p-Adic Monodromy of the Universal

Deformation of a HW-Cyclic Barsotti-Tate Group

Yichao Tian

Received: September 5, 2009

Communicated by Peter Schneider

Abstract. Let k be an algebraically closed field of characteristic p > 0, and G be a Barsotti-Tate over k. We denote by S the “algebraic” local moduli in characteristic p of G, by G the universal deformation of G over S, and by U ⊂ S the ordinary locus of G. The étale part of G over U gives rise to a monodromy representation ρ_{G} of the fundamental group of U on the Tate module of G. Motivated by a famous theorem of Igusa, we prove in this article that ρ_{G} is surjective if G is connected and HW-cyclic. This latter condition is equivalent to saying that Oort’s a-number of G equals 1, and it is satisfied by all connected one-dimensional Barsotti-Tate groups over k.

2000 Mathematics Subject Classification: 13D10, 14L05, 14H30, 14B12, 14D15, 14L15 Keywords and Phrases: Barsotti-Tate groups (p-divisible groups), padic monodromy representation, universal deformation, Hasse-Witt maps.

1. Introduction

1.1. A classical theorem of Igusa says that the monodromy representation associated with a versal family of ordinary elliptic curves in characteristic p > 0 is surjective [Igu, Ka2]. This important result has deep consequences in the theory of p-adic modular forms, and inpsired various generalizations. Faltings and Chai [Ch2, FC] extended it to the universal family over the moduli space of higher dimensional principally polarized ordinary abelian varieties in characteristic p, and Ekedahl [Eke] generalized it to the jacobian of the universal n-pointed curve in characteristic p, equipped with a symplectic level structure. Recently, Chai and Oort [CO] proved the maximality of the p-adic monodromy over each “central leaf” in the moduli space of abelian varieties which is not contained in the supersingular locus. We refer to Deligne-Ribet [DR] and Hida [Hid] for other generalizations to some moduli spaces of PEL-type and their arithmetic applications. Though it has been formulated in a global setting, the proof of Igusa’s theorem is purely local, and it has got also local generalizations. Gross [Gro] generalized it to one-dimensional formal 𝒪-modules over a complete discrete valuation ring of characteristic p, where 𝒪 is the integral closure of ℤ_{p} in a finite extension of ℚ_{p}. We refer to Chai [Ch2] and Achter-Norman [AN] for more results on local monodromy of Barsotti-Tate groups. Motivated by these results, it has been longly expected/conjectured that the monodromy of a versal family of ordinary Barsotti-Tate groups in characteristic p > 0 is maximal. The aim of this paper is to prove the surjectivity of the monodromy representation associated with the universal deformation in characteristic p of a certain class of Barsotti-Tate groups.

1.2. To describe our main result, we introduce first the notion of HW-cyclic Barsotti-Tate groups. Let k be an algebraically closed field of characteristic p > 0, and G be a Barsotti-Tate group over k. We denote by G^{∨} the Serre dual of G, and by Lie(G^{∨}) its Lie algebra. The Frobenius homomorphism of G (or dually the Verschiebung of G^{∨}) induces a semi-linear endomorphism φ_{G} on Lie(G^{∨}), called the Hasse-Witt map of G (2.6.1). We say that G is HW-cyclic, if c = dim(G^{∨}) ≥ 1 and there is a v ∈ Lie(G^{∨}) such that v, φ_{G}(v), ⋯, φ_{G}^{c−1}(v) form a basis of Lie(G^{∨}) over k (4.1). We prove in 4.7 that G is HW-cyclic and nonordinary if and only if the a-number of G, defined previously by Oort, equals 1. Basic examples of HW-cyclic Barsotti-Tate groups are given as follows. Let r, s be relatively prime integers such that 0 ≤ s ≤ r and r ≠ 0, λ = s/r, G^{λ} be the Barsotti-Tate group over k whose (contravariant) Dieudonné module is generated by an element e over the non-commutative Dieudonné ring with the relation (F^{r−s} − V^{s}) · e = 0 (4.10). It is easy to see that G^{λ} is HW-cyclic for any 0 < λ < 1. Any connected Barsotti-Tate group over k of dimension 1 and height h is isomorphic to G^{1/h} [Dem, Chap.IV §8]. Let G be a Barsotti-Tate group of dimension d and height c + d over k; assume c ≥ 1. We denote by S the “algebraic” local moduli of G in characteristic p, and by G be the universal deformation of G over S (cf. 3.8). The scheme S is affine of ring R ≃ k[[(t_{i,j})_{1≤i≤c,1≤j≤d}]], and the Barsotti-Tate group G is obtained by algebraizing the formal universal deformation of G over Spf(R) (3.7). Let U be the ordinary locus of G (i.e. the open subscheme of S parametrizing the ordinary fibers of G), and \overline{η} a geometric point over the generic point of U. For any integer n ≥ 1, we denote by G(n) the kernel of the multiplication by p^{n} on G, and by

T_{p}(G, \overline{η}) = \underleftarrow{lim}_{n} G(n)(\overline{η})

the Tate module of G at \overline{η}. This is a free ℤ_{p}-module of rank c. We consider the monodromy representation attached to the étale part of G over U

(1.2.1)  ρ_{G} : π_{1}(U, \overline{η}) → Aut_{ℤ_{p}}(T_{p}(G, \overline{η})) ≃ GL_{c}(ℤ_{p}).

The aim of this paper is to prove the following :

p-Adic Monodromy of a Barsotti-Tate Group 399

Theorem 1.3. If G is connected and HW-cyclic, then the monodromy representation ρ_{G} is surjective.

Igusa’s theorem mentioned above corresponds to Theorem 1.3 for G = G^{1/2} (cf. 5.7). My interest in the p-adic monodromy problem started with the second part of my PhD thesis [Ti1], where I guessed 1.3 for G = G^{λ} with 0 < λ < 1 and proved it for G^{1/3}. After I posted the manuscript on ArXiv [Ti2], Strauch proved the one-dimensional case of 1.3 by using Drinfeld’s level structures [Str, Theorem 2.1]. Later on, Lau [Lau] proved 1.3 without the assumption that G is HW-cyclic. By using the Newton stratification of the universal deformation space of G due to Oort, Lau reduced the higher dimensional case to the one-dimensional case treated by Strauch. In fact, Strauch and Lau considered more generally the monodromy representation over each p-rank stratum of the universal deformation space. In this paper, we provide first a different proof of the one-dimensional case of 1.3. Our approach is purely characteristic p, while Strauch used Drinfeld’s level structure in characteristic 0. Then by following Lau’s strategy, we give a new (and easier) argument to reduce the general case of 1.3 to the one-dimensional case for HW-cyclic groups. The essential part of our argument is a versality criterion by Hasse-Witt maps of deformations of a connected one-dimensional Barsotti-Tate group (Prop. 4.11). This criterion can be considered as a generalization of another theorem of Igusa which claims that the Hasse invariant of a versal family of elliptic curves in characteristic p has simple zeros. Compared with Strauch’s approach, our characteristic p approach has the advantage of giving also results on the monodromy of Barsotti-Tate groups over a discrete valuation ring of characteristic p.

1.4. Let A = k[[π]] be the ring of formal power series over k in the variable π, K its fraction field, and v the valuation on K normalized by v(π) = 1. We fix an algebraic closure \overline{K} of K, and let K^{sep} be the separable closure of K contained in \overline{K}, I be the Galois group of K^{sep} over K, I_{p} ⊂ I be the wild inertia subgroup, and I_{t} = I/I_{p} the tame inertia group. For every integer n ≥ 1, there is a canonical surjective character θ_{p^{n}−1} : I_{t} → 𝔽_{p^{n}}^{×} (5.2), where 𝔽_{p^{n}} is the finite subfield of k with p^{n} elements. We put S = Spec(A). Let G be a Barsotti-Tate group over S, G^{∨} be its Serre dual, Lie(G^{∨}) the Lie algebra of G^{∨}, and φ_{G} the Hasse-Witt map of G, i.e. the semi-linear endomorphism of Lie(G^{∨}) induced by the Frobenius of G. We define h(G) to be the valuation of the determinant of a matrix of φ_{G}, and call it the Hasse invariant of G (5.4). We see easily that h(G) = 0 if and only if G is ordinary over S, and h(G) < ∞ if and only if G is generically ordinary. If G is connected of height 2 and dimension 1, then h(G) = 1 is equivalent to that G is versal (5.7).

Proposition 1.5. Let S = Spec(A) be as above, G be a connected HW-cyclic Barsotti-Tate group with Hasse invariant h(G) = 1, and G(1) the kernel of the multiplication by p on G. Then the action of I on G(1)(\overline{K}) is tame; moverover, G(1)(\overline{K}) is an 𝔽_{p^{c}}-vector space of dimension 1 on which the induced action of I_{t} is given by the surjective character θ_{p^{c}−1} : I_{t} → 𝔽_{p^{c}}^{×}.

This proposition is an analog in characteristic p of Serre’s result [Se3, Prop. 9] on the tameness of the monodromy associated with one-dimensional formal groups over a trait of mixed characteristic. We refer to 5.8 for the proof of this proposition and more results on the p-adic monodromy of HW-cyclic Barsotti-Tate groups over a trait in characteristic p.

1.6. This paper is organized as follows. In Section 2, we review some well known facts on ordinary Barsotti-Tate groups. Section 3 contains some preliminaries on the Dieudonné theory and the deformation theory of Barsotti-Tate groups. In Section 4, after establishing some basic properties of HW-cyclic groups, we give the fundamental relation between the versality of a Barsotti-Tate group and the coefficients of its Hasse-Witt matrix (Prop. 4.11). Section 5 is devoted to the study of the monodromy of a HW-cyclic Barsotti-Tate group over a complete trait of characteristic p. Section 6 is totally elementary, and contains a criterion (6.3) for the surjectivity of a homomorphism from a profinite group to GL_{n}(ℤ_{p}). Section 7 is the heart of this work, and it contains a proof of Theorem 1.3 in the one-dimensional case. Finally in Section 8, we follow Lau’s strategy and complete the proof of 1.3 by reducing the general case to the one-dimensional case treated in Section 7. The proof in Section 7 of 1.3 in the one-dimensional case is based on an induction on the height n + 1 ≥ 2 of G. The case n = 1 is just the classical Igusa’s theorem (5.7). For n ≥ 2, by lemmas 6.3 and 6.5, it suffices to prove the following two statements: (a) the image of reduction modulo p of ρ_{G} contains a non-split Cartan subgroup; (b) under a suitable basis, the image of ρ_{G} contains

all matrix of the form (\begin{matrix}B & b \\ 0 & 1 \end{matrix}) with B ∈ GL_{n−1}(ℤ_{p}) and b ∈ M_{(n−1)×1}(ℤ_{p}).

The first statement follows easily from 1.5 by considering a certain base change of G to a complete discrete valuation ring. To prove (b), we consider the formal completion Spec(R′) of the localization of the local moduli S = Spec(R) of G at the generic point of the locus where the universal deformation G has p-rank ≤ 1 (7.4). The ring R′ is a complete regular ring of dimension n − 1, and the Barsotti-Tate group 𝒢′ = G ⊗_{R} R′ has a connected part of height n and an étale part of height 1. Let K_{0} be the residue field of R′, and \overline{K}_{0} an algebraic closure of K_{0}. In order to apply the induction hypothesis, we consider the set of k-algebra homomorphisms σ : R′ → R̃′ = \overline{K}_{0}[[t_{1}, ⋯, t_{n−1}]] lifting the natural inclusion K_{0} → \overline{K}_{0}. The key point is that, the natural map σ ↦ 𝒢_{R̃′,σ} = 𝒢′⊗_{R′,σ} R̃′ gives a bijection between the set of such σ’s and the set

of deformations of 𝒢_{\overline{K}_{0}} = 𝒢′⊗_{R′} \overline{K}_{0} to R̃′; moreover, we can compute explicitly the Hasse-Witt map of the connected component 𝒢_{R̃′,σ}^{∘} of 𝒢_{R̃′,σ} (Lemma 7.8). From the versality criterion for one-dimensional Barsotti-Tate groups in terms of the Hasse-Witt map established in Section 4 (Prop. 4.11), it follows immediately that there exists a σ such that the Barsotti-Tate group 𝒢_{R̃′,σ}^{∘}, which is connected and one-dimensional of height n, is the universal deformation of its closed fiber. We fix such a σ. Then the set of all σ′ with 𝒢_{R̃′,σ′}^{∘} ≃ 𝒢_{R̃′,σ}^{∘} as deformations of their common closed fiber is actually a group isomorphic to Ext1 ; R̃′(ℚ_{p}/ℤ_{p}, 𝒢_{R̃′,σ}^{∘}) (Prop. 3.10). Let σ_{1} be the element corresponding

to neutral element in Ext1 ; R̃′(ℚ_{p}/ℤ_{p}, 𝒢_{R̃′,σ}^{∘}). Applying the induction hypothesis to 𝒢_{R̃′,σ_{1}}^{∘}, we see that the monodromy group of 𝒢_{R̃′,σ_{1}}, hence that of G, con-

tains the subgroup (\begin{matrix}GL_{n−1}(ℤ_{p}) & 0 \\ 0 & 1 \end{matrix}) under a suitable basis of the Tate module

(7.5.3). In order to conclude the proof, we need another σ_{2} such that 𝒢_{R̃′,σ_{2}} has the same connected component as 𝒢_{R̃′,σ_{1}}, and that the induced extension between the Tate module of the étale part of 𝒢_{R̃′,σ_{2}} and that of 𝒢_{R′,σ_{2}}^{∘} is nontrivial after reduction modulo p (see 7.5 and 7.5.4). To verify the existence of such a σ_{2}, we reduce the problem to a similar situation over a complete trait of characteristic p (see 7.9), and we use a criterion of non-triviality of extensions by Hasse-Witt maps (5.12).

1.7. Acknowledgement. This paper is an expanded version of the second part of my Ph.D. thesis at University Paris 13. I would like to express my great gratitude to my thesis advisor Prof. A. Abbes for his encouragement during this work, and also for his various helpful comments on earlier versions of this paper. I also thank heartily E. Lau, F. Oort and M. Strauch for interesting discussions and valuable suggestions.

1.8. Notations. Let S be a scheme of characteristic p > 0. A BT-group over S stands for a Barsotti-Tate group over S. Let G be a commutative finite group scheme (resp. a BT-group) over S. We denote by G^{∨} its Cartier dual (resp. its Serre dual), by ω_{G} the sheaf of invariant differentials of G over S, and by Lie(G) the sheaf of Lie algebras of G. If S = Spec(A) is affine and there is no risk of confusions, we also use ω_{G} and Lie(G) to denote the corresponding A-modules of global sections. We put G^{(p)} the pull-back of G by the absolute Frobenius of S, F_{G}: G → G^{(p)} the Frobenius homomorphism and V_{G}: G^{(p)} → G the Verschiebung homomorphism. If G is a BT-group and n an integer ≥ 1, we denote by G(n) the kernel of the multiplication by p^{n} on G; we have G^{∨}(n) = (G^{∨})(n) by definition. For an 𝒪_{S}-module M, we denote by M^{(p)} = 𝒪_{S} ⊗_{F_{S}} M the scalar extension of M by the absolute Frobenius of 𝒪_{S}. If φ : M → N be a semi-linear homomorphism of 𝒪_{S}-modules, we denote by φ̃: M^{(p)} → N the linearization of φ, i.e. we have φ̃(λ ⊗ x) = λ · φ(x), where λ (resp. x) is a local section of 𝒪_{S} (resp. of M). Starting from Section 5, k will denote an algebraically closed field of characteristic p > 0.

2. Review of ordinary Barsotti-Tate groups

In this section, S denotes a scheme of characteristic p > 0.

2.1. Let G be a commutative group scheme, locally free of finite type over S. We have a canonical isomorphism of coherent 𝒪_{S}-modules [Ill, 2.1]

(2.1.1)  Lie(G^{∨}) ≃ ℋ om _{S_{fppf}} (G, 𝔾_{a}),

where ℋ om _{S_{fppf}} is the sheaf of homomorphisms in the category of abelian fppf-sheaves over S, and 𝔾_{a} is the additive group scheme. Since 𝔾_{a}^{(p)} ≃ 𝔾_{a}, the Frobenius homomorphism of 𝔾_{a} induces an endomorphism

(2.1.2)  φ_{G} : Lie(G^{∨}) → Lie(G^{∨}),

semi-linear with respect to the absolute Frobenius map F_{S} : 𝒪_{S} → 𝒪_{S}; we call it the Hasse-Witt map of G. By the functoriality of Frobenius, φ_{G} is also the canonical map induced by the Frobenius of G, or dually by the Verschiebung of G^{∨}.

