geom of the Zariski closure of the image of the monodromy representation ρ ℓ,geom : This generalises work of Chai [7] in the Siegel case and others [21,51] in the PEL case.
In the body of the paper, we work with the integral models of Shimura varieties of Hodge type of level K p ⊂ G(Q p ) constructed in [29].Here K p is not required to be hyperspecial, for example it is allowed to be any (connected) parahoric subgroup.Our results, namely Theorem 3.2.5 and Corollary 3.2.6, are proved under the assumption that Hypothesis 2.3.1 holds.This hypothesis holds for example when G Qp is quasi-split and has no factors of type D, or when K p is hyperspecial.
We also prove results about irreducible components of smooth locally closed subvarieties that are stable under the prime-to-p Hecke operators, in the style of [7,Proposition 4.4], see Theorem 3.4.10.These results will be used to prove irreducibility of Ekedahl-Oort strata in an upcoming version of [19].
1.2.1.An overview of the proof of Theorem II.Since Z • ⊂ Sh G is defined over a finite field k we can write it as Z • k ⊗ k F p .We can then consider the Zariski closure M of the image of An argument from [7] proves that M is (isomorphic to) a normal subgroup of G Q ℓ .If G ad Q ℓ was a simple-group, then we would be done if we could show that M was not central in G Q ℓ .However, in general there are no primes ℓ such that G ad Q ℓ is simple and so at this point we have to deviate from the strategy of [7].
Instead, we control M by studying the centraliser I x,ℓ ⊂ G Q ℓ of the image of Frobenius elements Frob x ∈ π ét 1 (Z • k , z) corresponding to points x ∈ Z k (F q ).Since the paper [29] makes an in-depth study of these Frobenius elements, we can make use of their results about these centralisers.For example, if x is not contained in the basic locus, then Frob x is not central.To get more precise results, we need to know that the element Frob x ∈ G(Q ℓ ) is defined over Q, which is what Hypothesis 2.3.1 makes precise.
In this way we can show that M ⊂ G Q ℓ is a normal subgroup that surjects onto G ad Q ℓ .The result about M geom ⊂ M will be deduced from this.1.3.A sketch of the proof of Theorem I. Let x ∈ Sh G (F p ) be an ordinary point, and let Z be the Zariski closure inside Sh G,ord of the prime-to-p Hecke orbit of x.Let y ∈ Z(F p ) be a smooth point of Z. Recall that it follows from the theory of Serre-Tate coordinates that the formal completion A /y g,n of A g,n at y is a formal torus.A special case of the main result of [47] tells us that S /y := Sh /y G,Fp ⊂ A /y g,n is a formal subtorus.Work of Chai on the deformation theory of ordinary p-divisible groups [5] tells us that the dimension of the smallest formal subtorus of S /y containing Z /y , is encoded in the unipotent radical of the p-adic monodromy group of the isocrystal M associated to the universal abelian variety A over Z.
Using Theorem II and results of D'Addezio, [12,13], we compute the monodromy group of M over Z.It follows from this computation that the smallest formal subtorus of S /y containing Z /y is equal to S /y .We conclude by proving that the formal completion Z /y is a formal subtorus of S /y .By the rigidity theorem for p-divisible formal groups of Chai [9], it suffices to give a representation-theoretic description of the Dieudonné module of S /y .Unfortunately, the description of the subtorus S /y coming out of the work of [47] does not readily lend itself to understanding its Dieudonné module.
Instead, we give a different proof that S /y is a subtorus of A /y g,n .We do this by giving a new description of Serre-Tate coordinates in terms of actions of formal unipotent groups on Rapoport-Zink spaces, generalising results of Howe [20] in the case g = 1.Once we have this perspective, the results of [24] give an explicit description of the Dieudonné module of the torus A /y g,n as well as the Dieudonné module of the subtorus S /y .1.4.Outline.Sections 2 and 3 form the first part of the paper and work in a more general setting than the rest of the paper.In Section 2 we introduce the integral models of Shimura varieties of Hodge type constructed in [29].We recall results and notation from loc. cit., in particular, about the Frobenius elements and their centralisers associated to F p -points of these models.In Section 3 we prove monodromy theorems for Hecke-stable subvarieties of the special fibers of these integral models, combining results of [29] with ideas of [7].
Section 4 is a standalone section on Serre-Tate coordinates.In it, we show that the classical Serre-Tate coordinates, as described in [22], can be reinterpreted using actions of unipotent formal groups as in [20].This section should be of independent interest.
In Section 5, we specialise to the smooth canonical integral models of Shimura varieties of Hodge type at hyperspecial level, and we moreover assume that the ordinary locus is nonempty.We reprove a result of [47], which states that the formal completion of the ordinary locus gives a subtorus of the Serre-Tate torus, and give a group-theoretic description of its Dieudonné module.At the end of this section we also give a short interlude on strongly nontrivial actions of algebraic groups on isocrystals, which we will need to confirm the hypotheses of the rigidity theorem of [9].
In Section 6, we put everything together and prove Theorem I. We end by deducing a result for Shimura varieties of abelian type.1.5.Acknowledgements.It is clear that our approach owes a substantial intellectual debt to the work of Chai and Oort, and in fact our main results were conceived after reading a remark in [8].We are very grateful to the referee for pointing out a serious error in a previous version, for a very detailed reading of a second version and for many helpful comments.We thank Sean Cotner for helpful discussions.

Integral models of Shimura varieties of Hodge type
Let (G, X) be a Shimura datum of Hodge type.In this section we follow [29,Sec. 1.3] and construct integral models for the Shimura varieties associated to (G, X) in a very general situation.The main goal is to introduce various Frobenius elements γ x,m,ℓ ∈ G(Q ℓ ) associated to F q m -points of these integral models, and to discuss result of [29] about their centralisers I x,m,ℓ .We end by introducing Hypothesis 2.3.1, which will be assumed throughout Section 3, and prove that it holds under minor assumptions.2.0.1.Hodge cocharacters.If (G, X) is a Shimura datum, then for each x ∈ X there is a cocharacter [29,Sec. 1.2.3] for the precise definition.The G(C)-conjugacy class of µ x does not depend on the choice of x and we will write {µ X } for this conjugacy class, and denote it by {µ} if X is clear from context.This conjugacy class of cocharacters is defined over a number field E ⊂ C, called the reflex field.
2.1.The construction of integral models.For a symplectic space (V, ψ) over Q we write G V := GSp(V, ψ) for the group of symplectic similitudes of V over Q.It admits a Shimura datum H V consisting of the union of the Siegel upper and lower half spaces.Let (G, X) be a Shimura datum of Hodge type with reflex field E and let (G, X) → (G V , H V ) be a Hodge embedding.
Fix a prime p and choose a Z (p) -lattice V (p) ⊂ V on which ψ is Z p -valued, and write , and similarly write K p for the stabiliser of V p in G(Q p ). 1 For every sufficiently small compact open subgroup K p ⊂ G(A p f ) we can find by varying the symplectic space and the Hodge embedding contains all stabilisers of vertices in the extended Bruhat-Tits building of G Qp .It is moreover explained in loc.cit.that this collection is stable under finite intersections.
of Shimura varieties of level K = K p K p and K = K p K p , respectively.We let S K over Z (p) be the moduli-theoretic integral model of Sh K (G V , H V ); it is a moduli space of polarised abelian schemes (A, λ) up to prime-to-p isogeny with level K p -structure.Fix a prime v | p of E and let . This construction is compatible with changing the level away from p and we define Then as discussed in [29, Sec.2.1], the transition maps in both inverse systems are finite étale and moreover G(A p f ) acts on S Kp .Let k = F q be the residue field of O E,(v) , and write Sh G,Kp for the special fiber of S Kp and Sh G,K p Kp for the special fiber of S K p Kp ; these are both schemes over k and G(A p f ) acts on Sh G,Kp .We will write Sh G V ,K p Kp for the special fiber of Let V p be the prime-to-p adelic Tate module of the universal abelian variety f -multiple of the Weil pairing.Here A p f denotes the pro-étale sheaf associated to the topological group A p f .
2.1.1.Tensors.Write V ⊗ for the direct sum of V ⊗n ⊗ (V * ) ⊗m for all pairs of integers m ≥ 0, n ≥ 0. We will also use this notation later for modules over commutative rings and modules over sheaves of rings.
As in [29,Sec. 1.3.4],we fix tensors {s α ∈ V } ⊂ V ⊗ such that G is their pointwise stabiliser in GL(V ).Then as explained in [29,Sec. 1.3.4,Sec. 2.1.2],there are global sections such that if we restrict the isomorphism ǫ via S Kp → S Kp we get an isomorphism In particular, for each x ∈ S Kp (F p ) the stabiliser of the tensors 2.1.2.We will use Zp to denote the p-typical Witt vectors W (F p ) of F p and we set Qp = Zp [1/p].We let σ : Zp → Zp be the automorphism induced by Frobenius on F p , and also denote by σ the induced automorphism of Qp .Let x ∈ Sh G,K p Kp (F p ) and let D x be the rational contravariant Dieudonné module of the pdivisible group A x [p ∞ ] of the abelian variety A x , equipped with its Frobenius φ.By [29,Prop. 1.3.7]there are φ-invariant tensors {s α,cris } ⊂ D ⊗ x and in [29,Sec. 1.3.8] it is argued that there is an isomorphism Qp ⊗ V → D x sending 1 ⊗ s α to s α,cris .See the statement of [29,Prop. 1.3.7]for a characterisation of the tensors s α,cris .
Under such an isomorphism, the Frobenius φ corresponds to an element b x ∈ G( Qp ), which is well defined up to σ-conjugacy, where σ : G( Qp ) → G( Qp ) is induced by σ : Qp → Qp .In other words, we can associate to φ a well defined element [b x ] of the Kottwitz set B(G) = B(G Qp ) of [33].By [29,Lem. 1.3.9] the element [b x ] is contained in the neutral acceptable set B(G, {µ −1 }) consisting of the {µ −1 }-admissible elements defined in [29,Sec. 1.1.5].Here we use {µ} to denote the G(Q p ) conjugacy class of cocharacters induced by the place v of E, where we recall that {µ} was introduced in Section 2.0.1.
It follows from [29,Thm. 1.3.14]that there are locally closed subschemes Sh Here we are using the partial order on B(G, {µ 2.2.Centralisers.Let x ∈ Sh G,Kp (F p ) and choose a sufficiently divisible integer m such that the image of x in Sh G,K p Kp (F p ) is defined over F q m .Then the geometric q m -Frobenius Frob q m acts on V p via tensor-preserving automorphisms and therefore determines an element γ p x,m ∈ G(A p f ), which depends on x and m.For ℓ = p there is an element δ where we write q = p r and where σ denotes the Frobenius on G(Q q m ).
We define to be the centraliser of γ p x,m , which does not depend on m as long as m is sufficiently divisible.We similarly define I x,ℓ ⊂ G Q ℓ for ℓ = p to be the centraliser of the projection γ x,m,ℓ of γ p x,m to G Q ℓ for sufficiently divisible m.We define I x,m,p to be the algebraic group over Q p whose functor of points is given by where σ is induced by σ : G(Q q m ) → G(Q q m ).As explained in [29,Sec. 2.1.7],the base change I x,m,p ⊗ Q q m is naturally identified with the centraliser of the semisimple element γ x,m,p in G(Q q m ), and I x,m,p is thus reductive.We similarly define J δx,m by its functor of points

Consider the decomposition
of G ad into simple groups over Q.Let δ x,m,i and γ x,m,p,i be the images of δ x,m and γ x,m,p in G i (Q q m ).

Lemma 2.2.2.
There is a product decomposition where I x,m,p,i represents the functor on Q p -algebras sending R to Similarly there is a product decomposition where J δ x,m,i represents the functor on Q p -algebras sending R to Proof.Consider the commutative diagram Since the kernel of the bottom map is central and the bottom map is surjective, it follows that the natural map I x,p → n i=1 I x,m,p,i is surjective.The kernel is given by the intersection of I x,m,p with the kernel of the bottom map and thus has the following functor of points: This forces g = σ(g) and so g ∈ Z G (R) ⊂ Z G (Q q m ⊗ Qp R).The same proof shows that there is a product decomposition J δx,m /Z G ≃ n i=1 J δ x,m,i .Note that I x,m,p,i ⊗ Q q m can be identified with the centraliser of γ x,m,p,i in G i,Q q m as in the beginning of Section 2.2.The centraliser of γ x,m,p,i ∈ G( Qp ) does not depend on m for m sufficiently divisible, and thus the group I x,m,p does not depend on m for m sufficiently divisible.We will write I x,p for the group I x,m,p for sufficiently divisible m and similarly I x,i,p for the group I x,m,p,i .We will identify I x,p ⊗ Qp with the centraliser of γ x,m,p in G( Qp ) for sufficiently divisible m and similarly identify I x,i,p with the centraliser of γ x,m,p,i in G i ( Qp ).

Let
x ∈ Sh G,Kp (F p ) and let Aut(A x ) be the algebraic group over Q with functor of points Following [29, Sec.2.1.3],we define I p x to be the largest closed subgroup of Aut(A x ) that fixes the tensors s α,A p f and I x ⊂ I p x to be the largest closed subgroup that also fixes the tensors s α,cris .There are natural maps I x,Q ℓ → I x,ℓ for all (including ℓ = p), see [29,Sec. 2.1.8]for the ℓ = p case.
The groups I x,ℓ are connected reductive subgroups of G Q ℓ and in fact Levi subgroups over Q ℓ .By [29, Cor.2.1.9]for all ℓ (including ℓ = p) the natural map is an isomorphism.This induces closed immersion of groups I x,Qp → J δx,m for some sufficiently divisible m.

2.3.
An assumption.We will need to assume the following hypothesis to prove our main monodromy theorems in Section 3.
Hypothesis 2.3.1.For all points x ∈ Sh G,Kp (F p ) and for sufficiently divisible m depending on x, there is an element γ x,m ∈ G(Q) that is conjugate to γ x,m,ℓ in G(Q ℓ ) for all ℓ (including ℓ = p).Moreover the G(Q)-conjugacy class of γ x,m is stable under the action of Gal(Q/Q).
If the G(Q)-conjugacy class of γ x,m contains an element of G(Q), then it is clearly Galois stable.However the converse does not necessarily hold.Lemma 2.3.2.The hypothesis holds when K p is hyperspecial.
Proof.If K p is hyperspecial, then [28,Cor. 2.3.1]tells us that there is an element γ x,m ∈ G(Q) that is conjugate to γ x,m,ℓ in G(Q ℓ ) for all ℓ (including ℓ = p).
Remark 2.3.3.By [29, Cor.2.2.14], an element γ x,m ∈ G(Q) satisfying the requirements of Hypothesis 2.3.1 exists when G Qp is quasi-split and has no factors of type D. If K p is very special, the group G Qp is tamely ramified and satisfies p ∤ #π 1 (G der ) and π 1 (G) I is torsion free, where I ⊂ Gal(Q p /Q p ) is the inertia group, then the existence of an element γ x,m ∈ G(Q) satisfying the requirements of Hypothesis 2.3.1 follows from Theorem I of [19] If K p is a very special parahoric subgroup and the triple (G, X, K p ) is acceptable in the sense of [31, Def.5.2.6, Def.5.2.9], then [31,Thm. 6.1.4]proves the existence of an element γ x,m ∈ G(Q) satisfying the requirements of Hypothesis 2.3.1.
Remark 2.3.4.When G Qp is not quasi-split, one should probably not expect that the G(Q)conjugacy class of γ x,m always contains an element of G(Q).This is because CM lifts do not exist in general when G Qp is not quasi-split.However, we expect Hypothesis 2.3.1 to hold in full generality.
For example, let x ∈ Sh G,Kp (F p ) be a point corresponding to the good reduction of an abelian variety defined over a number field and assume that p > 2. Then [31,Thm. 7.2.4]tells us that there is an element γ x,m ∈ G(Q) satisfying the requirements of Hypothesis 2.3.1.2.3.5.We end by deducing a consequence of Hypothesis 2.3.1 that will be used in Section 3. Let G * denote the quasi-split inner form of G over Q and let Ψ : Lemma 2.3.6.Suppose that Hypothesis 2.3.1 holds and let γ x,m ∈ G(Q) be the element that is guaranteed to exist by that Hypothesis.Then for sufficiently divisible m the element Proof.If m is sufficiently divisible, then the centraliser of γ x,m is connected because this is true for γ x,m,ℓ and the formation of centralisers commutes with base change.Since G * is quasi-split and the element Ψ(γ x,m ) is semisimple with connected centraliser, we may apply [32,Thm. 4.7.(2)] which tells us that the G * (Q)-conjugacy class of Ψ(γ x,m ) contains an element of G * (Q).