2.2. By a commutative p-Lie algebra over S, we mean a pair (L, φ), where L is an 𝒪_{S}-module locally free of finite type, and φ : L → L is a semi-linear endomorphism with respect to the absolute Frobenius F_{S} : 𝒪_{S} → 𝒪_{S}. When there is no risk of confusions, we omit φ from the notation. We denote by p-𝔏ie_{S} the category of commutative p-Lie algebras over S. Let (L, φ) be an object of p-𝔏ie_{S}. We denote by

𝒰 (L) = Sym(L) = ⊕_{n≥0} Sym^{n}(L),

the symmetric algebra of L over 𝒪_{S}. Let ℐ_{p}(L) be the ideal sheaf of 𝒰 (L) defined, for an open subset V ⊂ S, by

Γ(V, ℐ_{p}(L)) = {x^{⊗p} − φ(x) ; x ∈ Γ(V, 𝒰 (L))},

where x^{⊗p} = x ⊗ x ⊗ ⋯ ⊗ x ∈ Γ(V, Sym^{p}(L)). We put 𝒰_{p}(L) = 𝒰 (L)/ℐ_{p}(L), and call it the p-enveloping algebra of (L, φ). We endow 𝒰_{p}(L) with the structure of a Hopf-algebra with the comultiplication given by Δ(x) = 1 ⊗ x + x ⊗ 1 and the coinverse given by i(x) = −x. Let G be a commutative group scheme, locally free of finite type over S. We say that G is of coheight one if the Verschiebung V_{G} : G^{(p)} → G is the zero homomorphism. We denote by 𝔊V_{S} the category of such objects. For an object G of 𝔊V_{S}, the Frobenius F_{G^{∨}} of G^{∨} is zero, so the Lie algebra Lie(G^{∨}) is locally free of finite type over 𝒪_{S} ([DG] VII_{A} Théo. 7.4(iii)). The Hasse-Witt map of G (2.1.2) endows Lie(G^{∨}) with a commutative p-Lie algebra structure over S.

Proposition 2.3 ([DG] VII_{A}, Théo. 7.2 et 7.4). The functor 𝔊V_{S} → p-𝔏ie_{S} defined by G ↦ Lie(G^{∨}) is an anti-equivalence of categories; a quasi-inverse is given by (L, φ) ↦ Spec(𝒰_{p}(L)).

2.4. Assume S = Spec(A) affine. Let (L, φ) be an object of p-𝔏ie_{S} such that L is free of rank n over 𝒪_{S}, (e_{1}, ⋯, e_{n}) be a basis of L over 𝒪_{S}, (h_{ij})_{1≤i,j≤n} be the matrix of φ under the basis (e_{1}, ⋯, e_{n}), i.e. φ(e_{j}) = ∑_{i=1}^{n} h_{ij}e_{i} for 1 ≤ j ≤ n. Then the group scheme attached to (L, φ) is explicitly given by

Spec(𝒰_{p}(L)) = Spec(A[X_{1}, ⋯, X_{n}]/(X_{j}^{p} − ∑_{i=1}^{n} h_{ij}X_{i})_{1≤j≤n}),

with the comultiplication Δ(X_{j}) = 1 ⊗ X_{j} + X_{j} ⊗ 1. By the Jacobian criterion of étaleness [EGA, IV_{0} 22.6.7], the finite group scheme Spec(𝒰_{p}(L)) is étale over S if and only if the matrix (h_{ij})_{1≤i,j≤n} is invertible. This condition is equivalent to that the linearization of φ is an isomorphism.

Corollary 2.5. An object G of 𝔊V_{S} is étale over S, if and only if the linearization of its Hasse-Witt map (2.1.2) is an isomorphism.

Proof. The problem being local over S, we may assume S affine and L = Lie(G^{∨}) free over 𝒪_{S}. By Theorem 2.3, G is isomorphic to Spec(𝒰_{p}(L)), and we conclude by the last remark of 2.4. □

2.6. Let G be a BT-group over S of height c + d and dimension d. The Lie algebra Lie(G^{∨}) is an 𝒪_{S}-module locally free of rank c, and canonically identified with Lie(G^{∨}(1))([BBM] 3.3.2). We define the Hasse-Witt map of G

(2.6.1)  φ_{G} : Lie(G^{∨}) → Lie(G^{∨})

to be that of G(1) (2.1.2).

2.7. Let k be a field of characteristic p > 0, G be a BT-group over k. Recall that we have a canonical exact sequence of BT-groups over k

(2.7.1)  0 → G^{∘} → G → G^{ét} → 0

with G^{∘} connected and G^{ét} étale ([Dem] Chap.II, §7). This induces an exact sequence of Lie algebras

(2.7.2)  0 → Lie(G^{ét∨}) → Lie(G^{∨}) → Lie(G^{∘∨}) → 0,

compatible with Hasse-Witt maps.

Proposition 2.8. Let k be a field of characteristic p > 0, G be a BT-group over k. Then Lie(G^{ét∨}) is the unique maximal k-subspace V of Lie(G^{∨}) with the following properties:

(a) V is stable under φ_{G};

(b) the restriction of φ_{G} to V is injective.

Proof. It is clear that Lie(G^{ét∨}) satisfies property (a). We note that the Verschiebung of G^{ét}(1) vanishes; so G^{ét}(1) is in the category 𝔊V_{Spec(k)}. Since k is a field, 2.5 implies that the restriction of φ_{G} to Lie(G^{ét∨}), which coincides with φ_{G^{ét}}, is injective. This proves that Lie(G^{ét∨}) verifies (b). Conversely, let V be an arbitrary k-subspace of Lie(G^{∨}) with properties (a) and (b). We have to show that V ⊂ Lie(G^{ét∨}). Let σ be the Frobenius endomorphism of k. If M is a k-vector space, for each integer n ≥ 1, we put M^{(p^{n})} = k ⊗_{σ^{n}} M, i.e. we have 1 ⊗ ax = σ^{n}(a) ⊗ x in k ⊗_{σ^{n}} M for a ∈ k, x ∈ M. Since φ_{G}|_{V} : V → V is injective by assumption, the linearization φ̃_{G}^{n}|_{V^{(p^{n})}} : V^{(p^{n})} → V of φ_{G}^{n}|_{V} is injective (hence bijective) for any n ≥ 1. We have V = φ̃_{G}^{n}(V^{(p^{n})}). Since G^{∘} is connected, there is an integer n ≥ 1 such that the n-th iterated Frobenius F_{G^{∘}(1)}^{n} : G^{∘}(1) → G^{∘}(1)^{(p^{n})} vanishes. Hence by definition, the linearized

n-iterated Hasse-Witt map φ̃_{G^{∘}}^{n} : Lie(G^{∘∨})^{(p^{n})} → Lie(G^{∘∨}) is zero. By the compatibility of Hasse-Witt maps, we have φ̃_{G}^{n}(Lie(G^{∨})^{(p^{n})}) ⊂ Lie(G^{ét∨}); in particular, we have V = φ̃_{G}^{n}(V^{(p^{n})}) ⊂ Lie(G^{ét∨}). This completes the proof. □

Corollary 2.9. Let k be a field of characteristic p > 0, G be a BT-group over k. Then G is connected if and only if φ_{G} is nilpotent.

Proof. In the proof of the proposition, we have seen that the Hasse-Witt map of the connected part of G is nilpotent. So the “only if” part is verified. Conversely, if φ_{G} is nilpotent, Lie(G^{ét∨}) is zero by the proposition. Therefore G is connected. □

Definition 2.10. Let S be a scheme of characteristic p > 0, G be a BTgroup over S. We say that G is ordinary if there exists an exact sequence of BT-groups over S

(2.10.1)  0 → G^{mult} → G → G^{ét} → 0,

such that G^{mult} is multiplicative and G^{ét} is étale.

We note that when it exists, the exact sequence (2.10.1) is unique up to a unique isomorphism, because there is no non-trivial homomorphisms between a multiplicative BT-group and an étale one in characteristic p > 0. The property of being ordinary is clearly stable under arbitrary base change and Serre duality. If S is the spectrum of a field of characteristic p > 0, G is ordinary if and only if its connected part G^{∘} is of multiplicative type.

Proposition 2.11. Let G be a BT-group over S. The following conditions are equivalent:

(a) G is ordinary over S.

(b) For every x ∈ S, the fiber G_{x} = G ⊗_{S} κ(x) is ordinary over κ(x).

(c) The finite group scheme Ker V_{G} is étale over S. (c’) The finite group scheme Ker F_{G} is of multiplicative type over S.

(d) The linearization of the Hasse-Witt map φ_{G} is an isomorphism.

First, we prove the following lemmas.

Lemma 2.12. Let T be a scheme, H be a commutative group scheme locally free of finite type over T. Then H is étale ( resp. of multiplicative type) over T if and only if, for every x ∈ T, the fiber H ⊗_{T} κ(x) is étale ( resp. of multiplicative type) over κ(x).

Proof. We will consider only the étale case; the multiplicative case follows by duality. Since H is T -flat, it is étale over T if and only if it is unramified over T. By [EGA, IV 17.4.2], this condition is equivalent to that H ⊗_{T} κ(x) is unramified over κ(x) for every point x ∈ T. Hence the conclusion follows. □

Lemma 2.13. Let G be a BT-group over S. Then Ker V_{G} is an object of the category 𝔊V_{S}, i.e. it is locally free of finite type over S, and its Verschiebung is zero. Moreover, we have a canonical isomorphism (Ker V_{G})^{∨} ≃ Ker F_{G^{∨}}, which induces an isomorphism of Lie algebras Lie((Ker V_{G})^{∨}) ≃ Lie(Ker F_{G^{∨}}) = Lie(G^{∨}), and the Hasse-Witt map (2.1.2) of Ker V_{G} is identified with φ_{G} (2.6.1).

Proof. The group scheme Ker V_{G} is locally free of finite type over S ([Ill] 1.3(b)), and we have a commutative diagram

\xymatrix{(Ker V_{G})^{(p)} \ar[r]^{V_{Ker V_{G}}} \ar@{^{(}->}[d] & Ker V_{G} \ar@{^{(}->}[d] \\ (G^{(p)})^{(p)} \ar[r]^{V_{G^{(p)}}} & G^{(p)}}

By the functoriality of Verschiebung, we have V_{G^{(p)}} = (V_{G})^{(p)} and Ker V_{G^{(p)}} = (Ker V_{G})^{(p)}. Hence the composition of the left vertical arrow with V_{G^{(p)}} vanishes, and the Verschiebung of Ker V_{G} is zero. By Cartier duality, we have (Ker V_{G})^{∨} = Coker(F_{G^{∨}(1)}). Moreover, the exact sequence

⋯ → G^{∨}(1) ⟶^{F_{G^{∨}(1)}} (G^{∨}(1))^{(p)} ⟶^{V_{G^{∨}(1)}} G^{∨}(1) → ⋯,

induces a canonical isomorphism

(2.13.1)  Coker(F_{G^{∨}(1)}) ⟶^{∼} Im(V_{G^{∨}(1)}) = Ker F_{G^{∨}(1)} = Ker F_{G^{∨}}.

Hence, we deduce that

(2.13.2)  (Ker V_{G})^{∨} ≃ Coker(F_{G^{∨}(1)}) ⟶^{∼} Ker F_{G^{∨}} ↪ G^{∨}(1).

Since the natural injection Ker F_{G^{∨}} → G^{∨}(1) induces an isomorphism of Lie algebras, we get

(2.13.3)  Lie((Ker V_{G})^{∨}) ≃ Lie(Ker F_{G^{∨}}) = Lie(G^{∨}(1)) = Lie(G^{∨}).

It remains to prove the compatibility of the Hasse-Witt maps with (2.13.3). We note that the dual of the morphism (2.13.2) is the canonical map F : G(1) → Ker V_{G} = Im(F_{G(1)}) induced by F_{G(1)}. Hence by (2.1.1), the isomorphism (2.13.3) is identified with the functorial map

ℋ om _{S_{fppf}} (Ker V_{G}, 𝔾_{a}) → ℋ om _{S_{fppf}} (G(1), 𝔾_{a})

induced by F, and its compatibility with the Hasse-Witt maps follows easily from the definition (2.1.2). □

Proof of 2.11. (a)⇒(b). Indeed, the ordinarity of G is stable by base change. (b)⇒(c). By Lemma 2.12, it suffices to verify that for every point x ∈ S, the fiber (Ker V_{G}) ⊗_{S} κ(x) ≃ Ker V_{G_{x}} is étale over κ(x). Since G_{x} is assumed to be ordinary, its connected part (G_{x})^{∘} is multiplicative. Hence, the Verschiebung of (G_{x})^{∘} is an isomorphism, and Ker V_{G_{x}} is canonically isomorphic to Ker V_{G_{x}^{ét}} ⊂ (G_{x}^{ét})^{(p)} ≃ (G_{x}^{(p)})^{ét}, so our assertion follows.

(c) ⇔ (d). It follows immediately from Lemma 2.13 and Corollary 2.5. (c)⇔(c’). By 2.12, we may assume that S is the spectrum of a field. So the category of commutative finite group schemes over S is abelian. We will just prove (c)⇒(c’); the converse can be proved by duality. We have a fundamental short exact sequence of finite group schemes

(2.13.4)  0 → Ker F_{G} → G(1) ⟶^{F} Ker V_{G} → 0,

where F is induced by F_{G(1)}, That induces a commutative diagram

\xymatrix{0 \ar[r] & (Ker F_{G})^{(p)} \ar[d]^{V′} \ar[r] & (G(1))^{(p)} \ar[r]^{F^{(p)}} \ar[d]^{V_{G(1)}} & (Ker V_{G})^{(p)} \ar[r] \ar[d]^{V′′} & 0 \\ 0 \ar[r] & Ker F_{G} \ar[r] & G(1) \ar[r]^{F} & Ker V_{G} \ar[r] & 0}

where vertical arrows are the Verschiebung homomorphisms. We have seen that V′′ = 0 (2.13). Therefore, by the snake lemma, we have a long exact sequence

(2.13.5)

0 → Ker V′ → Ker V_{G(1)} ⟶^{α} (Ker V_{G})^{(p)} →

→ Coker V′ → Coker V_{G(1)} ⟶^{β} Ker V_{G} → 0,

where the map α is the Frobenius of Ker V_{G} and β is the composed isomorphism

Coker(V_{G(1)}) ≃ G(1)/ Ker F_{G(1)} ⟶^{∼} Im(F_{G(1)}) ≃ Ker V_{G}.

Then condition (c) is equivalent to that α is an isomorphism; it implies that Ker V′ = Coker V′ = 0, i.e. the Verschiebung of Ker F_{G} is an isomorphism, and hence (c’). (c)⇒(a). For every integer n > 0, we denote by F_{G}^{n} the composed homomorphism

G ⟶^{F_{G}} G^{(p)} ⟶^{F_{G^{(p)}}} ⋯ ⟶^{F_{G^{(p^{n−1})}}} G^{(p^{n})},

and by V_{G}^{n} the composed homomorphism

G^{(p^{n})} ⟶^{V_{G^{(p^{n−1})}}} G^{(p^{n−1})} ⟶^{V_{G^{(p^{n−2})}}} ⋯ ⟶^{V_{G}} G;

F_{G}^{n} and V_{G}^{n} are isogenies of BT-groups. From the relation V_{G}^{n} ∘ F_{G}^{n} = p^{n}, we deduce an exact sequence

(2.13.6)  0 → Ker F_{G}^{n} → G(n) ⟶^{F^{n}} Ker V_{G}^{n} → 0,

where F^{n} is induced by F_{G}^{n}. For 1 ≤ j < n, we have a commutative diagram

(2.13.7)  \xymatrix{G^{(p^{n})} \ar[rr]^{V_{G^{(p^{j})}}^{n−j}} \ar[dr]_{V_{G}^{n}} &  & G^{(p^{j})} \ar[dl]^{V_{G}^{j}} \\  & G.}

One notices that Ker V_{G^{(p^{j})}}^{n−j} = (Ker V_{G}^{n−j})^{(p^{j})} by the functoriality of Verschiebung . Since all maps in (2.13.7) are isogenies, we have an exact sequence

(2.13.8)  0 → (Ker V_{G}^{n−j})^{(p^{j})} ⟶^{i′_{n−j,n}} Ker V_{G}^{n} ⟶^{p_{n,j}} Ker V_{G}^{j} → 0.

Therefore, condition (c) implies by induction that Ker V_{G}^{n} is an étale group scheme over S. Hence the j-th iteration of the Frobenius Ker V_{G}^{n−j} → (Ker V_{G}^{n−j})^{(p^{j})} is an isomorphism, and Ker V_{G}^{n−j} is identified with a closed subgroup scheme of Ker V_{G}^{n} by the composed map

i_{n−j,n} : Ker V_{G}^{n−j} ⟶^{∼} (Ker V_{G}^{n−j})^{(p^{j})} ⟶^{i′_{n−j,n}} Ker V_{G}^{n}.

We claim that the kernel of the multiplication by p^{n−j} on Ker V_{G}^{n} is Ker V_{G}^{n−j}. Indeed, from the relation p^{n−j} · Id_{G^{(p^{n})}} = F_{G^{(p^{j})}}^{n−j} ∘ V_{G^{(p^{j})}}^{n−j}, we deduce a commutative diagram (without dotted arrows)

(2.13.9)  \xymatrix{Ker V_{G}^{n} \ar[rr] \ar[dd]_{p^{n−j}} \ar@{-->}[dr]^{p_{n,j}} &  & G^{(p^{n})} \ar[dd]_{p^{n−j}} \ar[dr]^{V_{G^{(p^{j})}}^{n−j}} \\  & Ker V_{G}^{j} \ar@{-->}[rr] \ar@{-->}[dl]^{i_{j,n}} &  & G^{(p^{j})} \ar[dl]^{F_{G^{(p^{j})}}^{n−j}} \\ Ker V_{G}^{n} \ar[rr] &  & G^{(p^{n})}.}

It follows from (2.13.8) that the subgroup Ker V_{G}^{n} of G^{(p^{n})} is sent by V_{G^{(p^{j})}}^{n−j} onto Ker V_{G}^{j}. Therefore diagram (2.13.9) remains commutative when completed by the dotted arrows, hence our claim. It follows from the claim that (Ker V_{G}^{n})_{n≥1} constitutes an étale BT-group over S, denoted by G^{ét}. By duality, we have an exact sequence

(2.13.10)  0 → Ker F_{G}^{j} → Ker F_{G}^{n} → (Ker F_{G}^{n−j})^{(p^{j})} → 0.