Monodromy of Hecke-invariant subvarieties
In this section we prove an ℓ-adic monodromy theorem in the style of Chai [7] (c.f.[21,51]) for prime-to-p Hecke stable subvarieties of Shimura varieties of Hodge type in characteristic p.We expect the results in this section to be of independent interest, at least beyond the hyperspecial case that we will use in the rest of this article.
In Section 3.1 we establish formal properties of subvarieties Z of Shimura varieties of Hodge type in characteristic p that are stable under prime-to-p Hecke operators.Using techniques from [7] we prove that the ℓ-adic monodromy groups of the universal abelian variety over such Z are normal subgroups of G Q ℓ , this is stated as Corollary 3.1.16.
In Section 3.2 we use the results from [29] in combination with Hypothesis 2.3.1 to prove Theorem 3.2.5 and Corollary 3.2.6; the latter is a generalisation of Theorem II.In Section 3.3 we combine this theorem with results of D'Addezio [13] to deduce results about the p-adic monodromy groups of the universal abelian variety over Hecke stable subvarieties.
Finally, in Section 3.4 we prove results about irreducible components of Hecke stable subvarieties in the style of [7,Prop. 4.5.4].We will not use these results in the rest of this article and so this section can safely be skipped for the reader only interested in the proof of Theorem I.
3.1.Arithmetic monodromy groups I. Let the notation be as in Section 2. In this section we are going to study arithmetic monodromy groups of Hecke stable subvarieties of Sh G,K p Kp .For maximal generality, we do not assume that these are defined over k = F q and so from now on we will implicitly base change the Shimura variety Sh G,K p Kp to an unspecified finite extension of k, which we will also denote by k.
The morphism π : Sh G,Kp → Sh G,K p Kp is a pro-étale K p -torsor over Sh G,K p Kp such that the action of K p ⊂ G(A p f ) extends to an action of G(A p f ).Let Z ⊂ Sh G,K p Kp be a locally closed subscheme and let Z be the inverse image of Z under π.We say that Z is stable under the prime-to-p-Hecke operators, For the rest of this section ℓ will be used to denote a prime number not equal to p.For such ℓ we let K ℓ be the image of be the induced pro-étale K ℓ -torsor.For Z ⊂ Sh G,K p Kp a locally closed subscheme we will write Z ℓ for the inverse image of Z under π ℓ .We say that Z is stable under the ℓ-adic Hecke operators, -stable for all ℓ = p.All the results in this section will be stated for smooth Z, and the following lemma will be used to reduce to the smooth case in the proof of Theorem I. Lemma 3.1.1.Let Z ⊂ Sh G,K p Kp be a locally closed subscheme that is stable under the action of G(A p f ) (respectively G(Q ℓ )), then the smooth locus U ⊂ Z is also stable under this action.
Proof.For g ∈ G(A p f ) and K p ⊂ G(A p f ) there is a finite étale correspondence Sh G,(K p ∩gK p g −1 )Kp and the assumption that Z is stable under the action of g is equivalent to the statement that the inverse image of Z under p 1 is the same as the inverse image of Z under g • p 2 for all choices of K p .Because all the maps in the diagram are finite étale, the same is true for the smooth locus U of Z. Therefore the inverse image Ũ of U under π is stable under the action of g ∈ G(A p f ).
Proof.This follows in the same way as in the proof of Lemma 3.1.1from the fact that the primeto-p Hecke correspondences are finite étale; indeed finite étale maps are open and closed, and thus take closures to closures.
3.1.3.Some general topology.Let {X i } i∈I be a countably indexed cofiltered inverse system of finite type schemes over a field k with surjective affine transition maps.Let X = lim ← −i X i be the inverse limit, it is a nonempty quasicompact scheme by [49,Lem. 01Z2].Recall that for a quasicompact scheme Y there is a profinite topological space π 0 (Y ) of connected components of Y .
Proof.The left hand side of (3.1.2) is a profinite topological space by [49,Lem. 0906] and the right hand side of (3.1.2) is visibly an inverse limit of finite sets.Hence both sides are compact Hausdorff topological spaces and to show that the map is a homeomorphism it suffices to show that it is a bijection.
To show that the natural map is a bijection, we construct an explicit inverse.Any compatible system of connected components {V i } i∈I of {X i } i∈I has nonempty and quasicompact inverse limit V ⊂ X by [49, Lem.0A2W].To prove that V is connected we suppose that there are nonempty open and closed subsets W and ] tells us that we can find i and (nonempty In particular, the subsets Z and Z ′ are disjoint nonempty open subsets of V i , which gives us a contradiction since V i is connected. We have produced a map lim ← −i π 0 (X i ) → π 0 (X) and it is not hard to check that it is an inverse of the natural map from the lemma; this concludes the proof.
Proof.The existence of the action follows from the existence of the action on Z (resp.Z ℓ ).The continuity follows from the continuity of the action of K p on lim ← −K p Z K p (resp. the continuity of the action of K ℓ on Z ℓ ) and Lemma 3.1.4.
The following lemma is only a slight generalisation of [7, Lem.2.8], but we include a proof for the benefit of the reader.Lemma 3.1.6.Let X be a second-countable compact Hausdorff topological space with a transitive and continuous action of a locally profinite topological group G. Let x ∈ X with stabiliser G x ⊂ G, then the orbit map Proof.We can write G as the increasing union of countably many compact open sets, for example by using finite unions of cosets of a compact open subgroup K ⊂ G. Since the quotient map G → G/G x is open for any topological group, it follows that G/G x can be written as the increasing union of countably many compact open subsets.
Since the orbit map is surjective, the topological space X can be written as a countable union of compact subsets O(U ) for U ⊂ G/G x compact open.Because X is second-countable it is metrisable by Urysohn's metrisation theorem and thus the Baire category theorem tells us that there exists a compact open subset ) is a continuous bijection between compact Hausdorff topological spaces and hence a homeomorphism.Now note that G acts transitively on both G/G x and on X. Hence by moving around V we see that any point of y ∈ G/G x has an open neighborhood V y such that the natural map O : V y → O(V y ) is a homeomorphism, and we conclude that O is a homeomorphism.
where the bracket notation means the commutator of two Lie subalgebras and where M is the Zariski closure of M .
Proof.The group M is an ℓ-adic Lie group by [17,Prop. 2.3] and the morphism M → H(Q ℓ ) is a morphism of ℓ-adic Lie groups by [17,Prop. 2.2].This implies that there is an induced morphism on Lie algebras Lie is Zariski dense, it follows that the smallest algebraic subgroup of H whose Lie algebra contains Lie M is equal to M; indeed, if there is a smaller algebraic subgroup M ′ ⊂ M with Lie M ⊂ Lie M ′ , then we see using the ℓ-adic exponential map that there is a finite index subgroup of M contained in M ′ (Q ℓ ).This contradicts the fact that M is Zariski dense in M.
The fact that the smallest algebraic subgroup of H whose Lie algebra contains Lie M is equal to M is expressed as a(Lie M ) = Lie M in the notation of [2,Sec. 7.1].By [2,Cor. 7.9] we have the following equality of Lie subalgebras of Lie H Lemma 3.1.9.Let M be a semisimple algebraic group over Q ℓ and let M ⊂ M(Q ℓ ) be subgroup closed in the ℓ-adic topology.If M equipped with the subspace topology is compact and Proof.It follows from Lemma 3.1.8that M is an ℓ-adic Lie group, that M → M(Q ℓ ) is a morphism of ℓ-adic Lie groups and that the Lie algebra of M is equal to the Lie algebra of M, since M is semisimple.Now we can use the exponential map for ℓ-adic Lie groups to show that 3.1.10.The main theorem of Galois theory for schemes tells us that the category of finite-étale covers of a smooth connected scheme Z over k is equivalent to the category of finite sets equipped with a continuous action of π ét 1 (Z, z).Under this equivalence, a finite étale cover f : Y → Z is sent to the finite set f −1 (z) equipped with its action of π ét 1 (Z, z).In particular, the set of connected components of Y is in bijection with the set of orbits of π ét 1 (Z, z) on f −1 (z).If f : Y → Z is a countably indexed inverse limit of finite étale covers f i : Y i → Z with surjective transition maps, then we can associate to f the profinite set equipped with its natural continuous action of π ét 1 (Z, z).By Lemma 3.1.4it follows that the profinite set of orbits of π ét 1 (Z, z) on f −1 (z) is homeomorphic to the topological space of connected components of Y .
3.1.11.Now let Z be a smooth G(Q ℓ )-stable locally closed subscheme of Sh G,K p Kp , let Z • ⊂ Z be a connected component of Z, and let z ∈ Z • (F p ).Let π ℓ be as in (3.1.1)and write Z ℓ for the inverse image of Z under π ℓ as above; it is stable under the action of G(Q ℓ ) by assumption.Denote by By the Galois theory for schemes discussed above, the cover π ℓ : Z • ℓ → Z • corresponds to the profinite set π −1 ℓ (z) equipped with its natural action of π ét 1 (Z • , z).In particular, the set of connected components of Z • ℓ corresponds to the set of orbits of π ét 1 (Z • , z) on π −1 ℓ (z).Choose an element z ∈ π −1 ℓ (z).Then using the simply transitive action of K ℓ on π −1 ℓ (z) we can identify π −1 ℓ (z) with K ℓ ; under this identification the chosen element z is send to 1 ∈ K ℓ .This defines a continuous group homomorphism It is a closed topological subgroup by the continuity of the action and the fact that π 0 (Z ℓ ) is Hausdorff.Its intersection with K ℓ gives us the stabiliser of y in K ℓ .The action map gives us a continuous map with image the orbit Orb(y) of y.
Proof.The identification tells us that there are finitely many K ℓ -orbits on π 0 (Z ℓ ), and that each of them is open and closed.The G(Q ℓ )-orbit of a point y is then a union of finitely many K ℓ -orbits, and thus also open and closed.Lemma 3.1.12shows that Orb(y) is open and closed inside a second-countable profinite topological space.Therefore Orb(y) is profinite and second-countable.The result now follows from Lemma 3.1.6.
Let M be the image of ρ ℓ and let M be the neutral component of its Zariski closure inside G(Q ℓ ).Let ρ ℓ,geom be the restriction of ρ ℓ to ), let M geom be its image and let M geom be the neutral component of its Zariski closure inside G(Q ℓ ).Lemma 3.1.13.The groups M and M geom are connected reductive groups over Q ℓ .
Proof.There is a short exact sequence (e.g. by [13, where Q is a commutative algebraic group of multiplicative type and where M ′ geom is a closed subgroup of M with neutral component given by M geom .In particular, it follows that M is reductive if M geom is reductive.The representation is the monodromy representation of the (rational) ℓ-adic Tate module of the abelian scheme π : A → Sh G,K p Kp coming from the Hodge embedding Sh G,K p Kp → Sh G V ,K p Kp .This is Tate-module is an ℓ-adic sheaf F 0 on Z • which is pure of weight one.Then [15,Thm. 3.4.1.(iii)]tells us that the basechange F of F 0 to Z • Fp is semi-simple.This base change corresponds to the composition of ρ ℓ,geom with [15,Cor. 1.3.9]tells us that M geom is a semi-simple algebraic group, and thus that it is reductive.Proof.Let γ ∈ P y , then we want to show that γ normalises M.
Since conjugation by γ is a homeomorphism in the Zariski topology, we see that the Zariski closure of M ∩ U is moved under conjugation by γ into the Zariski closure of M .But since M ∩ U is an open subgroup of M it is also a closed subgroup and thus compact and thus of finite index in M .This means that the Zariski closure of M ∩ U and the Zariski closure of M have the same identity component, both of which are equal to M. Since conjugation preserves 1, this mean it sends M to M.
3.2.Arithmetic monodromy groups II.So far we have not excluded the possibility that M is contained in the center of G Q ℓ .In fact, this happens when Z is the supersingular locus inside the modular curve.Thus we will need additional assumptions on Z to prove that M is not central.
We will show, using the results of [29], that if Z contains a point x ∈ Z(F q m ) not contained in the smallest Newton stratum, then the image of Frob x under ρ ℓ is noncentral.If G ad Q ℓ were a simple group over Q ℓ , then this would force M to contain G der Q ℓ .However G ad is generally not a simple group over Q, and even if it were simple then there would generally be no primes ℓ where G ad Q ℓ is simple.To deal with these issues, we will make use of Hypothesis 2.3.1.

Recall that for a point
If m is sufficiently divisible, then its centraliser is equal to the group I x,ℓ .

The decomposition
Proof.Let m be sufficiently divisible and let γ x,m ∈ G(Q) be the element guaranteed to exist by Hypothesis 2.3.1.Let G * denote the quasi-split inner form of G over Q and let Ψ : G ⊗ Q → G * ⊗ Q be an inner twisting.Then by Lemma 2.3.6 there is an element γ By the classification of adjoint algebraic groups we can find number fields2 F 1 , • • • , F n and absolutely simple adjoint algebraic groups We have a similar decomposition for G * ,ad with H i replaced by its quasi-split inner form H * i and we will write G * i for the restriction of scalars from ) be the image of γ x,m and let C x,i ⊂ H * i be its centraliser.Then there is a product decomposition for sufficiently divisible m be as in Section 2.1.2.Then Lemma 2.2.1 shows that there is a product decomposition J b /Z G,Qp ≃ n i=1 J b i , where each J b,i is the twisted centraliser of the image b i of b in G i ( Qp ).Moreover the natural inclusion I x,p → J b induces closed immersions I x,i,p → J b i .As in [29, Sec.1.1.4],there is an inclusion J b,i, Qp → G i, Qp , identifying its image with the centraliser M ν b i of the fractional cocharacter can be identified with the centraliser of γ x,m,p for m sufficiently divisible, and it follows that In particular, the group I ′ x,i,Q p is conjugate to I x,i,p,Q p and therefore of the same dimension.
The upshot of the above discussion is that Dim To deduce the last statement of Proposition 3.2.3,we note that it suffices to show that the image of γ x,m,i in every simple factor of G i,Q ℓ is noncentral.For this, we fix i and a prime number ℓ.
Then we can write G * i,Q ℓ as a product indexed by primes p of The element γ x,m,i is noncentral in H * i (F i ) and thus also noncentral in H * i (F i,p ) for all primes p of F i dividing ℓ, and thus we are done.Remark 3.2.4.When b i is basic then the inclusion I x,i,Qp ⊂ J b,i should be an equality and the image of ρ ℓ (Frob x,m ) in G i (Q ℓ ) should be trivial for all ℓ = p.This is true when K p is very special, see the proof of [18,Prop. 5.2.10].
We now state and prove our main arithmetic monodromy theorem.Theorem 3.2.5.Let Z ⊂ Sh G,K p Kp be a smooth G(Q ℓ )-stable locally closed subvariety.Let Z • ⊂ Z be a connected component and choose a point z ∈ Z • (F p ).Let M be the neutral component of the Zariski closure of the image of for all sufficiently divisible m, where we recall that M is the image of ρ ℓ .Thus γ x,m,ℓ is contained in the Zariski closure of M and thus after replacing m by a power we may assume that γ x,m,ℓ ∈ M, the neutral component of the Zariski closure of M .
This image is moreover normal, so to show that it is equal to G i,Q ℓ it suffices to show that it maps nontrivially to every simple factor of G i,Q ℓ over Q ℓ .But this follows from the last line of the statement of Proposition 3.2.3.Corollary 3.2.6.Let Z ⊂ Sh G,K p Kp,Fp be a smooth G(Q ℓ )-stable locally closed subscheme as before, let Z • ⊂ Z be a connected component and fix z ∈ Z • (F p ).Let M geom be the neutral component of the Zariski closure of the image of Assume that Hypothesis 2.3.1 holds.Then M geom is a normal subgroup of G Q ℓ surjecting onto G i,Q ℓ for all i such that there is a point x ∈ Z • (F p ) with b x,i non-basic.If we can find such a point for all i, then Proof.The subscheme Z is defined over a finite extension of k, and so we can speak of its arithmetic monodromy group M. Theorem 3.2.5 tells us that M surjects onto G i,Q ℓ for all i such that there is a point x ∈ Z • (F p ) with b x,i non-basic.We now claim that M geom and M have the same image in G i,Q ℓ for all such i.
It follows from the short exact sequence (3.1.3)that M geom ⊂ M is a normal subgroup with abelian cokernel.Let M geom,i be the image of M geom in G i,Q ℓ and let M i be the image of M in G i,Q ℓ .Then M geom,i ⊂ M i is a normal subgroup with abelian cokernel.Given an integer i with 1 ≤ i ≤ n such that there is a point x ∈ Z • (F p ) with b x,i non-basic, then M i = G i,Q ℓ and therefore M i has no nontrivial abelian quotients.Thus it follows that the inclusion M geom,i ⊂ M i is an equality.
If we can find a point x with b x,i non-basic for all i, then M geom surjects onto G ad Q ℓ and is moreover semi-simple by [15,Cor. 1.3.7].It must therefore be equal to G der Q ℓ .
3.3.p-adic monodromy groups.In this subsection we record a consequence of Theorem 3.2.5 in combination with the main results of [12,13].
Recall the following notions from [13, Sec.2.2].Write F −Isoc(S) for the Q p -linear Tannakian category of F -isocrystals over a smooth finite type scheme S over F p and write F −Isoc † (S) for the Q p -linear Tannakian category of overconvergent F -isocrystals over S.There is a natural fully faithful embedding F −Isoc † (S) ⊂ F −Isoc(S) which sends an overconvergent F -isocrystal M † to the underlying F -isocrystal M. Similarly we write Isoc † (S) and Isoc(S) for the Qp -linear category of (overconvergent) isocrystals over S.There are natural faithful forgetful functors from (overconvergent) F -isocrystals to (overconvergent) isocrystals.

The morphism Sh
gives us an abelian scheme π : A → Sh G,K p Kp and we consider the F -isocrystal which is overconvergent by [16,Thm. 7].Then [29,Cor. 1.3.13]proves that there is an exact Q p -linear tensor functor (the p-adic realisation functor ) such that the representation G Qp → G V → GL V coming from the choice of Hodge embedding is sent to the F -isocrystal M. Lemma 3.3.2.This morphism factors via an exact Q p -linear tensor functor which we will also denote by Rel p .
Proof.Recall that we have chosen finitely many tensors s α ∈ V ⊗ cutting out G inside GL V , which correspond to morphisms s α : 1 → V ⊗ in Rep Qp G. Applying the tensor functor we get morphism of F -isocrystals Since M is overconvergent and the category of overconvergent isocrystals is stable under tensor products and duals and direct sums by [1, Rem.2.3.3.(iii)], it follows that M ⊗ is also overconvergent.Thus the morphisms (3.3.3)live in the full subcategory F −Isoc † (Sh G,K p Kp ) and by [30,Lem. 4.1.8]we get a Q p -linear exact tensor functor Given a smooth locally closed subscheme Z ⊂ Sh G,K p Kp,Fp and a point z ∈ Z(F p ), there are monodromy groups Mon(Z, M, z) ⊂ Mon(Z, M † , z), which are algebraic groups over Qp , see the introduction of [12].They are defined to be the Tannakian groups corresponding to the smallest Tannakian subcategory of Isoc(Z) respectively Isoc † (Z) containing M, via the fiber functor ω z ω z : Isoc(Z) → Isoc(F p ) = Vect Qp .
We will often omit the chosen point z from the notation since the monodromy group does not depend on z up to isomorphism.Corollary 3.3.3.Let Z ⊂ Sh G,K p Kp be a smooth locally closed subscheme and assume that there is a prime ℓ = p such that Z is G(Q ℓ )-stable.Suppose that Z • contains a point x such that b x,i is non-basic for all i.If Hypothesis 2.3.1 holds, then there is an equality of subgroups of G Qp Proof.Let ℓ be as in the assumptions of the statement of the Corollary.Then it follows from Theorem 3.2.6 that the ℓ-adic monodromy group M geom of the abelian variety over Z is equal to G der Q ℓ .It follows from [13, Thm.1.2.1] (c.f.[41]) that there is an isomorphism of algebraic groups ) is equal to its own derived subgroup and therefore contained in G der ⊗ Qp .This inclusion has to be an isomorphism for dimensions reasons, because both groups are connected.
From now on we will assume that Z is contained in a single Newton stratum Sh G,[b],K p Kp of Sh G,K p Kp .This means that for every representation W of G Qp the Newton polygon of Rel p (W ) is constant.As explained in [12, Sec.2.2] (c.f.[23]), this means that Rel p (W ) admits a (unique) slope filtration Rel p (W ) • .There is an induced slope filtration on ω(Rel p (W )) = W ⊗ Qp , which gives a fractional cocharacter λ W of GL W, Qp .Since this construction is functorial in W , it defines a fractional cocharacter λ of G Qp .
Recall from [29, Sec.In this section we will study irreducible components of Hecke stable subvarieties and prove results in the style of [7,Prop. 4.5.4]. 4 The results proved in this section will not be used in the rest of this article, but they will be used to prove irreducibility results for EKOR strata in an upcoming version of [19].
Let ρ : G sc → G der be the simply connected cover of the derived group G der of G and note that ρ induces an action of G sc (A p f ) on Sh G,Kp .From now on we will need another assumption: Hypothesis 3.4.1.For each finite extension F of the reflex field E and any place w of F extending v, the natural maps are isomorphisms.Proof.This is [37,Cor. 4.1.11].
Remark 3.4.3.The variety Sh G,K p Kp has geometrically integral connected components if K p is hyperspecial because then the integral models are smooth by work of Kisin [27].More generally the Kisin-Pappas integral models of [26] have geometrically integral connected components K p is very special.
Then Y ∞ is stable under the action of G sc (A f ).
Then Y ∞ is stable under the action of G sc (A p f ).Proof.We consider and we let Y ′ ∞ be a connected component of the left hand side mapping to Y ∞ .Then Y ′ ∞ is stable under the action of G sc (A f ) and thus Y ∞ is stable under the action of G sc (A p f ).
Lemma 3.4.7.Suppose that Hypothesis 3.4.1 holds and let Y ∞ ⊂ Sh G,Kp,Fp be a connected component.Then Y ∞ is stable under the action of G sc (A p f ).Proof.It suffices to prove this for Shimura varieties over C, because the connected component are defined over an algebraic closure E of the reflex field E and the result can be transported to the special fiber using Hypothesis 3.4.1.The result over C is proved in Corollary 3.4.6.
3.4.8.Let Z ⊂ Sh G,K p Kp be a G(A p f )-stable locally closed subscheme with inverse image Z ⊂ Sh G,Kp .A finite étale cover X → Z is called G(A p f )-equivariant if X := Z × Z X has an action of G(A p f ) making the natural map X → Z equivariant for the action of G(A p f ).If Hypothesis 3.4.1 is satisfied, then G sc (A p f ) acts on the fibers of π 0 ( X) → π 0 (Sh G,Kp,Fp ).
Proof.The assumption that X → Z is finite étale implies that π 0 ( X) → π 0 ( Z) is a finite map with discrete fibers, and therefore the action of G(A p f ) on π 0 ( X) is continuous because the action on π 0 ( Z) is, see Corollary 3.1.5.
Let Σ be a finite set of places of Q containing the infinite place, containing p, and containing all places ℓ where G ad ℓ has a compact factor.From now on we will work with G(A p f )-stable subvarieties Z defined over F p and with geometric monodromy groups.For a prime ℓ ∈ Σ we will write K ℓ for the image of K p → G(Q ℓ ) and π ℓ : Sh G,KpK p ,ℓ,Fp → Sh G,K p Kp,Fp for the induced K ℓ -torsor over Sh G,K p Kp,Fp .Lemma 3.4.11.Suppose that Hypothesis 3.4.1 holds and let Proof.It follows from profinite Galois theory for schemes, see Section 3.1.11,that the stabiliser for some point y ∈ Y (F p ).If we apply Corollary 3.2.6 and Lemma 3.1.9to Z = Sh G,K p Kp , it follows that this image contains a compact open subgroup of G der (Q ℓ ).
Proof of Theorem 3.4.10.We write Z ℓ → Z for the induced K ℓ torsor and X ℓ → Z ℓ for Z ℓ × Z X.
Then the action of G(A p f ) on X and Z induces an action of G(Q ℓ ) on X ℓ , and it suffices to show that G sc (Q ℓ ) acts trivially on the fibers of a ℓ : π 0 (X ℓ ) → π 0 Sh G,KpK p ,ℓ,Fp for all ℓ ∈ Σ.Let x ∈ π 0 (X ℓ ) and let Z • be a connected component of Z containing the image of x.Moreover let Z • ℓ ⊂ Z ℓ be the inverse image of Z • .Fix a point z ∈ Z • (F p ), then Hypothesis 2.3.1 and Corollary 3.2.6 tell us that the image of ρ ℓ : π ét 1 (Z • , z) → K ℓ is a compact subgroup M geom,ℓ whose Zariski closure M geom,ℓ has neutral component equal to G der Q ℓ .It follows from Lemma 3.1.9that the image of ρ ℓ contains a compact open subgroup V ℓ ⊂ G der (Q ℓ ).The upshot of this discussion is that the stabiliser in G(Q ℓ ) of a connected component of Z ℓ contains a compact open subgroup of G der (Q ℓ ) and this implies that the stabiliser in Then it follows from Hypothesis 3.4.1 and Lemma 3.4.11that Y ∞ → Y is a pro-étale torsor for a compact open subgroup U ℓ ⊂ G der , and from Lemma 3.4.7 that Y ∞ is stable under the action of G sc (Q ℓ ).
We will write X ∞ ⊂ X ℓ for the inverse image of Y ∞ in X ℓ and let X ′ ⊂ X be its image.Note that x ∈ π 0 (X ∞ ) by construction.Then X ∞ → X ′ is pro-étale U ℓ torsor and X ∞ is stable under the action of G sc (Q ℓ ).This action is moreover continuous by Lemma 3.4.9and the inclusion ) is finite and since the map U ′ ℓ → U ℓ has finite cokernel.This means that there are finitely many (open and closed) U ′ ℓ orbits on π 0 (X ∞ ).Therefore the G sc (Q ℓ ) orbit of x on π 0 (X ∞ ) is a union of finite many U ′ ℓ -orbits and thus closed; in particular it is compact Hausdorff.It then follows from Lemma 3.1.6that the G sc (Q ℓ ) orbit of x is homeomorphic to G sc (Q ℓ )/P x , where P x ⊂ G sc (Q ℓ ) is the stabiliser of x.In particular, it follows that G sc (Q ℓ )/P x is compact.
The group P x contains a compact open subgroup of G sc (Q ℓ ) because the stabiliser of x in G(Q ℓ ) contains a compact open subgroup of G der (Q ℓ ) and G sc (Q ℓ ) → G der (Q ℓ ) has finite fibers.This implies that G sc (Q ℓ )/P x has the discrete topology, and we conclude that G sc (Q ℓ )/P x is a finite set or equivalently P x is a finite index subgroup.The assumption that G sc Q ℓ has no compact factors implies, by [42, Thm.7.1, Thm.7.5], that the group G sc (Q ℓ ) has no finite index subgroups.Therefore G sc (Q ℓ )/P x is a singleton which is precisely what we wanted to prove.