Condition (c’) implies by induction that Ker F_{G}^{n} is of multiplicative type. Hence the j-th iteration of Verschiebung (Ker F_{G}^{n−j})^{(p^{j})} → Ker F_{G}^{n−j} is an isomorphism. We deduce from (2.13.10) that (Ker F_{G}^{n})_{n≥1} form a multiplicative BTgroup over S that we denote by G^{mult}. Then the exact sequences (2.13.6) give a decomposition of G of the form (2.10.1). □

Corollary 2.14. Let G be a BT-group over S, and S^{ord} be the locus in S of the points x ∈ S such that G_{x} = G ⊗_{S} κ(x) is ordinary over κ(x). Then S^{ord} is open in S, and the canonical inclusion S^{ord} → S is affine.

The open subscheme S^{ord} of S is called the ordinary locus of G.

3. Preliminaries on Dieudonné Theory and Deformation Theory

3.1. We will use freely the conventions of 1.8. Let S be a scheme of characteristic p > 0, G be a Barsotti-Tate group over S, and M(G) = 𝔻(G)_{(S,S)} be the coherent 𝒪_{S}-module obtained by evaluating the (contravariant) Dieudonné crystal of G at the trivial divided power immersion S ↪ S [BBM, 3.3.6]. Recall that M(G) is an 𝒪_{S}-module locally free of finite type satisfying the following properties:

(i) Let F_{M} : M(G)^{(p)} → M(G) and V_{M} : M(G) → M(G)^{(p)} be the 𝒪_{S}-linear maps induced respectively by the Frobenius and the Verschiebung of G. We have the following exact sequence:

⋯ → M(G)^{(p)} ⟶^{F_{M}} M(G) ⟶^{V_{M}} M(G)^{(p)} → · · · .

(ii) There is a connection ∇ : M(G) → M(G) ⊗_{𝒪_{S}} Ω_{S/𝔽_{p}}^{1} for which F_{M} and V_{M} are horizontal morphisms. (iii) We have two canonical filtrations on M(G) by 𝒪_{S}-modules locally free of finite type:

(3.1.1)  0 → ω_{G} → M(G) → Lie(G^{∨}) → 0,

called the Hodge filtration on M(G) [BBM, 3.3.5], and the conjugate filtration on M(G)

(3.1.2)  0 → Lie(G^{∨})^{(p)} ⟶^{ϕ_{G}} M(G) → ω_{G}^{(p)} → 0,

which is obtained by applying the Dieudonné functor to the exact sequence of finite group schemes 0 → Ker F_{G} → G(1) → Ker V_{G} → 0 [BBM, 4.3.1, 4.3.6, 4.3.11]. Moreover, we have the following commutative diagram (cf. [Ka1, 2.3.2 and 2.3.4]) (3.1.3)

[figure: 0; 0; 0; ω_{G}^{(p)}; ω_{G}; ψ_{G}; ω_{G}^{(p)}; M(G)^{(p)}; F_{M}; M(G); V_{M}; M(G)^{(p)}; ,; Lie(G^{∨})^{(p)}; ϕ_{G}; φ̃_{G}; Lie(G^{∨}); Lie(G^{∨})^{(p)}; 0; 0; 0]

where the columns are the Hodge filtrations and the anti-diagonal is the conjugate filtration. By functoriality, we see easily that φ̃_{G} above is nothing but the linearization of the Hasse-Witt map φ_{G} (2.6.1), and the morphism ψ_{G}^{∗} : Lie(G)^{(p)} → Lie(G), which is obtained by applying the functor ℋ om _{𝒪_{S}} (_, 𝒪_{S}) to ψ_{G}, is identified with the linearization φ_{G^{∨}} of φ_{G^{∨}}. The formation of these structures on M(G) commutes with arbitrary base changes of S. In the sequel, we will use (M(G), F_{M}, ∇) to emphasize these structures on M(G).

3.2. In the reminder of this section, k will denote an algebraically closed field of characteristic p > 0. Let S be a scheme formally smooth over k such that Ω_{S/𝔽_{p}}^{1} = Ω_{S/k}^{1} is an 𝒪_{S}-module locally free of finite type, e.g. S = Spec(A) with A a formally smooth k-algebra with a finite p-basis over k. Let G be a BT-group over S. We put KS to be the composed morphism

(3.2.1)  KS : ω_{G} → M(G) ⟶^{∇} M(G) ⊗_{𝒪_{S}} Ω_{S/k}^{1} ⟶^{pr} Lie(G^{∨}) ⊗_{𝒪_{S}} Ω_{S/k}^{1}

which is 𝒪_{S}-linear. We put 𝒯_{S/k} = ℋ om _{𝒪_{S}} (Ω_{S/k}^{1}, 𝒪_{S}), and define the Kodaira-Spencer map of G

(3.2.2)  Kod : 𝒯_{S/k} → ℋ om _{𝒪_{S}} (ω_{G}, Lie(G^{∨}))

to be the morphism induced by KS. We say that G is versal if Kod is surjective.

3.3. Let r be an integer ≥ 1, R = k[[t_{1}, ⋯, t_{r}]], 𝔪 be the maximal ideal of R. We put 𝒮 = Spf(R), S = Spec(R), and for each integer n ≥ 0, S_{n} = Spec(R/𝔪^{n+1}). By a BT-group 𝒢 over the formal scheme 𝒮, we mean a sequence of BT-groups (G_{n})_{n≥0} over (S_{n})_{n≥0} equipped with isomorphisms G_{n+1} ×_{S_{n+1}} S_{n} ≃ G_{n}.

According to [deJ, 2.4.4], the functor G ↦ (G×_{S} S_{n})_{n≥0} induces an equivalence of categories between the category of BT-groups over S and the category of BTgroups over 𝒮. For a BT-group 𝒢 over 𝒮, the corresponding BT-group G over S is called the algebraization of 𝒢. We say that 𝒢 is versal over 𝒮, if its algebraization G is versal over S. Since S is local, by Nakayama’s Lemma, 𝒢 or G is versal if and only if the reduction of Kod modulo the maximal ideal

(3.3.1)  Kod_{0} : 𝒯_{S/k} ⊗_{𝒪_{S}} k ⟶ Hom_{k}(ω_{G_{0}}, Lie(G_{0}^{∨}))

is surjective.

3.4. We recall briefly the deformation theory of a BT-group. Let 𝔄L_{k} be the category of local artinian k-algebras with residue field k. We notice that all morphisms of 𝔄L_{k} are local. A morphism A′ → A in 𝔄L_{k} is called a small extension, if it is surjective and its kernel I satisfies I · 𝔪_{A′} = 0, where 𝔪_{A′} is the maximal ideal of A′. Let G_{0} be a BT-group over k, and A an object of 𝔄L_{k}. A deformation of G_{0} over A is a pair (G, ϕ), where G is a BT-group over Spec(A) and ϕ is an isomorphism ϕ : G ⊗_{A} k ⟶^{∼} G_{0}. When there is no risk of confusions, we will denote a deformation (G, ϕ) simply by G. Two deformations (G, ϕ) and (G′, ϕ′) over A are isomorphic if there exists an isomorphism of BT-groups ψ : G ⟶^{∼} G′ over A such that ϕ = ϕ′ ∘ (ψ ⊗_{A} k). Let’s denote by D the functor which associates with each object A of 𝔄L_{k} the set of isomorphsm classes of deformations of G_{0} over A. If f : A → B is a morphism of 𝔄L_{k}, then the map D(f) : D(A) → D(B) is given by extension of scalars. We call D the deformation functor of G_{0} over 𝔄L_{k}.

Proposition 3.5 ([Ill], 4.8). Let G_{0} be a BT-group over k of dimension d and height c + d, D be the deformation functor of G_{0} over 𝔄L_{k}.

(i) Let A′ → A be a small extension in 𝔄L_{k} with ideal I, x = (G, ϕ) be an element in D(A), D_{x}(A′) be the subset of D(A′) with image x in D(A). Then the set D_{x}(A′) is a nonempty homogenous space under the group Hom_{k}(ω_{G_{0}}, Lie(G_{0}^{∨})) ⊗_{k} I. (ii) The functor D is pro-representable by a formally smooth formal scheme 𝒮 over k of relative dimension cd, i.e. 𝒮 = Spf(R) with R ≃ k[[(t_{ij})_{1≤i≤c,1≤j≤d}]], and there exists a unique deformation (𝒢 , ψ) of G_{0} over 𝒮 such that, for any object A of 𝔄L_{k} and any deformation (G, ϕ) of G_{0} over A, there is a unique homomorphism of local k-algebras φ : R → A with (G, ϕ) = D(φ)(𝒢 , ψ). (iii) Let 𝒯_{𝒮 /k} (0) = 𝒯_{𝒮 /k} ⊗_{𝒪_{𝒮}} k be the tangent space of 𝒮 at its unique closed point,

Kod_{0} : 𝒯_{𝒮 /k} (0) ⟶ Hom_{k}(ω_{G_{0}}, Lie(G_{0}^{∨}))

be the Kodaira-Spencer map of 𝒢 evaluated at the closed point of 𝒮. Then Kod_{0} is bijective, and it can be described as follows. For an element f ∈ 𝒯_{𝒮 /k} (0), i.e. a homomorphism of local k-algebras f : R → k[ϵ]/ϵ^{2}, Kod_{0}(f) is the difference of deformations

[𝒢 ⊗_{R} (k[ϵ]/ϵ^{2})] − [G_{0} ⊗_{k} (k[ϵ]/ϵ^{2})],

which is a well-defined element in Hom_{k}(ω_{G_{0}}, Lie(G_{0}^{∨})) by (i).

Remark 3.6. Let (e_{j})_{1≤j≤d} be a basis of ω_{G_{0}}, (f_{i})_{1≤i≤c} be a basis of Lie(G_{0}^{∨}). In view of 3.5(iii), we can choose a system of parameters (t_{ij})_{1≤i≤c,1≤j≤d} of 𝒮 such that

Kod_{0}(\frac{∂}{∂t_{ij}}) = e_{j}^{∗} ⊗ f_{i},

where (e_{j}^{∗})_{1≤j≤d} is the dual basis of (e_{j})_{1≤j≤d}. Moreover, if 𝔪 is the maximal ideal of R, the parameters t_{ij} are determined uniquely modulo 𝔪^{2}.

Corollary 3.7 (Algebraization of the universal deformation). The assumptions being those of (3.5), we put moreover S = Spec(R) and G the algebraization of the universal formal deformation 𝒢. Then the BT-group G is versal over S, and satisfies the following universal property: Let A be a noetherian complete local k-algebra with residue field k, G be a BT-group over A endowed with an isomorphism G ⊗_{A} k ≃ G_{0}. Then there exists a unique continuous homomorphism of local k-algebras φ : R → A such that G ≃ G⊗_{R}A.

Proof. By the last remark of 3.3, G is clearly versal. It remains to prove that it satisfies the universal property in the corollary. Let G be a deformation of G_{0} over a noetherian complete local k-algebra A with residue field k. We denote by 𝔪_{A} the maximal ideal of A, and put A_{n} = A/𝔪_{A}^{n+1} for each integer n ≥ 0. Then by 3.5(b), there exists a unique local homomorphism φ_{n} : R → A_{n} such that G ⊗ A_{n} ≃ G ⊗_{R} A_{n}. The φ_{n}’s form a projective system (φ_{n})_{n≥0}, whose projective limit φ : R → A answers the question. □

Definition 3.8. The notations are those of (3.7). We call S the local moduli in characteristic p of G_{0}, and G the universal deformation of G_{0} in characteristic p.

If there is no confusions, we will omit “in characteristic p” for short.

3.9. Let G be a BT-group over k, G^{∘} be its connected part, and G^{ét} be its étale part. Let r be the height of G^{ét}. Then we have G^{ét} ≃ (ℚ_{p}/ℤ_{p})^{r}, since k is algebraically closed. Let D_{G} (resp. D_{G^{∘}}) be the deformation functor of G (resp. G^{∘}) over 𝔄L_{k}. If A is an object in 𝔄L_{k} and 𝒢 is a deformation of G (resp. G^{∘}) over A, we denote by [𝒢] its isomorphism class in D_{G}(A) (resp. in D_{G^{∘}(A)}).

Proposition 3.10. The assumptions are as above, let Θ : D_{G} → D_{G^{∘}} be the morphism of functors that maps a deformation of G to its connected component.

(i) The morphism Θ is formally smooth of relative dimension r. (ii) Let A be an object of 𝔄L_{k}, and 𝒢^{∘} be a deformation of G^{∘} over A. Then the subset Θ_{A}^{−1}([𝒢^{∘}]) of D_{G}(A) is canonically identified with Ext_{A}^{1}(ℚ_{p}/ℤ_{p}, 𝒢^{∘})^{r}, where Ext_{A}^{1} means the group of extensions in the category of abelian fppfsheaves on Spec(A).

Proof. (i) Since D_{G} and D_{G^{∘}} are both pro-representable by a noetherian local complete k-algebra and formally smooth over k (3.5), by a formal completion version of [EGA, IV 17.11.1(d)], we only need to check that the tangent map

Θ_{k[ϵ]/ϵ^{2}} : D_{G}(k[ϵ]/ϵ^{2}) → D_{G^{∘}}(k[ϵ]/ϵ^{2}) is surjective with kernel of dimension r over k. By 3.5(iii), D_{G}(k[ϵ]/ϵ^{2}) (resp. D_{G^{∘}}(k[ϵ]/ϵ^{2})) is isomorphic to Hom_{k}(ω_{G}, Lie(G^{∨})) (resp. Hom_{k}(ω_{G^{∘}}, Lie(G^{∘∨}))) by the Kodaira-Spencer morphism. In view of the canonical isomorphism ω_{G} ≃ ω_{G^{∘}}, Θ_{k[ϵ]/ϵ^{2}} corresponds to the map

Θ′_{k[ϵ]/ϵ^{2}} : Hom_{k}(ω_{G}, Lie(G^{∨})) → Hom_{k}(ω_{G}, Lie(G^{∘∨}))

induced by the canonical surjection Lie(G^{∨}) → Lie(G^{∘∨}). It is clear that Θ′_{k[ϵ]/ϵ^{2}} is surjective of kernel Hom_{k}(ω_{G}, Lie(G^{ét∨})), which has dimension r over k. (ii) Since G^{ét} is isomorphic to (ℚ_{p}/ℤ_{p})^{r}, every element in Ext_{A}^{1}(ℚ_{p}/ℤ_{p}, 𝒢^{∘})^{r} defines clearly an element of D_{G}(A) with image [𝒢^{∘}] in D_{G^{∘}}(A). Conversely, for any 𝒢 ∈ D_{G}(A) with connected component isomorphic to 𝒢^{∘}, the isomorphism G^{ét} ≃ (ℚ_{p}/ℤ_{p})^{r} lifts uniquely to an isomorphism 𝒢^{ét} ≃ (ℚ_{p}/ℤ_{p})^{r} because A is henselian. The canonical exact sequence 0 → 𝒢^{∘} → 𝒢 → 𝒢^{ét} → 0 shows that 𝒢 comes from an element of Ext_{A}^{1}(ℚ_{p}/ℤ_{p}, 𝒢^{∘})^{r}.

□

4. HW-cyclic Barsotti-Tate Groups

Definition 4.1. Let S be a scheme of characteristic p > 0, G be a BT-group over S such that c = dim(G^{∨}) is constant. We say that G is HW-cyclic, if c ≥ 1 and there exists an element v ∈ Γ(S, Lie(G^{∨})) such that

v, φ_{G}(v), ⋯, φ_{G}^{c−1}(v)

generate Lie(G^{∨}) as an 𝒪_{S}-module, where φ_{G} is the Hasse-Witt map (2.6.1) of G.

Remark 4.2. It is clear that a BT-group G over S is HW-cyclic, if and only if Lie(G^{∨}) is free over 𝒪_{S} and there exists a basis of Lie(G^{∨}) over 𝒪_{S} under which φ_{G} is expressed by a matrix of the form

(4.2.1)  (\begin{matrix}0 & 0 & ⋯ & 0 & −a_{1} \\ 1 & 0 & ⋯ & 0 & −a_{2} \\ 0 & 1 & ⋯ & 0 & −a_{3} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & −a_{c} \end{matrix}) ,

where a_{i} ∈ Γ(S, 𝒪_{S}) for 1 ≤ i ≤ c.

Lemma 4.3. Let R be a local ring of characteristic p > 0, k be its residue field.

(i) A BT-group G over R is HW-cyclic if and only if so is G ⊗ k. (ii) Let 0 → G′ → G → G′′ → 0 be an exact sequence of BT-groups over R. If G is HW-cyclic, then so is G′. In particular, if R is henselian, the connected part of a HW-cyclic BT-group over R is HW-cyclic.

Proof. (i) The property of being HW-cyclic is clearly stable under arbitrary base changes, so the “only if” part is clear. Assume that G_{0} = G ⊗ k is HW-cyclic. Let \overline{v} be an element of Lie(G_{0}^{∨}) = Lie(G^{∨}) ⊗ k such that (\overline{v}, φ_{G_{0}}(\overline{v}), ⋯, φ_{G_{0}}^{c−1}(\overline{v})) is a basis of Lie(G_{0}^{∨}). Let v be any lift of \overline{v} in Lie(G^{∨}). Then by Nakayama’s lemma, (v, φ_{G}(v), ⋯, φ_{G}^{c−1}(v)) is a basis of Lie(G^{∨}). (ii) By statement (i), we may assume R = k. The exact sequence of BT-groups induces an exact sequence of Lie algebras

(4.3.1)  0 → Lie(G^{′′∨}) → Lie(G^{∨}) → Lie(G^{′∨}) → 0,

and the Hasse-Witt map φ_{G′} is induced by φ_{G} by functoriality. Assume that G is HW-cyclic and G^{∨} has dimension c. Let u be an element of Lie(G^{∨}) such that

u, φ_{G}(u), ⋯, φ_{G}^{c−1}(u)

form a basis of Lie(G^{∨}) over k. We denote by u′ the image of u in Lie(G^{′∨}). Let r ≤ c be the maximal integer such that the vectors

u′, φ_{G′}(u′), ⋯, φ_{G′}^{r−1}(u′)

are linearly independent over k. It is easy to see that they form a basis of the k-vector space Lie(G^{′∨}). Hence G′ is HW-cyclic. □

Lemma 4.4. Let S = Spec(R) be an affine scheme of characteristic p > 0, G be a HW-cyclic BT-group over R with c = dim(G^{∨}) constant, and

(\begin{matrix}0 & 0 & ⋯ & 0 & −a_{1} \\ 1 & 0 & ⋯ & 0 & −a_{2} \\ 0 & 1 & ⋯ & 0 & −a_{3} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & −a_{c} \end{matrix}) ∈ M_{c×c}(R),

be a matrix of φ_{G}. Put a_{c+1} = 1, and P (X) = ∑_{i=0}^{c} a_{i+1}X^{p^{i}} ∈ R[X].

(i) Let V_{G} : G^{(p)} → G be the Verschiebung homomorphism of G. Then Ker V_{G} is isomorphic to the group scheme Spec(R[X]/P (X)) with comultiplication given by X ↦ 1 ⊗ X + X ⊗ 1. (ii) Let x ∈ S, and G_{x} be the fibre of G at x. Put

(4.4.1)  i_{0}(x) = min_{0≤i≤c}{i; a_{i+1}(x) ≠ 0},

where a_{i}(x) denotes the image of a_{i} in the residue field of x. Then the étale part of G_{x} has height c − i_{0}(x), and the connected part of G_{x} has height d + i_{0}(x). In particular, G_{x} is connected if and only if a_{i}(x) = 0 for 1 ≤ i ≤ c.

Proof. (i) By 2.3 and 2.13, Ker V_{G} is isomorphic to the group scheme

Spec(R[X_{1}, …, X_{c}]/(X_{1}^{p} − X_{2}, ⋯, X_{c−1}^{p} − X_{c}, X_{c}^{p} + a_{1}X_{1} + ⋯ + a_{c}X_{c}))

with comultiplication Δ(X_{i}) = 1 ⊗ X_{i} + X_{i} ⊗ 1 for 1 ≤ i ≤ c. By sending (X_{1}, X_{2}, ⋯, X_{c}) ↦ (X, X^{p}, ⋯, X^{p^{c−1}}), we see that the above group scheme is isomorphic to Spec(R[X]/P (X)) with comultiplication Δ(X) = 1⊗X+X⊗1.

(ii) By base change, we may assume that S = x = Spec(k) and hence G = G_{x}. Let G(1) be the kernel of the multiplication by p on G. Then we have an exact sequence

0 → Ker F_{G} → G(1) → Ker V_{G} → 0.

Since Ker F_{G} is an infinitesimal group scheme over k, we have G(1)(\overline{k}) = (Ker V_{G})(\overline{k}), where \overline{k} is an algebraic closure of k. By the definition of i_{0}(x), we have P (X) = Q(X^{p^{i_{0}(x)}}), where Q(X) is an additive sepearable polynomial in k[X] with deg(Q) = p^{c−i_{0}(x)}. Hence the roots of P (X) in \overline{k} form an 𝔽_{p}-vector space of dimension c − i_{0}(x). By (i), (Ker V_{G})(\overline{k}) can be identified with the additive group consisting of the roots of P (X) in \overline{k}. Therefore, the étale part of G has height c − i_{0}(x), and the connected part of G has height d + i_{0}(x). □

4.5. Let k be a perfect field of characteristic p > 0, and α_{p} = Spec(k[X]/X^{p}) be the finite group scheme over k with comultiplication map Δ(X) = 1⊗X+X⊗1. Let G be a BT-group over k. Following Oort, we call

a(G) = dim_{k} Hom_{k_{fppf}} (α_{p}, G)

the a-number of G, where Hom_{k_{fppf}} means the homomorphisms in the category of abelian fppf-sheaves over k. Since the Frobenius of α_{p} vanishes, any morphism of α_{p} in G factorize through Ker(F_{G}). Therefore we have

Hom_{k_{fppf}} (α_{p}, G) = Hom_{k−gr}(α_{p}, Ker(F_{G}))
= Hom_{k−gr}(Ker(F_{G})^{∨}, α_{p})
= Hom_{p-𝔏ie_{k}}(Lie(α_{p}), Lie(Ker(F_{G}))),

where Hom_{k−gr} denotes the homomorphisms in the category of commutative group schemes over k, and the last equality uses Proposition 2.3. Since we have a canonical isomorphism Lie(Ker(F_{G})) ≃ Lie(G) and Lie(α_{p}) has dimension one over k with φ_{α_{p}} = 0, we get

(4.5.1)  a(G) = dim_{k}{x ∈ Lie(G)|φ_{G^{∨}}(x) = 0} = dim_{k} Ker(φ_{G^{∨}}).

Due to the perfectness of k, we have also a(G) = dim_{k} Ker(φ_{G^{∨}}), where φ_{G^{∨}} is the linearization of φ_{G^{∨}}. By Proposition 2.11, we see that a(G) = 0 if and only if G is ordinary.

Lemma 4.6. Let G be a BT-group over k, and G^{∨} its Serre dual. Then we have a(G) = a(G^{∨}).

Proof. Let ψ_{G} : ω_{G} → ω_{G}^{(p)} be the k-linear map induced by the Verschiebung of G. Then ψ_{G}^{∗}, the morphism obtained by applying the functor Hom_{k}(_, k) to ψ_{G}, is identified with φ_{G^{∨}}. By (4.5.1) and the exactitude of the functor Hom_{k}(_, k), we have a(G) = dim_{k} Ker(ψ_{G}^{∗}) = dim_{k} Coker(ψ_{G}). Using the additivity of dim_{k}, we get finally a(G) = dim_{k} Ker(ψ_{G}). By considering the commutative diagram (3.1.3), we have

a(G) = dim_{k}(ω_{G} ∩ ϕ_{G}(Lie(G^{∨})^{(p)})).

On the other hand, it follows also from (3.1.3) that

a(G^{∨}) = dim_{k} Ker(φ̃_{G}) = dim_{k}(ϕ_{G}(Lie(G^{∨})^{(p)}) ∩ ω_{G}).

The lemma now follows immediately.

□

Proposition 4.7. Let k be a perfect field of characteristic p > 0, G a BT-group over k. Consider the following conditions:

(i) G is HW-cyclic and non-ordinary; (ii) the connected part G^{∘} of G is HW-cyclic and not of multiplicative type; (iii) a(G^{∨}) = a(G) = 1. We have (i) ⇒ (ii) ⇔ (iii). If k is algebraically closed, we have moreover (ii) ⇒ (i).

Remark 4.8. In [Oo1, Lemma 2.2], Oort proved the following assertion, which is a generalization of (iii) ⇒ (ii): Let k be an algebraically closed field of characteristic p > 0, and G be a connected BT-group with a(G) = 1. Then there exists a basis of the Dieudonné module M of G over W (k), such that the action of Frobenius on M is given by a display-matrix of “normal form” in the sense of [Oo1, 2.1].

Proof. (i) ⇒ (ii) follows from 4.3(ii). (ii) ⇒ (iii). First, we note that a(G) = a(G^{∘}), so we may assume G connected. Since G is not of multiplicative type, we have c = dim(G^{∨}) ≥ 1. By Lemma 4.4(ii), there exists a basis of Lie(G^{∨}) over k under which φ_{G} is expressed by

(\begin{matrix}0 & 0 & ⋯ & 0 & 0 \\ 1 & 0 & ⋯ & 0 & 0 \\ 0 & 1 & ⋯ & 0 & 0 \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & 0 \end{matrix}) ∈ M_{c×c}(k).

According to (4.5.1), a(G^{∨}) equals to dim_{k} Ker(φ_{G}), i.e. the k-dimension of the solutions of the equation system in (x_{1}, ⋯, x_{c})

(\begin{matrix}0 & 0 & ⋯ & 0 & 0 \\ 1 & 0 & ⋯ & 0 & 0 \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & 0 \end{matrix}) (x_{1}^{p} ; x_{2}^{p} ; ⋮ ; x_{c}^{p}) = 0

The solutions (x_{1}, ⋯, x_{c}) form clearly a vector space over k of dimension 1, i.e. we have a(G^{∨}) = 1. (iii) ⇒ (ii). Let G^{ét} be the étale part of G. Since k is perfect, the exact sequence (2.7.1) splits [Dem, Chap. II §7]; so we have G ≃ G^{∘} × G^{ét}. We put M = Lie(G^{∨}), M_{1} = Lie(G^{∘∨}) and M_{2} = Lie(G^{ét∨}) for short. By 2.8 and 2.9, we have a decomposition M = M_{1} ⊕ M_{2}, such that M_{1}, M_{2} are stable under φ_{G}, and the action of φ_{G} is nilpotent on M_{1} and bijective on M_{2}. We note that a(G^{∘∨}) = a(G^{∘}) = a(G) = 1. By the last remark of 4.5, G^{∘} is not of multiplicative type, hence dim_{k} M_{1} = dim(G^{∘∨}) ≥ 1. It remains to prove that G^{∘} is HW-cyclic. Let n be the minimal integer such that φ_{G}^{n}(M_{1}) = 0. We have a strictly increasing filtration

0 ⊊ Ker(φ_{G}) ⊊ ⋯ ⊊ Ker(φ_{G}^{n}) = M_{1}.

If n = 1, then M_{1} is one-dimensional, hence G^{∘} is clearly HW-cyclic. Assume n ≥ 2. For 2 ≤ m ≤ n, φ_{G}^{m−1} induces an injective map

\overline{φ_{G}^{m−1}} : Ker(φ_{G}^{m})/ Ker(φ_{G}^{m−1}) ⟶ Ker(φ_{G}).

Since dim_{k} Ker(φ_{G}) = a(G^{∘∨}) = 1, \overline{φ_{G}^{m−1}} is necessarily bijective. So we have dim_{k} Ker(φ_{G}^{m}) = m for 1 ≤ m ≤ n. Let v be an element of M_{1} but not in Ker(φ_{G}^{n−1}). Then v, φ_{G}(v), ⋯, φ_{G}^{n−1}(v) are linearly independant, hence they form a basis of M_{1} over k. This proves that G^{∘} is HW-cyclic. Assume k algebraically closed. We prove that (ii) ⇒ (i). Noting that G is ordinary if and only if G^{∘} is of multiplicative type, we only need to check that G is HW-cyclic. We conserve the notations above. Since φ_{G} is bijective on M_{2} and k algebraically closed, there exists a basis (e_{1}, ⋯, e_{m}) of M_{2} such that φ_{G}(e_{i}) = e_{i} for 1 ≤ i ≤ m. Let v ∈ M_{1} but not in Ker(φ_{G}^{n−1}) as above, and put u = v + λ_{1}e_{1} + ⋯ λ_{m}e_{m}, where λ_{i}(1 ≤ i ≤ m) are some elements in k to be determined later. Then we have

( φ_{G}^{n}(u) ; ⋮ ; φ_{G}^{n+m−1}(u)) = (\begin{matrix}λ_{1}^{p^{n}} & ⋯ & λ_{m}^{p^{n}} \\ ⋮ & ... & ⋮ \\ λ_{1}^{p^{n+m−1}} & ⋯ & λ_{m}^{p^{n+m−1}} \end{matrix} ) ( e_{1} ; ⋮ ; e_{m}) .

Let L(λ_{1}, ⋯, λ_{m}) ∈ k[λ_{1}, ⋯, λ_{m}] be the determinant polynomial of the matrix on the right side. An elementary computation shows that the polynomial L(λ_{1}, ⋯, λ_{m}) is not null. We can choose λ_{1}, ⋯, λ_{m} ∈ k such that L(λ_{1}, ⋯, λ_{m}) ≠ 0 because k is algebraically closed. So φ_{G}^{n}(u), ⋯, φ_{G}^{n+m−1}(u) form a basis of M_{2} over k. Since

φ_{G}^{i}(u) ≡ φ_{G}^{i}(v) mod M_{2} for 0 ≤ i ≤ n,

by the choice of u, we see that {u, φ_{G}(u), ⋯, φ_{G}^{n+m−1}(u)} form a basis of M = Lie(G^{∨}) over k. □

By combining 4.6 and 4.7, we obtain the following

Corollary 4.9. Let k be an algebraically closed field of characteristic p > 0. Then a BT-group over k is HW-cyclic if and only if so is its Serre dual.

4.10. Examples. Let k be a perfect field, W (k) be the ring of Witt vectors with coefficients in k, and σ be the Frobenius automorphism of W (k). Let s, r be relatively prime integers such that 0 ≤ s ≤ r and r ≠ 0; put λ = \frac{s}{r}. We consider the Dieudonné module M^{λ} ≃ W (k)[F, V]/(F^{r−s} − V^{s}), where W (k)[F, V] is the non-commutative ring with relations F V = V F = p, F a = σ(a)F and V σ(a) = aV for all a ∈ W (k). We note that M^{λ} is free of rank r over W (k) and M^{λ}/V M^{λ} ≃ k[F]/F^{r−s}. By the contravariant Dieudonné theory, M^{λ} corresponds to a BT-group G^{λ} over k of height r with Lie(G^{λ∨}) = M^{λ}/V M^{λ}. We see easily that G^{λ} is HW-cyclic, and we call it the elementary BT-group of slope λ. We note that G^{0} ≃ ℚ_{p}/ℤ_{p}, G^{1} ≃ μ_{p^{∞}}, and (G^{λ})^{∨} ≃ G^{1−λ} for 0 ≤ λ ≤ 1. Assume k algebraically closed. Then by the Dieudonné-Manin’s classification of isocrystals [Dem, Chap.IV §4], any BT-group over k is isogenous to a finite product of G^{λ}’s; moreover, any connected one-dimensional BT-group over k of height r is necessarily isomorphic to G^{1/r} [Dem, Chap.IV §8], hence in particular HW-cyclic.

Proposition 4.11. Let k be an algebraically closed field of characteristic p > 0, R be a noetherian complete regular local k-algebra with residue field k, and S = Spec(R). Let G be a connected HW-cyclic BT-group over R of dimension d ≥ 1 and height c + d,

𝔥 = (\begin{matrix}0 & 0 & ⋯ & 0 & −a_{1} \\ 1 & 0 & ⋯ & 0 & −a_{2} \\ 0 & 1 & ⋯ & 0 & −a_{3} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & −a_{c} \end{matrix}) ∈ M_{c×c}(R)

be a matrix of φ_{G}.

(i) If G is versal over S, then {a_{1}, ⋯, a_{c}} is a subset of a regular system of parameters of R. (ii) Assume that d = 1. The converse of (i) is also true, i.e. if {a_{1}, ⋯, a_{c}} is a subset of a regular system of parameters of R then G is versal over S. Furthermore, G is the universal deformation of its special fiber if and only if {a_{1}, ⋯, a_{c}} is a system of regular parameters of R.

Proof. Let (M(G), F_{M}, ∇) be the finite free 𝒪_{S}-module equipped with a semilinear endomorphism F_{M} and a connection ∇ : M(G) → M(G) ⊗_{𝒪_{S}} Ω_{S/k}^{1}, obtained by evaluating the Dieudonné crystal of G at the trivial immersion S ↪ S (cf. 3.1). Recall that we have a commutative diagram

(4.11.1)  \xymatrix{M(G)^{(p)} \ar[r]^{F_{M}} \ar[d]_{pr} & M(G) \ar[d]^{pr} \\ Lie(G^{∨})^{(p)} \ar[r]^{φ̃_{G}} \ar@{^{(}->}[ur]^{ϕ_{G}} & Lie(G^{∨}),}

where ϕ_{G} is universally injective (3.1.3). Let {v_{1}, ⋯, v_{c}} be a basis of Lie(G^{∨}) over 𝒪_{S} under which φ_{G} is expressed by 𝔥, i.e. we have φ_{G}^{i−1}(v_{1}) = v_{i} for 1 ≤ i ≤ c and φ_{G}^{c}(v_{1}) = φ_{G}(v_{c}) = − ∑_{i=1}^{c} a_{i}v_{i}. Let f_{1} be a lift of v_{1} to Γ(S, M(G)), and put f_{i+1} = ϕ_{G}(v_{i}^{(p)}) for 1 ≤ i ≤ c − 1, where v_{i}^{(p)} = 1 ⊗ v_{i} ∈ Γ(S, Lie(G^{∨})^{(p)}). The image of f_{i} in Γ(S, Lie(G^{∨})) is thus v_{i} for 1 ≤ i ≤ c by (4.11.1). We put

(4.11.2)  e_{1} = ϕ_{G}(v_{c}^{(p)}) + a_{1}f_{1} + ⋯ + a_{c}f_{c} ∈ Γ(S, M(G)).