Serre-Tate coordinates and unipotent group actions
In this section we show that the classical Serre-Tate coordinates, as described in [22], can be reinterpreted using the action of a unipotent formal group, as in [20].Our results are more-orless a direct generalisation of the results of [20], except that we construct the action of unipotent formal groups using Rapoport-Zink spaces, while in loc.cit.this action is constructed using Igusa varieties.
In Section 4.1, we recall the classical theory of Serre-Tate coordinates following [22], which shows that the formal deformation space Def(Y ) of an ordinary p-divisible group Y over F p has the structure of a commutative formal group.We then compute the scheme-theoretic p-adic Tate-module of the p-divisible group H 0,1 associated to this formal group.In Section 4.2 we use Rapoport-Zink spaces to describe an action of the universal cover H0,1 of H 0,1 on the formal scheme Def(Y ) associated to Def(Y ).In Section 4.3 we identify this action with the projection from the universal cover to H 0,1 followed by the left-translation action of H 0,1 on Def(Y ).4.0.1.We consider the category Art of Artin local Zp -algebras R such that the natural map F p → R/m R is an isomorphism.Here m R is the unique maximal ideal of R and we write α : R → F p for the composition of the natural map R → R/m with the inverse of the natural isomorphism F p → R/m.Note that α is functorial for morphisms in Art.We similarly consider the category Nilp of Zp -algebras in which p n = 0 for some n.The category Art is naturally a full subcategory of Nilp.
For a p-divisible group G over an algebra R ∈ Nilp we define the p-adic Tate module to be the functor where X is a p-divisible group over Spec R and β : X ⊗ R,α F p → Y F p is an isomorphism.This functor is (pro)-representable by a formally smooth formal scheme Def(Y ) of relative dimension g 2 over Spf Zp .By [22, Thm.2.1], this functor lifts to a functor valued in abelian groups such that the formal group Def(Y ) is a p-divisible. 5here is a canonical direct sum decomposition Y = Y 0 ⊕ Y 1 where Y 0 is the maximal étale quotient and where Y 1 is equal to the formal completion of Y at the origin.Since Y 0 is étale there is a unique lift to a p-divisible formal group over Zp , which we will denote by Y can 0 .Similarly Y 1 has a unique lift to a p-divisible formal group over Zp , for example because the Serre dual of Y 1 is étale.We will denote this lift by Y can Let S be the complete Noetherian local Zp -algebra representing Def(Y ) on Art.Then the abelian group structure on Def(Y ) induces a (continuous) co-commutative Hopf algebra structure on S. In particular the formal scheme Def(Y ) := Spf S, considered as a functor on Nilp, has the structure of a formal group.We will write H 0,1 In [22, p. 152] it is explained that Def(Y ) is isomorphic to the functor (on Art) sending R to and that because T p Y can 0 is an inverse limit of étale group schemes, the natural map ).The p n -torsion of this group is given by ) of functors on Nilp.It is straightforward to check that these isomorphisms are compatible with increasing n, which concludes the proof. .
Moreover the functors Aut(Y i ) are pro-étale group schemes which are non-canonically isomorphic to the group schemes associated to the profinite group GL g (Z p ).The functors Aut( Ỹi ) are noncanonically isomorphic to the group schemes associated to the locally profinite group GL g (Q p ).Let H0,1 be the universal cover of H 0,1 .Then by the discussion after [4, Def.4.1.1],we can identify the fpqc sheaves Moreover, by the proof of [4,Prop. 4.1.2],there is a short exact sequence of fpqc sheaves 0 → T p H 0,1 → H0,1 → H 0,1 → 0. By Lemma 4.1.2,we can identify this with Note that Hom (Y 0 , Y 1 )[1/p] is isomorphic to H 0,1 , and thus representable by a formal scheme by [45,Prop. 3.1.3.(iii)] as above.In particular, this means that Aut( Ỹ ) is representable by a formal scheme.

4.2.1.
Let RZ Y be the Rapoport-Zink space associated to Y .It is defined to be the functor on Nilp sending R to the set of isomorphism classes of pairs (X, f ) where X is a p-divisible group over Spec R and f : X Y R is a quasi-isogeny (or equivalently, by [22,Lemma 1.1.3.3], a quasi-isogeny The functor RZ Y is representable by a formally smooth formal scheme over Spf Zp by [44,Thm. 2.16].The group functor Aut( Ỹ ) acts on RZ Y via postcomposition, where we note that an automorphism of Ỹ is the same thing as a self quasi-isogeny of Y .
Let y be the F p -point of RZ Y corresponding to the identity map Y → Y and let RZ /y Y ⊂ RZ Y be the formal completion of RZ Y in {y}, in the sense of considered as a formal algebraic space as in [49,Tag0GVR].By definition this is the subfunctor of RZ Y corresponding to those morphisms Spec R → RZ Y that factor through {y} on the level of topological spaces.In other words, it consists of those morphisms Spec R → RZ Y such that the induced morphism Spec In terms of the moduli description, this means that we are looking at those quasi-isogenies f : X Y R such that: There is a (necessarily unique) isomorphism β : Now restrict this moduli description to the full subcategory Art ⊂ Nilp.Then RZ /y Y can be described as the functor on Art sending (R, α) to the set of isomorphisms classes of triples (X, β, f ), where X is a p-divisible group over R equipped with an isomorphism β : X ⊗ R,α F p → Y and where f is a quasi-isogeny f : X Y R such that (4.2.1) commutes.Proof.The commutativity of (4.2.1) expresses the fact that f is a lift of the quasi-isogeny Y → Y given by the identity.But since quasi-isogenies lift uniquely by [22,Lem. 1.1.3.3], the data of f is superfluous and we see that the forgetful map RZ preserves the identity quasi-isogeny Y → Y and therefore acts on Def(Y ).In particular, the profinite group acts on Def(Y ).This induces an action of Aut(Y )(F p ) on Def(Y ) because F p is an object of Art ⊂ Nilp.
Corollary 4.2.3.This action sends a pair (X, β) ∈ Def(Y )(R), where X is a p-divisible group over Spec R and β : Proof.This follows from Lemma 4.2.2 and the uniqueness of the isomorphism β : The goal of the rest of this section is to prove the following theorem, our proof of which was heavily inspired by the proof of [20, Thm.6.2.1], which deals with the g = 1 case.Proposition 4.2.5.The action ãRZ factors through an action of H 0,1 via the natural quotient map H0,1 → H 0,1 .Moreover the induced action of H 0,1 is given by a ST .
which induce isomorphisms of functors on Art p be the standard basis vectors, then we can in fact identify with coordinates q i,j for 1 ≤ i, j ≤ g and similarly corresponds to elements q i,j ∈ 1 + m R , and the corresponding deformation of Y is the p-divisible group X q corresponding to the pushout of (see [22, p. 152]) given by x i → (q i,1 , • • • , q i,g ).In fact, there is a pushout diagram 0 and so we can also think of X q as the quotient of µ ⊕g p ∞ ,R ⊕ Z[1/p] ⊕g by the image of the map Let N be an integer such that q p N i,j = 1 for all i, j, which exists since R is Artinian.Then the isogeny and the induced quasi-isogeny X q,Fp = X 1,Fp → X 1,Fp is given by p N .It follows that the quasiisogeny p −N f q,N is the unique quasi-isogeny lifting the identity X q,Fp = X 1,Fp → X 1,Fp .Let us write q ∈ RZ Y (R) for p −N f q,N : X q Y R .
A morphism Spec R → H 0,1 corresponds to elements ζ i,j ∈ 1 + m R .The left translation action of H 0,1 via the Serre-Tate action is given by a ST (ζ, q) = ζq, where (ζq) i,j = (ζ i,j • q i,j ) and where (ζ i,j • q i,j ) denotes the multiplication in µ p ∞ (R) = 1 + m R .In terms of p-divisible groups, this correspond to the p-divisible group X ζq .We will write ζq ∈ RZ Y (R) for the element corresponding to X ζq .
Proof of Proposition 4.2.5.By definition of the action ãRZ , it suffices to show that for every fpqc cover Spec R → Spec R and every lift There is a universal such lift over the fpqc cover R′ given by formally adjoining all the p-power roots of all ζ i,j , and it suffices to prove the result for this choice of R′.To be precise, we let where with transition maps defined by Y i,j → Y p i,j or equivalently by ζ By fpqc descent, we can furthermore pass to the fpqc cover R′ → R given by also formally adjoining all the p-power roots of all the coordinates q i,j .Recall that X q, R is defined as the quotient of by the image of the map h q which sends the standard basis element x i to The p-divisible group X ζq, R is defined similarly but then using the map h ζq .The compatible sequence of p-power roots of ζ defined by ζ defines a map .
It is straightforward to check that this map satisfies and that it thus induces an isomorphism on quotients Choose N sufficiently large such that ζ p N i,j = 1 and q p N i,j = 1 for all i, j.Let be the unique quasi-isogeny lifting the identity map X ζq,Fp = Y F p → Y F p as described in (4.3.1).To prove the proposition it suffices to show that the following diagram commutes: Here ξ ζ is given by the matrix The diagram of quasi-isogenies (4.3.2) is obtained from the diagram (4.3.3) by quotienting by the subgroups and formally inverting certain powers of p.It follows that (4.3.2) is commutative.

The formal neighborhood of an ordinary point
The goal of this section is to give Serre-Tate coordinates for the formal completions of points in the ordinary locus of Shimura varieties of Hodge type.
In Section 5.1 we specialise to the smooth canonical integral models of Shimura varieties of Hodge type at hyperspecial level, and we moreover assume that the ordinary locus is nonempty.In Section 5.2 we recall a small amount of covariant Dieudonné theory for semiperfect rings, following [45].
In Section 5.4 we prove that the formal completion of the ordinary locus gives a subtorus of the Serre-Tate torus, reproving a special case of [47,Thm. 1.1].We also give a group-theoretic description the Dieudonné module of the associated p-divisible group.In Section 5.5 we introduce strongly nontrivial actions of algebraic groups on isocrystals, which we will need to confirm the hypotheses of the rigidity theorem of [9].5.1.Integral models at hyperspecial level.Let the notation be as in Section 2. In particular, we have a Shimura datum (G, X) of Hodge type with reflex field E, a prime p and a prime v of E above p.Moreover there is a symplectic space V and a Hodge embedding (G, X) → (G V , H V ) and for every sufficiently small K p ⊂ G(A p f ) there is a sufficiently small K p ⊂ G V (A p f ) and a finite morphism Recall that there is a Z (p) -lattice V (p) of V on which the symplectic form is Z (p) -valued, and recall that we have defined K p ⊂ G(Q p ) to be its stabiliser.From now on we will assume that K p is a hyperspecial subgroup, in which case S K is the canonical integral model of Sh K p Kp (G, X) over O Ev .Moreover the main theorem of [52] tells us that the map (5.1.1)is a closed immersion.

The
. All the results of Section 2 still go through with this choice of tensors.
For x ∈ Sh G,K p Kp (F p ) we have seen in Section 2.1.1 that there are tensors ) be the integral contravariant Dieudonné module.Then as explained in [47,Sec. 6.3], we can choose the tensors {s α,cris } to lie inside It is moreover explained there that there is an isomorphism The maximal element [b] is known as the µ-ordinary element, and the maximal Newton stratum is known as the µ-ordinary locus.
Proof.The µ-ordinary locus and the ordinary locus are equal in this case by the proof of [35,Cor. 4.3.2],as explained in [35,Rem. 4.3.3].The density of the µ-ordinary locus is [50, Thm.1.1], see [29,Thm. 3] for a published reference.5.1.4.Let x ∈ Sh G (F p ) be an ordinary point and consider the closed immersions of formal neighbourhoods (considered as functor on the category Nilp Zp of Zp -algebras where p is nilpotent) Let A be the universal abelian scheme over S GSp and let X = A[p ∞ ] be associated p-divisible group over S GSp .Let Def(A x ) be the formal deformation space of the abelian variety A x , that is, the formal scheme representing the functor Def(A x ) on the category Art of deformations of the abelian variety A x .Similarly let Def(Y ) be the deformation space of the p-divisible group X x =: Y .There are natural morphisms The first is a closed immersion by the moduli description of S GSp , and the second morphism is an isomorphism by [22,Thm. 1.2.1].Now [47, Thm 1.1] (c.f.[40] for closely related results) implies that closed formal subscheme is a p-divisible formal subgroup.The goal of this section is to compute the Dieudonné module of Sh /x G .We do this by giving a new proof that Sh /x G ⊂ Def(Y ) is a p-divisible formal subgroup, using the methods of Section 4 and results of [24].5.2.Some covariant Dieudonné theory.