The image of e_{1} in Γ(S, Lie(G^{∨})) is φ_{G}(v_{c}) + ∑_{i=1}^{c} a_{i}v_{i} = 0; so we have e_{1} ∈ Γ(S, ω_{G}). By 4.4(ii), we notice that a_{1}, ⋯, a_{c} belong to the maximal ideal

𝔪_{R} of R, as G is connected. Hence, we have \overline{e_{1}} = \overline{ϕ_{G}(v_{c}^{(p)})}, where for a Rmodule M and x ∈ M, we denote by \overline{x} the canonical image of x in M ⊗ k. Since ϕ_{G} commutes with base change and is universally injective, we get \overline{e_{1}} = \overline{ϕ_{G}(v_{c}^{(p)})} = ϕ_{G⊗k}(\overline{v_{c}^{(p)}}) ≠ 0. Therefore, we can choose e_{2}, ⋯, e_{d} ∈ Γ(S, ω_{G}) such that (e_{1}, ⋯, e_{d}) becomes a basis of ω_{G} over 𝒪_{S}, so (e_{1}, ⋯, e_{d}, f_{1}, ⋯, f_{c}) is a basis of M(G). Since F_{M} is horizontal for the connection ∇ (cf. 3.1(ii)), we have

∇(ϕ_{G}(v_{c}^{(p)})) = ∇(F_{M}(f_{c}^{(p)})) = 0.

In view of (4.11.2), we get

∇(e_{1}) = ∑_{i=1}^{c} f_{i} ⊗ da_{i} + ∑_{i=1}^{c} a_{i}∇(f_{i})

≡ ∑_{i=1}^{c} f_{i} ⊗ da_{i} (mod 𝔪_{R}). (4.11.3)

Let KS_{0} and Kod_{0} be respectively the reductions modulo 𝔪_{R} of (3.2.1) and (3.2.2). Since (\overline{v_{i}})_{1≤i≤c} is a base of Lie(G^{∨}) ⊗ k, we can write

KS_{0}(e_{j}) = ∑_{i=1}^{c} \overline{v_{i}} ⊗ θ_{i,j} for 1 ≤ j ≤ d,

where θ_{i,j} ∈ Ω_{S/k} ⊗ k. From (4.11.3), we deduce that θ_{i,1} = da_{i}. By the definition of Kod_{0}, we have

(4.11.4)  Kod_{0}(∂) = ∑_{j=1}^{d} ∑_{i=1}^{c} < ∂, θ_{i,j} > \overline{e_{j}}^{∗} ⊗ \overline{v_{i}}

where ∂ ∈ 𝒯_{S/k} ⊗ k, < •, • > is the canonical pairing between 𝒯_{S/k} ⊗ k and Ω_{S/k}^{1} ⊗ k, and (\overline{e_{i}}^{∗})_{1≤i≤d} denotes the dual basis of (\overline{e_{i}})_{1≤i≤d}. Now assume that G is versal over S, i.e. Kod_{0} is surjective by definition (3.2). In particular, there are ∂_{1}, ⋯, ∂_{c} ∈ 𝒯_{S/k} ⊗ k such that Kod_{0}(∂_{i}) = \overline{e_{1}}^{∗} ⊗ v_{i} for 1 ≤ i ≤ c, i.e. we have

(4.11.5)  < ∂_{i}, da_{j} >= \begin{cases}1 & if i = j \\ 0 & if i ≠ j \end{cases} for 1 ≤ i, j ≤ c,

and

< ∂_{i}, θ_{j,ℓ} >= 0 for 1 ≤ i, j ≤ c, 2 ≤ ℓ ≤ d.

From (4.11.5), we see easily that da_{1}, ⋯, da_{c} are linearly independent in Ω_{S/k}⊗ k ≃ 𝔪_{R}/𝔪_{R}^{2}; therefore, (a_{1}, ⋯, a_{c}) is a part of a regular system of parameters of R. Statement (i) is proved.

For statement (ii), we assume d = 1 and that (a_{1}, ⋯, a_{c}) is a part of a regular system of parameters of R. Then the formula (4.11.4) is simplified as

Kod_{0}(∂) = ∑_{i=1}^{c} < ∂, da_{i} > \overline{e_{1}}^{∗} ⊗ \overline{v_{i}}.

Since da_{1}, ⋯, da_{c} are linearly independent in Ω_{S/k}^{1} ⊗k, there exist ∂_{1}, ⋯, ∂_{c} ∈ 𝒯_{S/k} ⊗k such that (4.11.5) holds, i.e. (\overline{e_{1}}^{∗} ⊗\overline{v_{i}})_{1≤i≤c} are in the image of Kod_{0}. But the elements (\overline{e_{1}}^{∗}⊗\overline{v_{i}})_{1≤i≤c} form already a basis of ℋ om _{𝒪_{S}} (ω_{G}, Lie(G^{∨}))⊗ k. So Kod_{0} is surjective, and hence G is versal over S by Nakayama’s lemma. Let G_{0} be the special fiber of G. It remains to prove that when d = 1, G is the universal deformation of G_{0} if and only if dim(S) = c and G is versal over S. Let S be the local moduli in characteristic p of G_{0}. By the universal property of G (3.7), there exists a unique morphism f : S → S such that G ≃ G ×_{S} S. Since S and S are local complete regular schemes over k with residue field k of the same dimension, f is an isomorphism if and only if the tangent map of f at the closed point of S, denoted by T_{f}, is an isomorphism. By the functoriality of Kodaira-Spencer maps (3.2.2), we have a commutative diagram

[figure: 𝒯_{S/k} ⊗_{𝒪_{S}} k; T_{f}; ^{Kod{S \atop 0}}; Hom_{k}(ω_{G_{0}}, Lie(G_{0}^{∨})); 𝒯_{S/k} ⊗_{𝒪_{S}} k; _{Kod{S \atop 0}}; Hom_{k}(ω_{G_{0}}, Lie(G_{0}^{∨}))] ,

where horizontal arrows are the Kodaira-Spencer maps evaluated at the closed points (3.3.1). Since Kod_{0}^{S} and Kod_{0}^{S} are isomorphisms according to the first part of this propostion, we deduce that so is T_{f}. This completes the proof. □

5. Monodromy of a HW-cyclic BT-group over a Complete Trait of Characteristic p > 0

5.1. Let k be an algebraically closed field of characteristic p > 0, A be a complete discrete valuation ring of characteristic p, with residue field k and fraction field K. We put S = Spec(A), and denote by s its closed point, by η its generic point. Let \overline{K} be an algebraic closure of K, K^{sep} be the maximal separable extension of K contained in \overline{K}, K^{t} be the maximal tamely ramified extension of K contained in K^{sep}. We put I = Gal(K^{sep}/K), I_{p} = Gal(K^{sep}/K^{t}) and I_{t} = I/I_{p} = Gal(K^{t}/K). Let π be a uniformizer of A; so we have A ≃ k[[π]]. Let v be the valuation on K normalized by v(π) = 1; we denote also by v the unique extension of v to \overline{K}. For every α ∈ ℚ, we denote by 𝔪_{α} (resp. by 𝔪_{α}^{+}) the set of elements x ∈ K^{sep} such that v(x) ≥ α (resp. v(x) > α). We put

(5.1.1)  V_{α} = 𝔪_{α}/𝔪_{α}^{+},

which is a k-vector space of dimension 1 equipped with a continuous action of the Galois group I.

5.2. First, we recall some properties of the inertia groups I_{p} and I_{t} [Se1, Chap. IV]. The subgroup I_{p}, called the wild inertia subgroup, is the unique maximal pro-p-group contained in I and hence normal in I. The quotient I_{t} = I/I_{p} is a commutative profinite group, called the tame inertia group. We have a canonical isomorphism

(5.2.1)  θ : I_{t} ⟶^{∼} \underleftarrow{lim}_{(d,p)=1} μ_{d},

where the projective system is taken over positive integers prime to p, μ_{d} is the group of d-th roots of unity in k, and the transition maps μ_{m} → μ_{d} are given by ζ ↦ ζ^{m/d}, whenever d divides m. We denote by θ_{d} : I_{t} → μ_{d} the projection induced by (5.2.1). Let q be a power of p, 𝔽_{q} be the finite subfield of k with q elements. Then μ_{q−1} = 𝔽_{q}^{×}, and we can write θ_{q−1} : I_{t} → 𝔽_{q}^{×}. The character θ_{d} is characterized by the following property.

Proposition 5.3 ([Se3] Prop.7). Let a, d be relatively prime positive integers with d prime to p. Then the natural action of I_{p} on the k-vector space V_{a/d} (5.1.1) is trivial, and the induced action of I_{t} on V_{a/d} is given by the character (θ_{d})^{a} : I_{t} → μ_{d}. In particular, if q is a power of p, the action of I_{t} on V_{1/(q−1)} is given by the character θ_{q−1} : I_{t} → 𝔽_{q}^{×} and any I-equivariant 𝔽_{p}-subspace of V_{1/(q−1)} is an 𝔽_{q}-vector space.

5.4. Let G be a BT-group over S. We define h(G) to be the valuation of the determinant of a matrix of φ_{G} if dim(G^{∨}) ≥ 1, and h(G) = 0 if dim(G^{∨}) = 0. We call h(G) the Hasse invariant of G.

(a) h(G) does not depend on the choice of the matrix representing φ_{G}. Indeed, let c be the rank of Lie(G^{∨}) over A, 𝔥 ∈ M_{c×c}(A) be a matrix of φ_{G}. Any other matrix representing φ_{G} can be written in the form U^{−1} · 𝔥 · U^{(p)}, where U ∈ GL_{c}(A), U^{−1} is the inverse of U, and U^{(p)} is the matrix obtained by applying the Frobenius map of A to the coefficients of U.

(b) By 2.11, the generic fiber G_{η} is ordinary if and only if h(G) < ∞; G is ordinary over T if and only h(G) = 0.

(c) Let 0 → G′ → G → G′′ → 0 be a short exact sequence of BT-groups over T, then we have h(G) = h(G′) + h(G′′). Indeed, the exact sequence of BT-groups induces a short exact sequence of Lie algebras (cf. [BBM] 3.3.2)

0 → Lie(G^{′′∨}) → Lie(G^{∨}) → Lie(G^{′∨}) → 0,

from which our assertion follows easily.

Proposition 5.5. Let G be a BT-group over S. Then we have h(G) = h(G^{∨}).

Proof. The proof is very similar to that of Lemma 4.6. First, we have

h(G) = leng(Lie(G^{∨})/φ̃_{G}(Lie(G^{∨})^{(p)})),

where φ̃_{G} is the linearization of φ_{G}, and “leng” means the length of a finite A-module (note that this formulae holds even if dim(G^{∨}) = 0). By the commutative diagram (3.1.3), we have

h(G) = leng M(G)/(ϕ_{G}(Lie(G^{∨})^{(p)}) + ω_{G}).

On the other hand, by applying the functor Hom_{A}(_, A) to the A-linear map φ_{G^{∨}} : Lie(G)^{(p)} → Lie(G), we obtain a map ψ_{G} : ω_{G} → ω_{G}^{(p)}. If U is a matrix of φ_{G^{∨}}, then the transpose of U, denoted by U^{t}, is a matrix of ψ_{G}. So we have

h(G^{∨}) = v(det(U)) = v(det(U^{t})) = leng(ω_{G}^{(p)}/ψ_{G}(ω_{G})).

By diagram 3.1.3, we get

h(G^{∨}) = leng M(G)/(ϕ_{G}(Lie(G^{∨})^{(p)}) + ω_{G}) = h(G).
□

5.6. Let G be a BT-group over S, c = dim(G^{∨}). We put

(5.6.1)  T_{p}(G) = lim_{⟵_{n}} G(n)(\overline{K})

the Tate module of G, where G(n) is the kernel of p^{n} : G → G. It is a free ℤ_{p}-module of rank ≤ c, and the equality holds if and only if the generic fiber G_{η} is ordinary. The Galois group I acts continuously on T_{p}(G). We are interested in the image of the monodromy representation

(5.6.2)  ρ : I = Gal(K^{sep}/K) → Aut_{ℤ_{p}}(T_{p}(G)).

We denote by

(5.6.3)  \overline{ρ} : I = Gal(K^{sep}/K) → Aut_{𝔽_{p}}(G(1)(\overline{K}))

its reduction mod p.

Theorem 5.7 (Reformulation of Igusa’s theorem). Let G be a connected BTgroup over S of height 2 and dimension 1. Then G is versal (3.2) if and only if h(G) = 1; moreover, if this condition is satisfied, the monodromy representation ρ : I → Aut_{ℤ_{p}}(T_{p}(G)) ≃ ℤ_{p}^{×} is surjective.

Proof. Since Lie(G^{∨}) is an 𝒪_{S}-module free of rank 1, the condition that h(G) = 1 is equivalent to that any matrix of φ_{G} is represented by a uniformizer of A. Hence the first part of this theorem follows from Proposition 4.11(ii). We follow [Ka2, Thm 4.3] to prove the surjectivity of ρ under the assumption that h(G) = 1. For each integer n ≥ 1, let

ρ_{n} : I → Aut_{ℤ/p^{n}ℤ}(G(n)(\overline{K})) ≃ (ℤ/p^{n}ℤ)^{×}

be the reduction mod p^{n} of ρ, K_{n} be the subfield of K^{sep} fixed by the kernel of ρ_{n}. Then ρ_{n} induces an injective homomorphism Gal(K_{n}/K) → (ℤ/p^{n}ℤ)^{×}. By taking projective limits, we are reduced to proving the surjectivity of ρ_{n} for every n ≥ 1. It suffices to verify that

| Im(ρ_{n})| = [K_{n} : K] ≥ p^{n−1}(p − 1)

(then the equality holds automatically).

We regard G as a formal group over S. Then by [Ka2, 3.6], there exists a parameter X of the formal group G normalized by the condition that [ξ](X) = ξ(X) for all (p − 1)-th root of unity ξ ∈ ℤ_{p}. For such a parameter, we have

[p](X) = a_{1}X^{p} + αX^{p^{2}} + ∑_{m≥2} c_{m}X^{p(1+m(p−1))} ∈ A[[X]],

where we have v(a_{1}) = h(G) = 1 by [Ka2, 3.6.1 and 3.6.5], and v(α) = 0, as G is of height 2. For each integer i ≥ 0, we put

V^{(p^{i})}(X) = a_{1}^{p^{i}}X + α^{p^{i}}X^{p} + ∑_{m≥2} c_{m}^{p^{i}}X^{1+m(p−1)} ∈ A[[X]];

then we have [p^{n}](X) = V^{(p^{n−1})} ∘ V^{(p^{n−2})} ∘ ⋯ ∘ V (X^{p^{n}}). Hence each point of G(n)(\overline{K}) is given by a sequence y_{1}, ⋯, y_{n} ∈ K^{sep} (or simply an element y_{n} ∈ K^{sep}) satisfying the equations

{V (y_{1}) = a_{1}y_{1} + αy_{1}^{p} + ⋯ = 0; ; V^{(p)}(y_{2}) = a_{1}^{p}y_{2} + α^{p}y_{2}^{p} + ⋯ = y_{1}; ; ⋮ ; V^{(p^{n−1})}(y_{n}) = a_{1}^{p^{n−1}}y_{n} + α^{p^{n−1}}y_{n}^{p} + ⋯ = y_{n−1}.

Let y_{n} ∈ K^{sep} be such that y_{1} ≠ 0. By considering the Newton polygons of the equations above, we verify that

v(y_{i}) = \frac{1}{p^{i−1}(p − 1)} for 1 ≤ i ≤ n.

In particular, the ramification index e(K_{n}/K) is at least p^{n−1}(p − 1). By the definition of K_{n}, the Galois group Gal(K^{sep}/K_{n}) must fix y_{n} ∈ K^{sep}, i.e. K_{n} is an extension of K(y_{n}). Therefore, we have [K_{n} : K] ≥ [K(y_{n}) : K] ≥ e(K(y_{n})/K) ≥ p^{n−1}(p − 1). □

Proposition 5.8. Let G be a HW-cyclic BT-group over S of height c + d and dimension d such that G ⊗ K is ordinary,

𝔥 = (\begin{matrix}0 & 0 & ⋯ & 0 & −a_{1} \\ 1 & 0 & ⋯ & 0 & −a_{2} \\ 0 & 1 & ⋯ & 0 & −a_{3} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & −a_{c} \end{matrix})

be a matrix of φ_{G}. Put q = p^{c}, a_{c+1} = 1, and P (X) = ∑_{i=0}^{c} a_{i+1}X^{p^{i}} ∈ A[X].

(i) Assume that G is connected and the Hasse invariant h(G) = 1. Then the representation \overline{ρ} (5.6.3) is tame, G(1)(\overline{K}) is endowed with the structure of an 𝔽_{q}-vector space of dimension 1, and the induced action of I_{t} is given by the character θ_{q−1} : I_{t} → 𝔽_{q}^{×}. (ii) Assume that c > 1, v(a_{i}) ≥ 2 for 1 ≤ i ≤ c − 1 and v(a_{c}) = 1. Then the order of Im(\overline{ρ}) is divisible by p^{c−1}(p − 1).

(iii) Put i_{0} = min_{0≤i≤c}{i; v(a_{i+1}) = 0}. Assume that there exists α ∈ k such that v(P (α)) = 1. Then we have i_{0} ≤ c − 1 and the order of Im(\overline{ρ}) is divisible by p^{i_{0}}.

Proof. Since G is generically ordinary, we have a_{1} ≠ 0 by 2.11(d). Hence P (X) ∈ K[X] is a separable polynomial. By 4.4, G(1)(\overline{K}) ≃ (Ker V_{G})(K^{sep}) is identified with the additive group consisting of the roots of P (X) in K^{sep}.

(i) By definition of the Hasse invariant, we have v(a_{1}) = h(G) = 1. By 4.4(ii), the assumption that G is connected is equivalent to saying v(a_{i}) ≥ 1 for 1 ≤ i ≤ c. From the Newton polygon of P (X), we deduce that all the non-zero roots of P (X) in K^{sep} have the same valuation 1/(q − 1). We denote by

ψ : G(1)(\overline{K}) → V_{1/(q−1)}

the map which sends each root x ∈ K^{sep} of P (X) to the class of x in V_{1/(q−1)} = 𝔪_{1/(q−1)}/𝔪_{1/(q−1)}^{+} (5.1.1). We remark that G(1)(\overline{K}) is an 𝔽_{p}-vector space of dimension c. Hence G(1)(\overline{K}) is automatically of dimension 1 over 𝔽_{q} once we know it is an 𝔽_{q}-vector space. By 5.3, it suffices to show that ψ is an injective I-equivariant homomorphism of groups. By 4.4(i), ψ is obviously an I-equivariant homomorphism of groups. Let x_{0} be a root of P (X), and put Q(y) = P (x_{0}y). Then the polynomial Q(y) has the form Q(y) = x_{0}^{q}Q_{1}(y), where

Q_{1}(y) = y^{q} + b_{c}y^{p^{c−1}} + ⋯ + b_{2}y^{p} + b_{1}y

with b_{i} = a_{i}/x_{0}^{(q−p^{i−1})} ∈ K^{sep}. We have v(b_{i}) > 0 for 2 ≤ i ≤ c and v(b_{1}) = 0. Let \overline{b}_{1} be the class of b_{1} in the residue field k = 𝔪_{0}/𝔪_{0}^{+}. Then the images of the roots of P (X) in V_{1/(q−1)} are x_{0}\overline{b}_{1}^{1/(q−1)}ζ, where ζ runs over the finite field 𝔽_{q}. Therefore, ψ is injective. (ii) By computing the slopes of the Newton polygon of P (X), we see that P (X) has p^{c−1}(p − 1) roots of valuation 1/(p^{c} − p^{c−1}). Let L be the sub-extension of K^{sep} obtained by adding to K all the roots of P (x). Then the ramification index e(L/K) is divisible by p^{c−1}(p − 1). Let L̃ be the sub-extension of K^{sep} fixed by the kernel of \overline{ρ} (5.6.3). The Galois group Gal(K^{sep}/L̃) fixes the roots of P (x) by definition. Hence we have L ⊂ L̃, and | Im(\overline{ρ})| = [L̃ : K] is divisible by [L : K]; in particular, it is divisible by p^{c−1}(p − 1). (iii) Note that the relation i_{0} ≤ c − 1 is equivalent to saying that G is not connected by 4.4(ii). Assume conversely i_{0} = c, i.e. G is connected. Then we would have

P (X) ≡ X^{q} mod (πA[X]).

But v(P (α)) = 1 implies that α^{p^{c}} ∈ πA, i.e. α = 0; hence we would have P (α) = 0, which contradicts the condition v(P (α)) = 1. We put Q(X) = P (X + α) = P (X) + P (α). As v(P (α)) = 1, then (0, 1) and (p^{i_{0}}, 0) are the first two break points of the Newton polygon of Q(X). Hence there exists p^{i_{0}} roots of Q(X) of valuation 1/p^{i_{0}}. Let L be the subextension of K in K^{sep} generated by the roots of P (X). The ramification index e(L/K) is divisible by p^{i_{0}}. As in the proof of (ii), if L̃ is the subextension of K^{sep} fixed by the kernel of \overline{ρ}, then it is an extension of L. Therefore, we have | Im(\overline{ρ})| = [L̃ : K] is divisible by [L : K], and in particular, divisible by p^{i_{0}}. □

5.9. Let G be a BT-group over S with connected part G^{∘}, and étale part G^{ét} of height r. We have a canonical exact sequence of I-modules

(5.9.1)  0 → G^{∘}(1)(\overline{K}) → G(1)(\overline{K}) → G^{ét}(1)(\overline{K}) → 0

giving rise to a class \overline{C} ∈ Ext_{𝔽_{p}[I]}^{1}(G^{ét}(1)(\overline{K}), G^{∘}(1)(\overline{K})), which vanishes if and only if (5.9.1) splits. Since I acts trivially on G^{ét}(1)(\overline{K}), we have an isomorphism of I-modules G^{ét}(1)(\overline{K}) ≃ 𝔽_{p}^{r}. Recall that for any 𝔽_{p}[I]-module M, we have a canonical isomorphism ([Se1] Chap.VII, §2)

Ext_{𝔽_{p}[I]}^{1}(𝔽_{p}, M) ≃ H^{1}(I, M).

Hence we deduce that

(5.9.2)  \overline{C} ∈ Ext_{𝔽_{p}[I]}^{1}(G^{ét}(1)(\overline{K}), G^{∘}(1)(\overline{K})) ≃ H^{1}(I, G^{∘}(1)(\overline{K}))^{r}.

Proposition 5.10. Let G be a HW-cyclic BT-group over S such that h(G) = 1, \overline{ρ} (5.6.3) be the representation of I on G(1)(\overline{K}). Then the cohomology class \overline{C} does not vanish if and only if the order of the group Im(\overline{ρ}) is divisible by p.

First, we prove the following result on cohomology of groups.

Lemma 5.11. Let F be a field, Γ be a commutative group, and χ : Γ → F^{×} be a non-trivial character of Γ. We denote by F (χ) an F -vector space of dimension 1 endowed with an action of Γ given by χ. Then we have H^{1}(Γ, F (χ)) = 0.

Proof. Let C be a 1-cocycle of Γ with values in F (χ). We prove that C is a 1-coboundary. For any g, h ∈ Γ, we have

C(gh) = C(g) + χ(g)C(h),
C(hg) = C(h) + χ(h)C(g).

Since Γ is commutative, it follows from the relation C(gh) = C(hg) that

(5.11.1)  (χ(g) − 1)C(h) = (χ(h) − 1)C(g).

If χ(g) ≠ 1 and χ(h) ≠ 1, then

\frac{1}{χ(g) − 1}C(g) = \frac{1}{χ(h) − 1}C(h).

Therefore, there exists x ∈ F (\overline{χ}) such that C(g) = (χ(g) − 1)x for all g ∈ Γ with χ(g) ≠ 1. If χ(g) = 1, we have also C(g) = 0 = (χ(g) − 1)x by (5.11.1). This shows that C is a 1-coboundary. □

Proof of 5.10. By 4.3(ii) and 5.4(c), the connected part G^{∘} of G is HW-cyclic with h(G^{∘}) = h(G) = 1. Assume that T_{p}(G^{∘}) has rank ℓ over ℤ_{p}, and T_{p}(G^{ét}) has rank r. Then by 5.8(a), G^{∘}(1)(\overline{K}) is an 𝔽_{q}-vector space of dimension 1 with q = p^{ℓ}, and the action of I on G^{∘}(1)(\overline{K}) factors through the character \overline{χ} : I →

I_{t} ⟶^{θ_{q−1}} 𝔽_{q}^{×}. We write G^{∘}(1)(\overline{K}) = 𝔽_{q}(\overline{χ}) for short. If the cohomology class \overline{C} is zero, then the exact sequence (5.9.1) splits, i.e. we have an isomorphism of Galois modules G(1)(\overline{K}) ≃ 𝔽_{q}(χ) ⊕ 𝔽_{p}^{r}. It is clear that the group Im(\overline{ρ}) has order q − 1. Conversely, if the cohomology class \overline{C} is not zero, we will show that there exists an element in Im(\overline{ρ}) of order p. We choose a basis adapted to the exact sequence (5.9.1) such that the action of g ∈ I is given by

(5.11.2)  \overline{ρ}(g) = (\begin{matrix}\overline{χ}(g) & \overline{C}(g) \\ 0 & 1_{r} \end{matrix}) ,

where 1_{r} is the unit matrix of type (r, r) with coefficients in 𝔽_{p}, and the map g ↦ \overline{C}(g) gives rise to a 1-cocycle representing the cohomology class \overline{C}. Let I_{1} be the kernel of \overline{χ} : I → 𝔽_{q}^{×}, Γ be the quotient I/I_{1}, so \overline{χ} induces an isomorphism \overline{χ} : Γ ⟶^{∼} 𝔽_{q}^{×}. We have an exact sequence

0 → H^{1}(Γ, 𝔽_{q}(\overline{χ}))^{r} ⟶^{Inf} H^{1}(I, 𝔽_{q}(\overline{χ}))^{r} ⟶^{Res} H^{1}(I_{1}, 𝔽_{q}(\overline{χ}))^{r},

where “Inf” and “Res” are respectively the inflation and restriction homomorphisms in group cohomology. Since H^{1}(Γ, 𝔽_{q}(\overline{χ}))^{r} = 0 by 5.11, the restriction of the cohomology class \overline{C} to H^{1}(I_{1}, 𝔽_{q}(\overline{χ}))^{r} is non-zero. Hence there exists h ∈ I_{1} such that \overline{C}(h) ≠ 0. As we have \overline{χ}(h) = 1, then

\overline{ρ}(h)^{p} = (\begin{matrix}1_{ℓ} & p\overline{C}(h) \\ 0 & 1_{r} \end{matrix}) = 1_{ℓ+r}.

Thus the order of \overline{ρ}(h) is p. □

Corollary 5.12. Let G be a HW-cyclic BT-group over S,

𝔥 = (\begin{matrix}0 & 0 & ⋯ & 0 & −a_{1} \\ 1 & 0 & ⋯ & 0 & −a_{2} \\ 0 & 1 & ⋯ & 0 & −a_{3} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & −a_{c} \end{matrix})

be a matrix of φ_{G}, P (X) = X^{p^{c}} + a_{c}X^{p^{c−1}} + ⋯ + a_{1}X ∈ A[X]. If h(G) = 1 and if there exists α ∈ k ⊂ A such that v(P (α)) = 1, then the cohomology class (5.9.2) is not zero, i.e. the extension of I-modules (5.9.1) does not split.

Proof. Since v(a_{1}) = h(G) = 1, the integer i_{0} defined in 5.8(iii) is at least 1. Then the corollary follows from 5.8(iii) and 5.10. □

6. Lemmas in Group Theory

In this section, we fix a prime number p ≥ 2 and an integer n ≥ 1.

6.1. Recall that the general linear group GL_{n}(ℤ_{p}) admits a natural exhaustive decreasing filtration by normal subgroups

GL_{n}(ℤ_{p}) ⊃ 1 + pM_{n}(ℤ_{p}) ⊃ ⋯ ⊃ 1 + p^{m}M_{n}(ℤ_{p}) ⊃ ⋯,

where M_{n}(ℤ_{p}) denotes the ring of matrix of type (n, n) with coefficients in ℤ_{p}. We endow GL_{n}(ℤ_{p}) with the topology for which (1 + p^{m}M_{n}(ℤ_{p}))_{m≥1} form a fundamental system of neighborhoods of 1. Then GL_{n}(ℤ_{p}) is a complete and separated topological group.

6.2. Let 𝔊 be a profinite group, ρ : 𝔊 → GL_{n}(ℤ_{p}) be a continuous homomorphism of topological groups. By taking inverse images, we obtain a decreasing filtration (F^{m}𝔊, m ∈ ℤ_{≥0}) on 𝔊 by open normal subgroups:

F^{0}𝔊 = 𝔊, and F^{m}𝔊 = ρ^{−1}(1 + p^{m}M_{n}(ℤ_{p})) for m ≥ 1.

Furthermore, the homomorphism ρ induces a sequence of injective homomorphisms of finite groups

ρ_{0} : F^{0}𝔊/F^{1}𝔊 ⟶ GL_{n}(𝔽_{p}) (6.2.1)

ρ_{m}: F^{m}𝔊/F^{m+1}𝔊 → M_{n}(𝔽_{p}), for m ≥ 1. (6.2.2)

Lemma 6.3. The homomorphism ρ is surjective if and only if the following conditions are satisfied:

(i) The homomorphism ρ_{0} is surjective. (ii) For every integer m ≥ 1, the subgroup Im(ρ_{m}) of M_{n}(𝔽_{p}) contains an element of the form

(\begin{matrix}x & 0 & ⋯ & 0 \\ 0 & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & 0 \end{matrix})

with x ≠ 0; or equivalently, there exists, for every m ≥ 1, an element g_{m} ∈ 𝔊 such that ρ(g_{m}) is of the form

(\begin{matrix}1 + p^{m}a_{1,1} & p^{m+1}a_{1,2} & ⋯ & p^{m+1}a_{1,n} \\ p^{m+1}a_{2,1} & 1 + p^{m+1}a_{2,2} & ⋯ & p^{m+1}a_{2,n} \\ ⋮ & ⋮ & ... & ⋮ \\ p^{m+1}a_{n,1} & p^{m+1}a_{n,2} & ⋯ & 1 + p^{m+1}a_{n,n} \end{matrix}) ,

where a_{i,j} ∈ ℤ_{p} for 1 ≤ i, j ≤ n and a_{1,1} is not divisible by p.

Proof. We notice first that ρ is surjective if and only if ρ_{m} is surjective for every m ≥ 0, because 𝔊 is complete and GL_{n}(ℤ_{p}) is separated [Bou, Chap. III §2 n^{∘}8 Cor.2 au Théo. 1]. The surjectivity of ρ_{0} is condition (i). Condition (ii) is clearly necessary. We prove that it implies the surjectivity of ρ_{m} for all m ≥ 1, under the assumption of (i). First, we remark that under condition (i), if A lies in Im(ρ_{m}), then for any U ∈ GL_{n}(𝔽_{p}) the conjuagate matrix U · A · U^{−1} lies also in Im(ρ_{m}). In fact, let Ã be a lift of A in M_{n}(ℤ_{p}) and Ũ ∈ GL_{n}(ℤ_{p}) a lift of U. By assumption, there exist g, h ∈ 𝔊 such that

ρ(g) ≡ 1+p^{m}Ã mod (1+p^{m+1}M_{n}(ℤ_{p})) and ρ(h) ≡ Ũ mod (1+pM_{n}(ℤ_{p})).

Therefore, we have ρ(hgh^{−1}) ≡ (1 + p^{m}Ũ · Ã · Ũ^{−1}) mod (1 + p^{m+1}M_{n}(ℤ_{p})). Hence hgh^{−1} ∈ F^{m}𝔊 and ρ_{m}(hgh^{−1}) = U · A · U^{−1}. For 1 ≤ i, j ≤ n, let E_{i,j} ∈ M_{n}(𝔽_{p}) be the matrix whose (i, j)-th entry is 0 and the other entries are 0. The matrices E_{i,j}(1 ≤ i, j ≤ n) form clearly a basis of M_{n}(𝔽_{p}) over 𝔽_{p}. To prove the surjectivity of ρ_{m}, we only need to verify that E_{i,j} ∈ Im(ρ_{m}) for 1 ≤ i, j ≤ n, because Im(ρ_{m}) is an 𝔽_{p}- subspace of M_{n}(𝔽_{p}). By assumption, we have E_{1,1} ∈ Im(ρ_{m}). For 2 ≤ i ≤ n, we put U_{i} = E_{1,i} − E_{i,1} + ∑_{j≠1,i} E_{j,j}. Then we have U_{i} ∈ GL_{n}(ℤ_{p}) and U_{i} · E_{1,1} · U_{i}^{−1} = E_{i,i} ∈ Im(ρ_{m}). For 1 ≤ i < j ≤ n, we put U_{i,j} = I + E_{i,j} where I is the unit matrix. Then we have U_{i,j} ·E_{i,i}·U_{i,j}^{−1} = E_{i,i}+E_{i,j} ∈ Im(ρ_{m}), and hence E_{i,j} ∈ Im(ρ_{m}). This completes the proof.

□

Remark 6.4. By using the arguments in [Se2, Chap. IV 3.4 Lemma 3], we can prove the following stronger form of Lemma 6.3: If p = 2, condition (i) and (ii) for m = 1, 2 are sufficient to guarantee the surjectivity of ρ; if p ≥ 3, then (i) and (ii) just for m = 1 suffice already.

A subgroup C of GL_{n}(𝔽_{p}) is called a non-split Cartan subgroup, if the subset C ∪{0} of the matrix algebra M_{n}(𝔽_{p}) is a field isomorphic to 𝔽_{p^{n}}; such a group is cyclic of order p^{n} − 1.

Lemma 6.5. Assume that n ≥ 2. We denote by H the subgroup of GL_{n}(𝔽_{p})

consisting of all the elements of the form (\begin{matrix}A & b \\ 0 & 1 \end{matrix}) , where A ∈ GL_{n−1}(𝔽_{p}) and

b = ( b_{1} ; ⋮ ; b_{n−1}) with b_{i} ∈ 𝔽_{p}(1 ≤ i ≤ n − 1). Let G be a subgroup of GL_{n}(𝔽_{p}).

Then G = GL_{n}(𝔽_{p}) if and only if G contains H and a non-split Cartan subgroup of GL_{n}(𝔽_{p}).