5.2.1.
A caveat.In the rest of this subsection we are going to recall some covariant Dieudonné theory for semiperfect rings following [45].The reason we do this is that the references [4] and [24] are written in this language.Moreover we feel that results such as Lemma 5.2.5 are most naturally stated using the covariant theory.
To avoid potential confusion, we will always write a subscript cov when using covariant Dieudonné theory.The covariant theory and the contravariant theory will interact only once, in Section 5.3, and we will warn the reader again there.

5.2.2.
Recall that an F p -algebra A is semiperfect if it is the quotient of a perfect F p -algebra B and that it is f-semiperfect if it is the quotient of a perfect F p -algebra by a finitely generated ideal.Let A be a semiperfect F p -algebra and let A cris (A) be Fontaine's ring of crystalline periods (see [45,Prop. 4.1.3])with ϕ : A cris (A) → A cris (A) induced by the absolute Frobenius on A. Definition 5.2.3.A covariant Dieudonné module over a semiperfect F p -algebra A is a pair (M, ϕ M ), where M is a finite locally free A cris (A)-module and where Remark 5.2.4.Usually one instead asks that The reasons for our conventions is that they agree with the conventions in [4] and [24].A p-divisible group G over A has a covariant6 Dieudonné module (D cov (G ), ϕ G ).For Spec A ′ → Spec A there is a canonical isomorphism Our covariant Dieudonné modules are normalised as in [4].In particular, this means that the covariant Dieudonné module of Q p /Z p over A is A cris (A) equipped with the trivial Frobenius, and the covariant Dieudonné module of µ p ∞ is A cris (A) equipped with Frobenius given by 1/p.This also means that the contravariant Dieudonné module is isomorphic to the dual of the covariant Dieudonné module, see [4, footnote on page 692].Now let G be a p-divisible group over F p with universal cover G in the sense of [45,Sec. 3.1].If we consider G as a functor on Nilp then it is a a filtered colimit of spectra of f-semiperfect F p -algebras by [45,Prop. 3.1.3.(iii)] and is thus determined by its restriction to the category of semiperfect F p -algebras.We can describe it explicitly on the category of f-semiperfect F p -algebras as follows: Lemma 5.2.5.There is a commutative diagram of natural transformation of functors on the category of f-semiperfect F p -algebras, which evaluated at an object A gives where ϕ is given by the diagonal Frobenius and where B + cris (A) := A cris (A)[1/p].Proof.Let A be f-semiperfect, then [45,Thm. 4.1.4]tells us that Dieudonné module functor over A is fully faithful after inverting p.There is a natural map where the latter bijection is induced by evaluation at 1.After inverting p we get a natural isomorphism Consider the special fiber H 0,1,Fp .Then by Lemma 4.1.2its p-adic Tate module is given by Hom (Y 0 , Y 1 ).Therefore by [4,Lem. 4.1.7,Lem. 4.1.8]we have an isomorphism where Hom denotes the internal hom in F -isocrystals and where () ≤0 denotes the slope at most 0 part of an F -isocrystal.
This map realises the source as the slope −1 part of the target.5.3.2.Write gl(V * ) for the Lie algebra of the algebraic group GL(V * ) ⊗ Qp and identify it with the vector space of endomorphisms of V * ⊗ Qp equipped with the commutator bracket.We can equip gl(V * ) with the structure of an F -isocrystal by letting Frobenius act by conjugation by b ∈ GL(V * ) ⊗ Qp .Let us write (gl(V * ), Ad bσ) for this isocrystal.
Lemma 5.3.3.Let C be a field of characteristic zero and let W be a finite dimensional C vector space.Let H ⊂ GL(W ) be a connected reductive group that is the stabiliser of a collection of tensors {t α } α∈A ⊂ W ⊗ .Then the Lie algebra h ⊂ gl(W ) is given by the subspace {H ∈ gl(W ) : H * (t α ) = 0 for all α ∈ A }.
Proof.The Lie algebra is given by the kernel of the map G(C[ǫ]/(ǫ 2 )) → G(C).Thus it consists of matrices of the form 1 + ǫM , where M ∈ gl(W ), such that for α ∈ A we have for the slope −1 subspace of the F -isocrystal (g, Ad bσ).Then by [24,Lem. 3.1.3]and its proof, there is an inclusion of p-divisible groups H G 0,1,Fp ⊂ H 0,1,Fp (5.3.3)inducing (5.3.2) upon taking rational covariant Dieudonné modules.Since both of these p-divisible groups have étale Serre duals, there is a unique lift H G 0,1 of H G 0,1,Fp to Zp and a unique lift H G 0,1 ⊂ H 0,1 of the inclusion (5.3.3).Lemma 5.3.4.Let A be an f-semiperfect F p -algebra.Then the inclusion identifies H G 0,1,Fp (A) with the subspace of those quasi-endomorphisms f : Y 0,A Y 1,A such that the induced quasi-endomorphism Proof.This follows from Lemma 5.2.5 in combination with Lemma 5.3.3.
The following lemma and its corollary essentially follow from [24,Prop. 3.2.4].However the construction there is incorrect because of the error in [24,Lem 3.1.3]pointed out above.Once the subgroup in the statement of Lemma 5.3.6 has been shown to exist with the properties proved in Corollary 5.3.7, the rest of the arguments in [24] go through without further changes.Lemma 5.3.6.There is a closed subgroup such that on A-points for f-semiperfect F p -algebras A, it is the subgroup of those quasi-isogenies We will call such quasi-isogenies tensor-preserving quasi-isogenies.Here we are using the map as the subgroup of lower triangular automorphisms of Ỹ .The condition that 1 0 f 1 = 1 + f satisfies (1 + f ) * (s α,cris ⊗ 1) = s α,cris ⊗ 1 is equivalent to the condition that f * (s α,cris ⊗ 1) = 0. Thus we see that the intersection Hom (Y 0 , Y 1 )[1/p] with Aut G ( Ỹ ) is given by . By Lemma 5.3.4,this is representable by a closed subgroup.
We can identify the group Aut( Ỹ1 ) Fp × Aut( Ỹ0 ) Fp with the locally profinite group scheme associated to the locally profinite group Aut( Ỹ )(F p ).Using Dieudonné theory, we can identify this locally profinite group with the σ-centraliser of b in GL(V * )( Qp ), where we recall that we have fixed an isomorphism V * (p) ⊗ Zp ≃ D cov (Y ) giving rise to b ∈ G( Qp ).The subgroup of tensor-preserving automorphisms of Ỹ over F p can be identified with J b (Q p ), the σ-centraliser of b ∈ G( Qp ), which is a closed subgroup.Note that J b (Q p ) ⊂ G( Qp ) stabilises (g, Ad bσ) −1 because it acts on g via automorphisms that preserve the slope decomposition.Using Lemma 5.3.4we see that the closed subgroup Fp , has the required properties over F p , and so we are done.
Since the R-points of H G 0,1 and J b (Q p ) both only depend on R/p, we see that described the unique lift to Zp .This identifies H G 0,1 with the neutral component Aut G ( Ỹ ) • of Aut G ( Ỹ ).
Corollary 5.3.7.The identity component is isomorphic to Spf S where S is the p-adic completion of Zp [[x ]] for some d.
Proof.This is true for H G 0,1,Fp because it is the universal cover of a p-divisible group, see [45, Cor.Let {λ} be a conjugacy class of (fractional) cocharacters of a connected reductive group H over an algebraically closed field C. Let T be a maximal torus, let λ be a representative of {λ} factoring through T and let B ⊃ T be a Borel.Let ρ ∈ X * (T ) be the half sum of the positive roots with respect to B. Then the pairing 2ρ, λ does not depend on the choice of T, B or λ, and we denote it by 2ρ, {λ} .
We can identify these groups with The inclusion H G 0,1 ⊂ H 0,1 induces an inclusion T p H G 0,1 ⊂ T p H 0,1 which induces (5.4.1) after inverting p.This implies that the action of H G 0,1 on S /x G factors through an action of H G 0,1 via the natural quotient map G .Then the associated orbit map gives a closed immersion This means that we similarly get a closed immersion Recall from Section 4 that there is an action of Aut( Ỹ ) on RZ Y .Recall from the discussion before Corollary 4.2.3, that Aut(Y )(F p ) ⊂ Aut( Ỹ ) preserves the F p points corresponding to the identity from Y to Y , and that this induces an action of the profinite group Aut(Y )(F p ) on Def(Y ).This action is described in Corollary 4.2.3.5.4.5.Recall that there are locally profinite groups for the compact open subgroup given by the intersection Then U p acts on H G 0,1 ⊂ Hom (Y 0 , Y 1 )[1/p] and preserves the action of T p H G 0,1 , and thus acts on the quotient H 0,1 ≃ S /x G .By Proposition 5.4.3 and the proof of Lemma 5.3.6, the induced action on rational Dieudonné modules can be identified with the natural action of U p ⊂ J b (Q p ) on (g, Ad bσ) −1 ⊂ (g, Ad bσ).
In order to apply the rigidity result of Chai [9] we need to understand this action.We will do this in more generality in the next section.5.5.Strongly non-trivial actions.Let G be a connected reductive group over Q p .Let b ∈ G( Qp ) be an element and consider the F -isocrystal (g, Ad bσ), where g = Lie G ⊗ Qp , equipped with its action of J b (Q p ).If we replace b by a σ-conjugate b ′ , then J b (Q p ) and J b ′ (Q p ) are conjugate in G( Qp ) and there is an isomorphism of isocrystals (g, Ad bσ) ≃ (g, Ad b ′ σ).
Let λ ∈ Q and let N λ ⊂ (g, Ad bσ) be the largest sub F -isocrystal of slope λ.Then because J b (Q p ) acts on (g, Ad bσ) via F -isocrystal automorphisms, it preserves the subspace N λ .Let us also denote by b the image of b in GL(g), then there is a homomorphism of algebraic groups where GL(g) b denotes the σ-centraliser of b in GL(g).There is a parabolic subgroup P (λ) ⊂ GL(g) consisting of automorphisms of g that preserve the slope filtration on the F -isocrystal (g, Ad bσ), and after potentially replacing b by a σ-conjugate, the image of b lands in P (λ).There is thus a group homomorphism where P (λ) b denotes the σ-centraliser of b in P (λ).Since N λ is a graded quotient of the slope filtration of the F -isocrystal (g, Ad bσ), there is an induced quotient map P (λ) → GL(N λ ) and this induces a group homomorphism where GL(N λ ) b denotes the σ-centraliser of b in GL(N λ ).Let E be the Q p -algebra of endomorphisms of the F -isocrystal N λ and let E × be the functor on Q p -algebras given by R → (R ⊗ E) × .Then there is a natural isomorphism E × ≃ GL(N λ ) b .
Let GL(E) be the general linear group of E considered as Q p -vector space and let E × → GL(E) be the natural map corresponding to the action of E on itself by left translation.Consider E as a Q p -linear representation of J b via J b → E × , then the goal of this section is to prove the following result: Proposition 5.5.1.Let T ⊂ J b be a maximal torus.If λ = 0, then the trivial representation of T does not occur in the representation of T given by E.
Proof of Proposition 5.5.1.After replacing b by a σ-conjugate we can arrange for it to satisfy (c.f.[33,Sec. 4 for some r.Here ν b is the Newton cocharacter of b, which is defined over Q p r .Let M ν b ⊂ G ⊗ Qp denote the centraliser of the cocharacter ν b .By [24, Prop.2.2.6], there is a unique isomorphism After tensoring up to Qp , there is a commutative diagram (where L reg is the left regular representation of GL(N λ ) on GL(End(N λ ))) If we show that the trivial representation of T ⊗ Qp does not occur in E ⊗ Qp , then it follows that the trivial representation of T does not occur in E. The representation W = End(N λ ) of J b, Qp defined by composition with the left regular representation is a direct sum of copies of the representation N λ .Therefore it suffices to show that the representation N λ of T ⊗ Qp does not contain the trivial representation.
We note that T ⊗ Qp =: T ′ is a maximal torus of M ν b acting on the associated graded of the slope filtration of the F -isocrystal (g, Ad bσ).Since ν b is a central cocharacter of M ν b by definition, we see that (rν b )(p) ∈ T ′ ( Qp ).To determine the slope decomposition of the F -isocrystal (g, Ad bσ), it suffices to determine the slope decomposition of the F r -isocrystal (g, (Ad bσ) r ) for some positive integer r.
Let C be an algebraic closure of Qp and consider the action of T ′ C on g C via the adjoint action of G C .Then we have a decomposition where Φ ⊂ X * (T ′ C ) consists of the simple roots of T C .There is a similar decomposition where Φ 0 ∈ X * (T ′ C ) I is the image of Φ and where I = Gal(C/ Qp ) is the inertia group.Now we choose an integer r with the following properties: The isomorphism is satisfied, and the decomposition (5.5.1) is defined over Q p r .Then each U α 0 is stable under the action of σ r and (Ad bσ) r acts on it by (rν b )(p)σ r .The operator Ad bσ moreover acts trivially on t ′ , and thus for nonzero λ we have that After basechanging to C, we see that

Thus T ′
C acts on N λ via a subset of the nontrivial characters given by the simple roots Φ ⊂ X * (T ′ C ), and therefore the trivial representation of T ′ does not occur in N λ and thus it does not occur in E.

Proof of the main theorems
There are two final ingredients that are introduced in this section.In Section 6.1, we prove the local stabiliser principle of Chai-Oort (c.f.[10, Thm.9.5]), which shows that the formal completion of the Zariski closure of a prime-to-p Hecke orbit is stable under the action of a large p-adic Lie group.In Section 6.2.2 we give a summary of results of [5], which relates Serre-Tate coordinates of families of ordinary abelian varieties to the p-adic monodromy groups of these abelian varieties.Then in Section 6.3 we put everything together to prove Theorem I.In Section 6.4 we prove Corollary 6.4.1, which is a generalisation of Theorem I to Shimura varieties of abelian type.
We will use the notation introduced in Section 2 and Section 5.1 and moreover we will keep track of the level again.Let x ∈ Sh G,K p Kp (F p ) and let x be a lift of x to Sh G,Kp (F p ). Then the prime-to-p Hecke orbit of x is defined to be the image H K p (x) ⊂ Sh G,K p Kp (F p ) of the orbit G(A p f ) • x ⊂ Sh G,Kp (F p ); it does not depend on the choice of lift x.For the rest of this section we fix x as above and we let Z ⊂ Sh G,ord,K p Kp be the closure of H K p (x), then Z is again G(A p f )-stable by Lemma 3.1.2.

6.1.2.
There is a G(A p f )-equivariant closed immersion (using the fact that we have a closed immersion at finite level by the main theorem of [52]) Sh G,Kp → Sh G V ,Kp , where G(A p f ) acts on the right hand side via the inclusion G(A p f ) → G V (A p f ).The space Sh G V ,Kp is a moduli space of polarised abelian varieties (A, λ) up to prime-to-p isogeny, equipped with an isomorphism V p A → V ⊗ A p f compatible with the polarisation.
Let z be a lift of z to Sh G,Kp (F p ) as above, which defines an inclusion The stabiliser in G V (A p f ) of z ∈ Sh G V ,Kp is given by End λ (A z ) × , which is the group of automorphisms of the abelian variety up to prime-to-p isogeny A that take λ to a Z × (p) multiple of λ.In order to prove Proposition 6.1.1,we first prove it for Sh GSp,K p Kp .See [10, Thm.9.5] for closely related results and arguments.

Let Sh
/z G V ,Kp be the formal completion of Sh G V ,Kp , considered as a formal algebraic space as in [49, Sec.0AIX], and restrict its functor of points to Artin local F p -algebras R with residue field isomorphic to F p .Then Sh /z G V ,Kp (R) is the set of isomorphism classes of polarised abelian varieties (A, λ) up to prime-to-p isogeny, equipped with an isomorphism ǫ : V p A → V ⊗ A p f compatible with the polarisation, such that after basechanging to F p we recover the point given by the image of z.
This means that there is a (necessarily unique) isomorphism β : A Fp → A z making the following diagram commute: The quadruple (A, λ, β, ǫ) is uniquely determined by (A, λ, β) because (pro-)étale sheaves on Artin local rings are determined by their restriction to the residue field.In particular, for all R ∈ Art the natural forgetful map is an isomorphism.This induces an action of End λ (A z ) × on Sh

/z
GSp,K p Kp that we will now identify.
6.1.4.Recall that there is an inclusion End λ (A z ) × ⊂ G V (A p f ) determined by z or rather ǫ z .This means that an automorphism f of A z acts on V ⊗ A p f in a way that makes the following diagram commute: Since End λ (A z ) × stabilises z, it acts on Sh /z G V ,Kp .This action can be described as follows: An automorphism f sends a triple (A, λ, ǫ) to (A, λ, f • ǫ).It is straightforward to check that the unique upgrade (A, λ, f • ǫ) to a quadruple (A, λ, β ′ , f • ǫ) is realised by taking β ′ = f • β.Therefore the induced action of End λ (A z ) × on Sh /z GSp,K p Kp is given by (A, λ, β) → (A, λ, f • β).6.1.5.Because deformations of abelian varieties are uniquely determined by deformations of their p-divisible groups, we can also identify Sh /z G V ,Kp (R) with the space of triples (X, λ, β) where (X, λ) is a polarised p-divisible group and β is an isomorphism (X, λ) F p → (A z [p ∞ ], λ z ).The action of End λ (A z ) × is then given by postcomposing β with f .There is a similar description of at finite level, and it follows that the natural map  in particular, the height of the isocrystal b is bounded from below by the dimension of Mon(Z /z , M, z).Since b has slope 1, it follows that the dimension of the p-divisible group associated to b + is also bounded from below by the dimension of Mon(Z /z , M, z).By Lemma 5.4.1, this unipotent radical is isomorphic to the unipotent radical of the parabolic subgroup P µ ⊂ G for any choice of representative µ of {µ}.This unipotent radical has dimension equal to 2ρ, {µ} (this notation was introduced after the statement of Lemma 5.4.1).Corollary 6.1.6tells us that Z /z is a formal subtorus.Applying Proposition 6.2.1 we see that the Krull dimension of O Z,z is bounded from below by 2ρ, {µ} .Since the Shimura variety Sh G,K p Kp also has dimension 2ρ, {µ} , we conclude that Because this is true for a dense set of points, it follows that Z is a union of connected components of Sh G,ord,K p Kp .By Lemma 5.1.3,the ordinary locus is dense and thus π 0 (Sh G,ord,K p Kp ) = π 0 (Sh G,K p Kp ).Since G(A p f ) acts transitively on π 0 (Sh G,Kp ), by [27, Lem.2.2.5] in combination with [37,Cor. 4.1.11],it follows that Z = Sh G,ord,K p Kp .We conclude that the prime-to-p Hecke orbit of x is dense in Sh G,K p Kp since Sh G,ord,K p Kp is dense in Sh G,K p Kp .

3. 1 . 7 .
Lie groups over ℓ-adic local fields.Recall that a topological group M is called an ℓ-adic Lie group if it admits the (necessarily unique) structure of an ℓ-adic Lie group, see [17, Def.2.1, Prop.2.2].If M is an ℓ-adic Lie group, then by definition there is a finite-dimensional Q ℓ -Lie algebra Lie M , an open neighborhood U ⊂ Lie M of the identity and an exponential map Exp : U → M that is a homeomorphism onto a compact open subgroup of M .For example for an algebraic group H over Q ℓ the topological group H(Q ℓ ) is an ℓ-adic Lie group with Lie algebra Lie H(Q ℓ ) = Lie H.Lemma 3.1.8.Let H be an algebraic group over Q ℓ and let M ⊂ H(Q ℓ ) be a subgroup closed in the ℓ-adic topology.Then M is an ℓ-adic Lie group and the morphism M → H(Q ℓ ) is a morphism of ℓ-adic Lie groups.Moreover, the induced Lie subalgebra Lie

1 . 1 . 2 ] 3 Lemma 3 . 3 . 4 . 4 3. 4 .
that associated to [b] ∈ B(G Qp ) there is a conjugacy class of fractional cocharacters {ν [b] } of G Qp that is defined over G Qp .The conjugacy class of λ agrees with {ν [b] }.Proof.To identify the conjugacy class of the cocharacter λ, it suffices to identify the conjugacy class of λ composed with all representations of G Qp .Then the lemma comes down proving that the (conjugacy class of the) Newton cocharacter of the isocrystal Rel p (W ) corresponding to a representation W of G Qp given by r : G Qp → GL(W ), is given by {r • ν [b] }.But this is true by construction of the Newton cocharacter ν [b] associated to [b], see [33, Sec.4].Under our assumption that Z is contained in a single Newton stratum Sh G,[b],K p Kp of Sh G,K p Kp we note that the monodromy group Mon(Z • , M) ⊂ G Qp of a connected component Z • of Z is contained in the parabolic subgroup P (λ) associated to λ, as explained in [12, Sec.4.1].Corollary 3.3.5.Let Z ⊂ Sh G,K p Kp be a smooth locally closed subscheme and assume that there is a prime ℓ = p such that Z is G(Q ℓ )-stable.Let Z • be a connected component of Z and suppose that Z • is contained in a single Q-non-basic Newton stratum Sh G,[b],K p Kp .If Hypothesis 2.3.1 holds, then the p-adic monodromy group Mon(Z • , M) ⊂ Mon(Z • , M † ) = G der Qp ⊂ G Qp 3 The conjugacy class {ν [b] } is defined over Qp, but it is not necessarily true that {ν [b] } has a representative defined over Qp unless G Qp is quasi-split is equal to the intersection G der Qp ∩ P (λ).In particular, the unipotent radical of Mon(Z • , M) is isomorphic to the unipotent radical of the parabolic subgroup P ν [b] of G Qp , for some representative ν [b] of {ν [b] }.Proof.The first assertion is a direct consequence of Corollary 3.3.3and [12, Thm 1.1.1].The second assertion follows from Lemma 3.3.Irreducible components of Hecke stable subvarieties.