Proof. The “only if” part is clear. For the “if” part, let C be a non-split Cartan subgroup contained in G. For a finite group Λ, we denote by |Λ| its order. An easy computation shows that |GL_{n}(𝔽_{p})| = |H| · |C|. So we just need to prove that U ∩C = {1}; since then we will have |GL_{n}(𝔽_{p})| = |G|, hence G = GL_{n}(𝔽_{p}). Let g ∈ H ∩ C, and P (T) ∈ 𝔽_{p}[T] be its characteristic polynomial. We fix an isomorphism C ≃ 𝔽_{p^{n}}^{×}, and let ζ ∈ 𝔽_{p^{n}}^{×} be the element corresponding to g. We have P (T) = ∏_{σ∈Gal(𝔽_{p^{n}} /𝔽_{p})}(T − σ(ζ)) in 𝔽_{p^{n}}[T]. On the other hand, the fact that g ∈ H implies that (T − 1) divises P (T). Therefore, we get ζ = 1, i.e. g = 1. □

Remark 6.6. E. Lau point out the following strengthened version of 6.5: When n ≥ 3, a subgroup G ⊂ GL_{n}(𝔽_{p}) coincides with GL_{n}(𝔽_{p}) if and only if G

contains a non-split Cartan subgroup and the subgroup (\begin{matrix}GL_{n−1}(𝔽_{p}) & 0 \\ 0 & 1 \end{matrix}). This

can be used to simplify the induction process in the proof of Theorem 7.3 when n ≥ 3.

7. Proof of Theorem 1.3 in the One-dimensional Case

7.1. We start with a general remark on the monodromy of BT-groups. Let X be a scheme, G be an ordinary BT-group over a scheme X, G^{ét} be its étale part (2.10.1). If \overline{η} is a geometric point of X, we denote by

T_{p}(G, \overline{η}) = \underleftarrow{lim}_{n} G(n)(\overline{η}) = \underleftarrow{lim}_{n} G^{ét}(n)(\overline{η})

the Tate module of G at \overline{η}, and by ρ(G) the monodromy representation of π_{1}(X, \overline{η}) on T_{p}(G, \overline{η}). Let f : Y → X be a morphism of schemes, \overline{ξ} be a geometric point of Y, G_{Y} = G ×_{X} Y. Then by the functoriality, we have a commutative diagram

(7.1.1)  [figure: π_{1}(Y, ξ); π_{1}(f); ρ(G_{Y} ); π_{1}(X, f(\overline{ξ})); ρ(G); Aut_{ℤ_{p}}(T_{p}(G_{Y} , \overline{ξ})); Aut_{ℤ_{p}}(T_{p}(G, f(\overline{ξ})))]

In particular, the monodromy of G_{Y} is a subgroup of the monodromy of G. In the sequel, diagram (7.1.1) will be refereed as the functoriality of monodromy for the BT-group G and the morphism f.

7.2. Let k be an algebraically closed field of characteristic p > 0, G be the unique connected BT-group over k of dimension 1 and height n + 1 ≥ 2 (4.10). We denote by S the algebraic local moduli of G in characteristic p, by G the universal deformation of G over S, and by U the ordinary locus of G over S (3.8). Recall that S is affine of ring R ≃ k[[t_{1}, ⋯, t_{n}]] (3.7), and that G and G are HW-cyclic (cf. 4.3(i) and 4.10). Let \overline{η} be a geometric point of U over its generic point. We put

T_{p}(G, \overline{η}) = \underleftarrow{lim}_{m∈ℤ_{≥1}} G(m)(\overline{η})

to be the Tate module of G at the point \overline{η}. This is a free ℤ_{p}-module of rank n. We have the monodromy representation

ρ_{n} : π_{1}(U, \overline{η}) → Aut_{ℤ_{p}}(T_{p}(G, \overline{η})) ≃ GL_{n}(ℤ_{p}).

The following is the one-dimensional case of Theorem 1.3.

Theorem 7.3. Under the above assumptions, the homomorphism ρ_{n} is surjective for n ≥ 1.

7.4. First, we assume n ≥ 2. By Proposition 4.11(ii), we may assume that

(7.4.1)  𝔥 = (\begin{matrix}0 & 0 & ⋯ & 0 & −t_{1} \\ 1 & 0 & ⋯ & 0 & −t_{2} \\ 0 & 1 & ⋯ & 0 & −t_{3} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & −t_{n} \end{matrix})

is a matrix of the Hasse-Witt map φ_{G}. Let 𝔭 be the prime ideal of R generated by t_{1}, ⋯, t_{n−1}. Then the closed subscheme of S defined by 𝔭 is just the locus where the p-rank of G is ≤ 1 by 4.4(ii). Let K_{0} ≃ k((t_{n})) be the fraction field of R/𝔭, R′ = R̂_{𝔭} be the completion of the localization of R at 𝔭, and 𝒢_{R′} = G ⊗_{R} R′. Since the natural map R → R′ is injective, for any a ∈ R, we will denote also by a its image in R′. Since the Hasse-Witt map commutes with base change, the image of 𝔥 in M_{n×n}(R′), denoted also by 𝔥, is a matrix of φ_{𝒢_{R′}} . We see easily that the étale part of 𝒢_{R′} has height 1 and its connected part 𝒢_{R′}^{∘} has height n. We have an exact sequence of BT-groups over R′

(7.4.2)  0 → 𝒢_{R′}^{∘} → 𝒢_{R′} → 𝒢_{R′}^{ét} → 0.

We fix an imbedding i : K_{0} → \overline{K}_{0} of K_{0} into an algebraically closed field. Put 𝒢_{\overline{K}_{0}}^{∗} = 𝒢_{R′}^{∗} ⊗ \overline{K}_{0} for ∗ = ∅, ét, ∘. We have 𝒢_{\overline{K}_{0}}^{ét} ≃ ℚ_{p}/ℤ_{p}, and 𝒢_{\overline{K}_{0}}^{∘} is the unique connected one-dimensional BT-group over \overline{K}_{0} of height n (cf. 4.10). We put R̃′ = \overline{K}_{0}[[x_{1}, ⋯, x_{n−1}]], and

(7.4.3) Σ = \{ring homomorphisms σ : R′ → R̃′ lifting R′ → K_{0} ⟶^{i} \overline{K}_{0}\}

Let σ ∈ Σ. We deduce from (7.4.2) by base change an exact sequence of BT-groups over R̃′

(7.4.4)  0 → 𝒢_{R̃′,σ}^{∘} → 𝒢_{R̃′,σ} → 𝒢_{R̃′,σ}^{ét} → 0,

where we have put 𝒢_{R̃′,σ}^{∗} = 𝒢_{R′}^{∗} ⊗_{σ} R̃′ for ∗ = ∘, ∅, ét. Due to the henselian

property of R̃′, the isomorphism 𝒢_{\overline{K}_{0}}^{ét} ≃ ℚ_{p}/ℤ_{p} lifts uniquely to an isomorphism

𝒢_{R̃′,σ}^{ét} ≃ ℚ_{p}/ℤ_{p} . Assume that 𝒢_{R̃′,σ}^{∘} is generically ordinary over S̃′ = Spec(R̃′).

Let Ũ′_{σ} ⊂ S̃′ be its ordinary locus, and \overline{x} be a geometric point over the generic point of Ũ′_{σ}. The exact sequence (7.4.4) induces an exact sequence of Tate modules

(7.4.5)  0 → T_{p}(𝒢_{R̃′,σ}^{∘}, \overline{x}) → T_{p}(𝒢_{R̃′,σ}, \overline{x}) → T_{p}(𝒢_{R̃′,σ}^{ét}, \overline{x}) → 0

compatible with the actions of π_{1}(Ũ′_{σ}, \overline{x}). Since we have T_{p}(𝒢_{R̃′,σ}^{ét}, \overline{x}) ≃ T_{p}(ℚ_{p}/ℤ_{p}, \overline{x}) = ℤ_{p}, this determines a cohomology class

(7.4.6) C_{σ} ∈ Ext_{ℤ_{p}[π_{1}(Ũ′_{σ},\overline{x})]}^{1}(ℤ_{p}, T_{p}(𝒢_{R̃′,σ}^{∘}, \overline{x})) ≃ H^{1}(π_{1}(Ũ′_{σ}, \overline{x}), T_{p}(𝒢_{R̃′,σ}^{∘}, \overline{x})).

We consider also the “mod-p version” of (7.4.5)

0 → 𝒢_{R̃′,σ}^{∘}(1)(\overline{x}) → 𝒢_{R̃′,σ}(1)(\overline{x}) → 𝔽_{p} → 0,

which determines a cohomology class

(7.4.7) \overline{C}_{σ} ∈ Ext_{𝔽_{p}[π_{1}(Ũ′_{σ},\overline{x})]}^{1}(𝔽_{p}, 𝒢_{R̃′,σ}^{∘}(1)(\overline{x})) ≃ H^{1}(π_{1}(Ũ′_{σ}, \overline{x}), 𝒢_{R̃′,σ}^{∘}(1)(\overline{x})).

It is clear that \overline{C}_{σ} is the image of C_{σ} by the canonical reduction map

H^{1}(π_{1}(Ũ′_{σ}, \overline{x}), T_{p}(𝒢_{R̃′,σ}^{∘}, \overline{x})) → H^{1}(π_{1}(Ũ′_{σ}, \overline{x}), 𝒢_{R̃′,σ}^{∘}(1)(\overline{x})).

Lemma 7.5. Under the above assumptions, there exist σ_{1}, σ_{2} ∈ Σ satisfying the following properties:

(i) We have 𝒢_{R̃′,σ_{1}}^{∘} = 𝒢_{R̃′,σ_{2}}^{∘}, and it is the universal deformation of 𝒢_{\overline{K}_{0}}^{∘}.

(ii) We have C_{σ_{1}} = 0 and \overline{C}_{σ_{2}} ≠ 0.

Before proving this lemma, we prove first Theorem 7.3.

Proof of 7.3. First, we notice that the monodromy of a BT-group is independent of the base point. So we can change \overline{η} to any geometric point of U when discussing the monodromy of G. We make an induction on the codimension n = dim(G^{∨}). The case of n = 1 is proved in Theorem 5.7. Assume that n ≥ 2 and the theorem is proved for n − 1. We denote by

\overline{ρ}_{n} : π_{1}(U, \overline{η}) → Aut_{𝔽_{p}}(G(1)(\overline{η})) ≃ GL_{n}(𝔽_{p})

the reduction of ρ_{n} modulo by p. By Lemma 6.3 and 6.5, to prove the surjectivity of ρ_{n}, we only need to verify the following conditions:

(a) Im(\overline{ρ}_{n}) contains a non-split Cartan subgroup of GL_{n}(𝔽_{p});

(b) Im(ρ_{n}) contains the subgroup H ⊂ GL_{n}(ℤ_{p}) consisting of all the elements

of the form (\begin{matrix}B & b \\ 0 & 1 \end{matrix}) ∈ GL_{n}(ℤ_{p}), with B ∈ GL_{n−1}(ℤ_{p}) and b ∈ M_{(n−1)×1}(ℤ_{p});

For condition (a), let A = k[[π]], T = Spec(A), ξ be its generic point, \overline{ξ} be a geometric point over ξ, and I = Gal(\overline{ξ}/ξ) be the absolute Galois group over ξ. We keep the notations of 7.4. Let f^{∗} : R → A be the homomorphism of k-algebras such that f^{∗}(t_{1}) = π and f^{∗}(t_{i}) = 0 for 2 ≤ i ≤ n. We denote by f : T → S the corresponding morphism of schemes, and put G_{T} = G ×_{S} T. By the functoriality of Hasse-Witt maps,

𝔥_{T} = (\begin{matrix}0 & 0 & ⋯ & 0 & −π \\ 1 & 0 & ⋯ & 0 & 0 \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & 0 \end{matrix})

is a matrix of φ_{G_{T}}. By definition 5.4, the Hasse invariant of G_{T} is h(G) = 1. Hence G_{T} is generically ordinary; so f(ξ) ∈ U. Let

\overline{ρ}_{T} : I = Gal(\overline{ξ}/ξ) → Aut_{𝔽_{p}}(G_{T} (1)(\overline{ξ}))

be the mod-p monodromy representation attached to G_{T}. Proposition 5.8(i) implies that Im(\overline{ρ}_{T}) is a non-split Cartan subgroup of GL_{n}(𝔽_{p}). On the other hand, by the functoriality of monodromy, we get Im(\overline{ρ}_{T}) ⊂ Im(\overline{ρ}_{n}). This verifies condition (a). To check condition (b), we consider the constructions in 7.4. Let S′ = Spec(R′), f : S′ → S be the morphism of schemes corresponding to the natural ring homomorphism R → R′, U′ be the ordinary locus of 𝒢_{R′}, and \overline{ξ} be a geometric point of U′. From (7.4.2), we deduce an exact sequence of Tate modules

(7.5.1)  0 → T_{p}(𝒢_{R′}^{∘}, \overline{ξ}) → T_{p}(𝒢_{R′}, \overline{ξ}) → T_{p}(𝒢_{R′}^{ét}, \overline{ξ}) → 0.

Let ρ_{𝒢′} : π_{1}(U′, \overline{ξ}) → Aut_{ℤ_{p}}(T_{p}(𝒢_{R′}, \overline{ξ})) ≃ GL_{n}(ℤ_{p}) be the monodromy represention of 𝒢_{R′}. Under any basis of T_{p}(𝒢_{R′}, \overline{ξ}) adapted to (7.5.1), the action of π_{1}(U′, \overline{ξ}) on T_{p}(𝒢_{R′}, \overline{ξ}) is given by

ρ_{𝒢_{R′}} : g ∈ π_{1}(U′, \overline{ξ}) ↦ (\begin{matrix}ρ_{𝒢_{R′}^{∘}} (g) & ∗ \\ 0 & ρ_{𝒢_{R′}^{ét}} (g), \end{matrix})

where g ↦ ρ_{𝒢_{R′}^{∘}} (g) ∈ GL_{n−1}(ℤ_{p}) (resp. g ↦ ρ_{𝒢_{R′}^{ét}} (g) ∈ ℤ_{p}^{×}) gives the action of π_{1}(U′, \overline{ξ}) on T_{p}(𝒢_{R′}^{∘}, \overline{ξ}) (resp. on T_{p}(𝒢_{R′}^{ét}, \overline{ξ})). Note that f(U′) ⊂ U. So by the functoriality of monodromy, we get Im(ρ_{𝒢′} ) ⊂ Im(ρ_{n}). To complete the proof of Theorem 7.3, it suffices to check condition (b) with ρ_{n} replaced by ρ_{𝒢_{R′}} under the induction hypothesis that 7.3 is valide for n−1. Let σ_{1}, σ_{2} : R′ → R̃′ be the homomorphisms given by 7.5. For i = 1, 2, we denote by f_{i} : S̃′ = Spec(R̃′) → S′ = Spec(R′) the morphism of schemes corresponding to σ_{i}, and put 𝒢_{i} = 𝒢_{R̃′,σ_{i}} = 𝒢_{R′} ⊗_{σ_{i}} R̃′ to simply the notations. By condition 7.5(i), we

can denote by 𝒢^{∘} the common connected component of 𝒢_{1} and 𝒢_{2}. Let Ũ′ ⊂ S̃′ be the ordinary locus of 𝒢^{∘}. Then we have f_{i}(Ũ′) ⊂ U′ for i = 1, 2. Let \overline{x} be a geometric point over the generic point of Ũ′. We have an exact sequence of Tate modules

(7.5.2)  0 → T_{p}(𝒢^{∘}, \overline{x}) → T_{p}(𝒢_{i}, \overline{x}) → T_{p}(ℚ_{p}/ℤ_{p}, \overline{x}) → 0

compatible with the actions of π_{1}(Ũ′, \overline{x}). We denote by

ρ_{𝒢_{i}} : π_{1}(Ũ′, \overline{x}) → Aut_{ℤ_{p}}(T_{p}(𝒢_{i}, \overline{x})) ≃ GL_{n}(ℤ_{p})

the monodromy representation of 𝒢_{i}. In a basis adapted to (7.5.2), the action of π_{1}(Ũ′, \overline{x}) on T_{p}(𝒢_{i}, \overline{x}) is given by

ρ_{𝒢_{i}} : g ↦ (\begin{matrix}ρ_{𝒢^{∘}} (g) & C_{σ_{i}}(g) \\ 0 & 1 \end{matrix}) ,

where ρ_{𝒢^{∘}} : π_{1}(Ũ′, \overline{x}) → GL_{n−1}(ℤ_{p}) is the monodromy representation of 𝒢^{∘}, and the cohomology class in H^{1}(π_{1}(Ũ′, \overline{x}), T_{p}(𝒢^{∘})) given by g ↦ C_{σ_{i}}(g) is nothing but the class defined in (7.4.6). By 7.5(i) and the induction hypothesis, ρ_{𝒢^{∘}} is surjective. Since the cohomology class C_{σ_{1}} = 0 by 7.5(ii), we may assume C_{σ_{1}}(g) = 0 for all g ∈ π_{1}(U′, \overline{x}). Therefore Im(ρ_{𝒢_{1}} ) contains all the matrix of

the form (\begin{matrix}B & 0 \\ 0 & 1 \end{matrix}) with B ∈ GL_{n−1}(ℤ_{p}). By the functoriality of monodromy,

Im(ρ_{𝒢_{R′}} ) contains Im(ρ_{𝒢_{1}} ). Hence we have

(7.5.3)  (\begin{matrix}GL_{n−1}(ℤ_{p}) & 0 \\ 0 & 1 \end{matrix}) ⊂ Im(ρ_{𝒢_{1}} ) ⊂ Im(ρ_{𝒢_{R′}} ).