3. 4 . 4 .
Connected components.The following result is well known.Lemma 3.4.5.Let Y ∞ be a connected component of the scheme

Theorem 3 . 4 . 10 .
Let X → Z be a G(A p f )-equivariant finite étale cover of a smooth G(A p f )stable locally closed subscheme Z ⊂ Sh G,K p Kp,Fp and suppose that each connected component of Z intersects a Q-non-basic Newton stratum.If Hypotheses 2.3.1 and 3.4.1 hold, then G sc (A Σ f ) acts trivially on the fibers of π 0 ( X) → π 0 (Sh G,Kp,Fp ).

1⊕ Y can 1 to
and we will use Y can := Y can 0 denote the canonical lift of Y to Zp .Let Y ∨ be the Serre-dual of Y and consider the free Z p -modules of rank g given by T p Y (F p ) and T p Y ∨ (F p ).By [22, Thm.2.1], the formal group Def(Y ) is isomorphic to the functor on Art sending R to hom

4. 2 .
Rapoport-Zink spaces and unipotent formal groups.Let Ỹ → Y be the universal cover of Y , defined as the inverse limit of the projective system lim ← − p:G→G Y.It is representable by a formal scheme by [45, Prop.3.1.3.(iii)].By the proof of [4, Proposition 4.2.11], the automorphism group functors of Y and Ỹ on Nilp can be described as follows

1 ,
see the beginning of Section 4.2 for the matrix notation.To show that this diagram commutes we consider the auxiliary commutative diagram(4.3.3)

5. 1 . 2 .
Let us now drop the level from the notation and write S G and S GSp respectively for the base changes of S K and S K respectively to Zp for some choice of O E,v → Zp .Similarly write Sh G for the special fiber of S G and Sh GSp for the special fiber of S GSp .Let Sh GSp,ord ⊂ Sh GSp be the dense open ordinary locus and define the ordinary locus of Sh G by Sh G,ord := Sh G ∩ Sh GSp,ord .It is an open subset which is nonempty if and only if E v = Q p , by [35, Cor.1.0.2].We will assume from now on that E v = Q p .Lemma 5.1.3.The ordinary locus Sh G,ord is open and dense and equal to the Newton stratum Sh G,[b],K p Kp for [b] ∈ B(G, {µ −1 }) the maximal element in the partial order introduced in Section 2.1.2.

. 3 .Y can 1 over
The Dieudonné module of the Serre-Tate torus.Let x ∈ Sh G,ord (F p ) be as above and let Y = A x [p ∞ ] be the corresponding p-divisible group.Recall from Section 4 that Y = Y 0 ⊕ Y 1 and that both Y 0 and Y 1 lift uniquely to p-divisible groups Y can 0 and Zp .Let Def(Y ) be the formal deformation space of Y , considered as a functor on Art together with its extension Def(Y ) to Nilp.We have seen that Def(Y ) has the structure of a p-divisible formal group, and we use H 0,1 to denote the corresponding p-divisible group over Spf Zp .
3.1.5,Sec.6.4].5.4.Serre-Tate coordinates for Hodge type Shimura varieties.Recall that x ∈ Sh G (F p ) is an ordinary point with associated element b = b x ∈ G( Qp ).Recall also from Section 2.1.1 that we have a G(Q p ) conjugacy class of cocharacters {µ} coming from the Shimura datum X and the fixed place v of E. Lemma 5.4.1.The conjugacy class of fractional cocharacters {ν [b] } defined by [b] is equal to {µ −1 }.Proof.The ordinary locus is equal to the µ-ordinary locus by Lemma 5.1.3.Therefore we have that {ν [b] } = {µ}, where {µ} is the Galois-average of {µ −1 }, see [47, Sec.2.1].But since G Qp is unramified and the local reflex field E v is equal to Q p there is a cocharacter µ defined over Q p inducing the conjugacy class of cocharacters {µ}.It follows that {µ} = {µ −1 }.

4 . 1 )
By Proposition 4.2.5, the action of H 0,1 on Def(Y ) factors through the natural action of H 0,1 on Def(Y ) by left translation, via the natural quotient map H 0,1 → H 0,1 .

G
, Prop.3.1.4],the formal scheme Def(Y ) G,Fp has dimension 2ρ, {ν [b] } , which is equal to 2ρ, {µ} by Lemma 5.4.1, which in turn is equal to the dimension of Sh /x G .It follows that the orbit map induces an isomorphism is a formal subgroup of Def(Y ) Fp satisfying the conclusions of the proposition.It remains to show that S /x G ⊂ Def(Y ) is a formal subgroup, which follows from [47, Thm.1.1].5.4.4.The action of automorphism groups.Let the notation be as in Section 5. Recall that we have fixed an isomorphism D cov (Y ) ≃ V * (p) ⊗ Zp sending s α ⊗ 1 to s α,cris .This gives us an element b ∈ G( Qp ) ⊂ GL(V * )( Qp ) corresponding to the Frobenius in D cov (Y )[1/p].

6. 1 .
Rigidity of Zariski closures of Hecke orbits.Let z ∈ Z(F p ) smooth point of Z and let I z (Q) be the group of self quasi-isogenies of z respecting the tensors, which was introduced in Section2.2.Let Y = A z [p ∞ ] and fix a choice of isomorphism D cov (Y ) ≃ V * (p) ⊗ Zp sending s α ⊗ 1 to s α,cris as in Section 5.3.This gives rise to an element b z = b ∈ G( Qp ) and we let U p ⊂ J b (Q p ) be the compact open subgroup introduced in Section 5.4.4.Let I z (Z (p) ) be the intersection of I z (Q) with U p inside J b (Q p ).We consider the closed immersion of formal neighbourhoods (where the notation is as in (5.1.2))Z/z ⊂ Sh /z G,K p Kp ⊂ Sh /z GSp,K p Kp .The goal of this section is to prove the following result.Proposition 6.1.1 (Local stabiliser principle).The closed subscheme Z /z ⊂ Sh /z G,K p Kp is stable under the action of I z (Z (p) ) via I z (Z (p) ) → U p .

Lemma 6 . 1 . 3 .
The stabiliser inside G(A p f ) of the point z is equal to I z (Z (p) ).Proof.By [29, Lem.2.1.4],the stabiliser is contained in I z (Z (p) ).The stabiliser in G V (A p f ) of the image of z in Sh G V ,Kp is equal to End λ (A z ) × and thus contains I z (Z (p) ).The result follows.

Sh
p Kp ⊂ Def(A z [p ∞ ]) is End λ (A z ) × -equivariant, where End λ (A z ) × acts on the RHS via the inclusion End λ (A z ) × ⊂ End(A z [p ∞ ]) × , followed by the natural action of End(A z [p ∞ ]) × on Def(A z [p ∞ )].Let Y = A z [p ∞ ]as above and write a + =: D cov (Y ) and a = a + [1/p].Write b + ⊂ a + for the covariant Dieudonné module of the p-divisible group associated to Z /z and b = b + [1/p].Then in the notation of [14, Sec.5.5] we have Z /z = Z(b + ).Now [14, Thm.5.5.3]tells us that there is an inclusion of algebraic groups over Qp Mon(Z /z , M, z) ⊂ U (b) := b ⊗ Qp G a .

6. 3 .
Monodromy and conclusion.Recall from Section 3.2.2 the maps B(G Qp ) → B(G ad Qp ) → n i=1 B(G i,Qp ).induced by the decomposition G ad = n i=1 G i of (2.2.1).Let [b ord ] ∈ B(G, {µ −1 }) be the σconjugacy class corresponding to the ordinary locus, and let [b ord,i ] be the image of [b ord ] in B(G i,Qp ).Lemma 6.3.1.For all i the element [b ord,i ] is non-basic.Proof.By the axioms of a Shimura datum, the G i (Q p )-conjugacy class of cocharacters {µ −1 i } induced by {µ −1 } is nontrivial for all i.By Lemma 5.4.1, we have an equality {µ −1 i } = {ν [b ord,i ] } and so the Newton cocharacter of [b ord,i ] is noncentral for all i.In other words, the σ-conjugacy class [b ord,i ] is non-basic for all i.Proof of Theorem I. Let x ∈ Sh G,K p Kp (F p ) be an ordinary point and let Z be the Zariski closure (inside Sh G,ord,K p Kp ) of its prime-to-p Hecke orbit.Then Z is G(A p f )-stable by Lemma 3.1.2and similarly its smooth locus Z sm ⊂ Z is G(A p f )-stable by Lemma 3.1.1.Let X be the p-divisible group over Z sm of the universal abelian variety and let M † be the associated overconvergent F -isocrystal, see Section 3.3.By Lemma 6.3.1, the element [b ord ] is Q-non-basic and by Lemma 2.3.2,we know that Hypothesis 2.3.1 is satisfied because K p is hyperspecial.Therefore Corollary 3.3.3tells us that the monodromy group of M † over Z sm is isomorphic to G der ⊗ Qp .Corollary 3.3.5 tells us that unipotent radical of the monodromy group of M over Z sm is isomorphic to the unipotent radical of the parabolic subgroup P ν [b] ⊂ G ⊗ Qp for any choice of ν [b] ∈ {ν [b] }.
3.1.14.Let N be the normaliser of M in G Q ℓ and let N • be its neutral component.The group N • is a connected reductive group because we are working with reductive groups in characteristic zero by e.g.[11, Prop.A.8.12].Lemma 3.1.15.The group P y is contained in N.
which is representable by a flat affine scheme over Spec R by Proposition [45, Prop.3.3.1].4.1.Classical Serre-Tate theory.Let Y be an ordinary p-divisible group of dimension g and height 2g over F p .In other words, let Y be a p-divisible group isomorphic to ) be the functor on Art sending (R, α) to the set of isomorphism classes of pairs (X, β) := lim − → Spf S[p n ] for the corresponding p-divisible group over Spf Zp .Note that it acts via left translation on Def(Y ); we will denote this action by a ST (for Serre-Tate).Let us prove the stronger assertion that there are isomorphisms H 0,1 [p n ] ≃ Hom (Y can 0 , Y can 1 )[p n ] for all n, compatible with changing n.Note that H 0,1 [p n ] is represented by the spectrum of an Artin local Zp -algebra.The same is true for Hom (Y can 0 , Y can 1 )[p n ], since Hom (Z/p n Z, µ p n ) ≃ µ p n .Thus it suffices to show that the functors H 0,1 [p n ] and Hom (Y can 0 , Y can 1 )[p n ] are isomorphic as functors on Art.
Lemma 4.1.2.The p-adic Tate module of H 0,1 is isomorphic to the sheaf Hom (Y