On the other hand, since the cohomology class \overline{C}_{σ_{2}} ≠ 0, there exists a g ∈ π_{1}(Ũ′, \overline{x}) such that b_{2} = \overline{C}_{σ_{2}}(g) ≠ 0. Hence the matrix ρ_{𝒢_{2}} (g) has the

form (\begin{matrix}B_{2} & b_{2} \\ 0 & 1 \end{matrix}) such that B_{2} ∈ GL_{n−1}(ℤ_{p}) and the image of b_{2} ∈ M_{1×n−1}(ℤ_{p}) in M_{1×n−1}(𝔽_{p}) is non-zero. By the functoriality of monodromy, we have

Im(ρ_{𝒢_{2}} ) ⊂ Im(ρ_{𝒢_{R′}} ); in particular, we have (\begin{matrix}B_{2} & b_{2} \\ 0 & 1 \end{matrix}) ∈ Im(ρ_{𝒢_{R′}} ). In view

of (7.5.3), we get

(7.5.4)  (\begin{matrix}GL_{n−1}(ℤ_{p}) & 0 \\ 0 & 1 \end{matrix}) (\begin{matrix}B_{2} & b_{2} \\ 0 & 1 \end{matrix}) (\begin{matrix}GL_{n−1}(ℤ_{p}) & 0 \\ 0 & 1 \end{matrix}) ⊂ Im(ρ_{𝒢_{R′}} ).

But the subset of GL_{n}(ℤ_{p}) on the left hand side is just the subgroup H described in condition (b). Therefore, condition (b) is verified for ρ_{𝒢_{R′}} , and the proof of 7.3 is complete.

The rest of this section is dedicated to the proof of Lemma 7.5.

Lemma 7.6. Let k be an algebraically closed field of characteristic p > 0, A be a noetherian henselian local k-algebra with residue field k, G be a BT-group over A, and G^{ét} be its étale part. Put

Lie(G^{∨})^{φ=1} = {x ∈ Lie(G^{∨}) such that φ_{G}(x) = x}.

Then Lie(G^{∨})^{φ=1} is an 𝔽_{p}-vector space of dimension equal to the rank of Lie(G^{ét∨}), and the A-submodule Lie(G^{ét∨}) of Lie(G^{∨}) is generated by Lie(G^{∨})^{φ=1}.

Proof. Let r be the rank of Lie(G^{ét∨}), G^{∘} be the connected part of G, and s be the height of Lie(G^{∘∨}). We have an exact sequence of A-modules

0 → Lie(G^{ét∨}) → Lie(G^{∨}) → Lie(G^{∘∨}) → 0,

compatible with Hasse-Witt maps. We choose a basis of Lie(G^{∨}) adapted to

this exact sequence, so that φ_{G} is expressed by a matrix of the form (\begin{matrix}U & W \\ 0 & V \end{matrix} )

with U ∈ M_{r×r}(A), V ∈ M_{s×s}(A), and W ∈ M_{r×s}(A). An element of

Lie(G^{∨})^{φ=1} is given by a vector (\begin{matrix}x \\ y \end{matrix}), where x = (x_{1} ; ⋮ ; x_{r}) and y = (y_{1} ; ⋮ ; y_{s}) with

x_{i}, y_{j} ∈ A, satisfying

(7.6.1)  (\begin{matrix}U & W \\ 0 & V \end{matrix} ) · (\begin{matrix}x^{(p)} \\ y^{(p)} \end{matrix}) = (\begin{matrix}x \\ y \end{matrix}) ⇔ \begin{cases}U · x^{(p)} + W · y^{(p)} = x \\ V · y^{(p)} = y. \end{cases}

where x^{(p)} (resp. y^{(p)}) is the vector obtained by applying a ↦ a^{p} to each x_{i}(1 ≤ i ≤ r) (resp. y_{j}(1 ≤ j ≤ s)). By 2.9, the Hasse-Witt map of the special fiber of G^{∘} is nilpotent. So there exists an integer N ≥ 1 such that φ_{G^{∘}}^{N}(Lie(G^{∘∨})) ⊂ 𝔪_{A} · Lie(G^{∘∨}), i.e. we have V · V^{(p)} ⋯ V^{(p^{N−1})} ≡ 0 (mod 𝔪_{A}). From the equation V · y^{(p)} = y, we deduce that

y = V · V^{(p)} ⋯ V^{(p^{N−1})} · y^{(p^{N})} ≡ 0 (mod 𝔪_{A}).

But this implies that y^{(p^{N})} ≡ 0 (mod 𝔪_{A}^{p^{N}}). Hence we get y = V · y^{(p)} ≡ 0 (mod 𝔪_{A}^{p^{N} +1}). Repeting this argument, we get finally y ≡ 0 (mod 𝔪_{A}^{ℓ}) for all integers ℓ ≥ 1, so y = 0. This implies that Lie(G^{∨})^{φ=1} ⊂ Lie(G^{ét∨}), and the equation (7.6.1) is simplified as U · x^{(p)} = x. Since the linearization of φ_{G^{ét}} is bijective by 2.11, we have U ∈ GL_{r}(A). Let \overline{U} be the image of U in GL_{r}(k), and Sol be the solutions of the equation \overline{U} · x^{(p)} = x. As k is algebraically closed, Sol is an 𝔽_{p}-space of dimension r, and Lie(G^{ét∨}) ⊗ k is generated by Sol (cf. [Ka2, Prop. 4.1]). By the henselian property of A, every elements in Sol lifts uniquely to a solution of U ·x^{(p)} = x, i.e. the reduction map Lie(G^{∨})^{φ=1} ⟶^{∼} Sol is bijective. By Nakayama’s lemma, Lie(G^{∨})^{φ=1} generates the A-module Lie(G^{ét∨}). □

7.7. We keep the notations of 7.4. Let Comp_{\overline{K}_{0}} be the category of noetherian complete local \overline{K}_{0}-algebras with residue field \overline{K}_{0}, D_{𝒢_{\overline{K}_{0}}} (resp. D_{𝒢_{\overline{K}_{0}}^{∘}} ) be the functor which associates to every object A of Comp_{\overline{K}_{0}} the set of isomorphsm classes of deformations of 𝒢_{\overline{K}_{0}} (resp. 𝒢_{\overline{K}_{0}}^{∘}) . If A is an object in Comp_{\overline{K}_{0}} and G is a deformation of 𝒢_{\overline{K}_{0}} (resp. 𝒢_{\overline{K}_{0}}^{∘}) over A, we denote by [G] its isomorphic class in D_{𝒢_{\overline{K}_{0}}} (A) (resp. in D_{𝒢_{\overline{K}_{0}}^{∘}} ).

Lemma 7.8. Let Σ be the set defined in (7.4.3).

(i) The morphism of sets Φ : Σ → D_{𝒢_{\overline{K}_{0}}} (R̃′) given by σ ↦ [𝒢_{R̃′,σ}] is bijective. (ii) Let σ ∈ Σ. Then there exists a basis of Lie(𝒢_{R̃′,σ}^{∘∨}) such that φ_{𝒢_{R̃′,σ}^{∘}} is represented by a matrix of the form

(7.8.1)  𝔥_{σ}^{∘} = (\begin{matrix}0 & 0 & ⋯ & 0 & a_{1} \\ 1 & 0 & ⋯ & 0 & a_{2} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & a_{n−1} \end{matrix})

with a_{i} ≡ α · σ(t_{i}) (mod 𝔪2 ; R̃′) for 1 ≤ i ≤ n − 1, where α ∈ R̃′^{×} and 𝔪R̃′ is the maximal ideal of R̃′. In particular, 𝒢_{R̃′,σ}^{∘} is the universal deformation of 𝒢_{\overline{K}_{0}}^{∘}

if and only if {σ(t_{1}), ⋯, σ(t_{n−1})} is a system of regular parameters of R̃′.

Proof. (i) We begin with a remark on the Kodaira-Spencer map of 𝒢_{R′}. Let 𝒯_{S/k} = ℋ om _{𝒪_{S}} (Ω_{S/k}^{1}, 𝒪_{S}) be the tangent sheaf of S. Since G is universal, the Kodaira-Spencer map (3.2.2)

Kod : 𝒯_{S/k} ⟶^{∼} ℋ om _{𝒪_{S}} (ω_{G}, Lie(G^{∨}))

is an isomorphism. By functoriality, this induces an isomorphism of R′-modules

(7.8.2)  Kod_{R′} : T_{R′/k} ⟶^{∼} Hom_{R′}(ω_{𝒢_{R′}} , Lie(𝒢_{R′}^{∨})),

where T_{R′/k} = Hom_{R′}(Ω_{R′/k}^{1}, R′) = Γ(S, 𝒯_{S/k}) ⊗_{R} R′. For each integer ν ≥ 0, we put R̃′_{ν} = R̃′/𝔪ν+1 ; R̃′, Σ_{ν} to be the set of liftings of R → K_{0} → \overline{K}_{0} to R → R̃′_{ν}, and Φ_{ν} : Σ_{ν} → D_{𝒢_{\overline{K}_{0}}} (R̃′_{ν}) to be the morphism of sets σ_{ν} ↦ [𝒢_{R′} ⊗_{σ_{ν}} R̃′_{ν}]. We prove by induction on ν that Φ_{ν} is bijective for all ν ≥ 0. This will complete the proof of (i). For ν = 0, the claim holds trivially. Assume that it holds for ν − 1 with ν ≥ 1. We have a commutative diagram

\xymatrix{Σ_{ν} \ar[d] \ar[r]^{Φ_{ν}} & D_{𝒢_{\overline{K}_{0}}} (R̃′_{ν}) \ar[d] \\ Σ_{ν−1} \ar[r]^{Φ_{ν−1}} & D_{𝒢_{\overline{K}_{0}}} (R̃′_{ν−1}),}

where the vertical arrows are the canonical reductions, and the lower arrow is an isomorphism by induction hypothesis. Let τ be an arbitrary element of Σ_{ν−1}. We denote by Σ_{ν,τ} ⊂ Σ_{ν} the preimage of τ, and by D_{Φ_{ν−1}(τ)}(R̃′_{ν}) ⊂ D_{𝒢_{\overline{K}_{0}}} (R̃′_{ν}) the preimage of Φ_{ν−1}(τ). It suffices to prove that Φ_{ν} induces a

bijection between Σ_{ν,τ} and D_{Φ_{ν−1}(τ)}(R̃′_{ν}). Let I_{ν} = 𝔪ν ; R̃′/𝔪ν+1 ; R̃′ be the ideal of

the reduction map R̃′_{ν} → R̃′_{ν−1}. By [EGA, 0_{IV} 21.2.5 and 21.9.4], we have Ω_{R′/k}^{1} ≃ Ω̂_{R′/k}^{1}, and they are free over A of rank n. By [EGA, 0_{IV} 20.1.3], Σ_{ν,τ} is a (nonempty) homogenous space under the group

Hom_{K_{0}}(Ω_{R′/k}^{1} ⊗_{R′} K_{0}, I_{ν}) = T_{R′/k} ⊗_{R′} I_{ν}.

On the other hand, according to 3.5(i), D_{Φ_{ν−1}(τ)}(R̃′_{ν}) is a homogenous space under the group

Hom_{\overline{K}_{0}}(ω_{𝒢_{\overline{K}_{0}}} , Lie(𝒢_{\overline{K}_{0}}^{∨})) ⊗_{\overline{K}_{0}} I_{ν} = Hom_{R′}(ω_{𝒢_{R′}} , Lie(𝒢_{R′}^{∨})) ⊗_{R′} I_{ν}.

Moreover, it is easy to check that the morphism of sets Φ_{ν} : Σ_{ν,τ} → D_{Φ_{ν−1}(τ)}(R̃′_{ν}) is compatible with the homomorphism of groups

Kod_{R′} ⊗_{R′} Id : T_{R′/k} ⊗_{R′} I_{ν} → Hom_{R′}(ω_{𝒢_{R′}} , Lie(𝒢_{R′}^{∨})) ⊗_{R′} I_{ν},

where Kod_{R′} is the Kodaira-Spencer map (7.8.2) associated to 𝒢_{R′}. The bijectivity of Φ_{ν} now follows from the fact that Kod_{R′} is an isomorphism. (ii) The second part of the statement follows immediately from 4.11. It remains to compute the Hasse-Witt map of 𝒢_{R̃′,σ}^{∘}. We determine first the submodule

Lie(𝒢_{R̃′,σ}^{ét∨}) of Lie(𝒢_{R̃′,σ}^{∨}). We choose a basis of Lie(G^{∨}) over 𝒪_{S} such that φ_{G} is expressed by the matrix 𝔥 (7.4.1). As 𝒢_{R̃′,σ} derives from G by base change

R → R′ ⟶^{σ} R̃′, there exists a basis (e_{1}, ⋯, e_{n}) of Lie(𝒢_{R̃′,σ}^{∨}) such that φ_{𝒢_{R̃′,σ}} is expressed by

𝔥^{σ} = (\begin{matrix}0 & 0 & ⋯ & 0 & −σ(t_{1}) \\ 1 & 0 & ⋯ & 0 & −σ(t_{2}) \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & −σ(t_{n}) \end{matrix}) .

By Lemma 7.6, Lie(𝒢_{R̃′,σ}^{ét∨}) is generated by Lie(𝒢_{R̃′,σ}^{∨})^{φ=1}. If ∑_{i=1}^{n} x_{n}e_{n} ∈

Lie(𝒢_{R̃′,σ}^{∨})^{φ=1} with x_{i} ∈ R̃′ for 1 ≤ i ≤ n, then (x_{i})_{1≤i≤n} must satisfy the equation 𝔥^{σ} · (x_{1}^{p} ; ⋮ ; x_{n}^{p}) = (x_{1} ; ⋮ ; x_{n}); or equivalently,

(7.8.3)  \begin{cases}x_{1} = −σ(t_{1})x_{n}^{p} \\ x_{2} = −σ(t_{2})x_{n}^{p} − σ(t_{1})^{p}x_{n}^{p^{2}} \\ ⋯ \\ x_{n−1} = −σ(t_{n−1})x_{n}^{p} − ⋯ − σ(t_{1})^{p^{n−2}}x_{n}^{p^{n−1}} \\ σ(t_{1})^{p^{n−1}}x_{n}^{p^{n}} + σ(t_{2})^{p^{n−2}}x_{n}^{p^{n−1}} + ⋯ + σ(t_{n})x_{n}^{p} + x_{n} = 0. \end{cases}

We note that σ(t_{i}) ∈ 𝔪R̃′ for 1 ≤ i ≤ n − 1 and σ(t_{n}) ∈ R̃′^{×} with image i(t_{n}) ∈ \overline{K}_{0}, where i : K_{0} → \overline{K}_{0} is the fixed immbedding. By Hensel’s lemma, every solution in \overline{K}_{0} of the equation i(t_{n})x_{n}^{p}+x_{n} = 0 lifts uniquely to a solution of (7.8.3). As Lie(𝒢_{R̃′,σ}^{ét∨}) has rank 1, by Lemma 7.6, these are all the solutions of (7.8.3). Let (λ_{1}, ⋯, λ_{n}) be a non-zero solution of (7.8.3). We have

(7.8.4)  λ_{n} ∈ R̃′^{×} and λ_{i} ≡ −λ_{n}^{p}σ(t_{i}) (mod 𝔪2 ; R̃′).

We put v = λ_{1}e_{1} + ⋯ + λ_{n}e_{n}; so v is a basis of Lie(𝒢_{R̃′,σ}^{ét∨}) by 7.6. For 1 ≤ i ≤ n, let f_{i} be the image of e_{i} in Lie(𝒢_{R̃′,σ}^{∘∨}). Then f_{1}, ⋯, f_{n} clearly generate Lie(𝒢_{R̃′,σ}^{∘∨}). By the explicit description above of Lie(𝒢_{R̃′,σ}^{ét∨}), we have f_{n} = −λ_{n}^{−1}(λ_{1}f_{1} ⋯+λ_{n−1}f_{n−1}). Hence f_{1}, ⋯, f_{n−1} form a basis of Lie(𝒢_{R̃′,σ}^{∘∨}). By the functoriality of Hasse-Witt maps, we have φ_{𝒢 ∘ ; R̃′} (f_{i}) = f_{i+1} for 1 ≤ i ≤ n − 1, or equivalently,

φ_{𝒢_{R̃′,σ}^{∘}} (f_{1}, ⋯, f_{n−1}) = (f_{1}, ⋯, f_{n−1}) · (\begin{matrix}0 & 0 & ⋯ & 0 & −λ_{n}^{−1}λ_{1} \\ 1 & 0 & ⋯ & 0 & −λ_{n}^{−1}λ_{2} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & −λ_{n}^{−1}λ_{n−1} \end{matrix}) .

In view of (7.8.4), we see that the above matrix has the form of (7.8.1) by setting α = λ_{n}^{p−1} ∈ R̃′^{×}. The second part of statement (ii) follows immediately from Proposition 4.11(ii) and the description above of φ_{𝒢_{R̃′,σ}^{∘}} . □

Now we can turn to the proof of 7.5.

7.9. Proof of Lemma 7.5. First, suppose that we have found a σ_{2} ∈ Σ such that \overline{C}_{σ_{2}} ≠ 0 and 𝒢_{R̃′,σ_{2}}^{∘} is the universal deformation of 𝒢_{\overline{K}_{0}}^{∘}. Since Φ : Σ ⟶^{∼} D_{𝒢_{\overline{K}_{0}}} (R̃′) is bijective by 7.8(i), there exists a σ_{1} ∈ Σ corresponding to the deformation [𝒢_{R̃′,σ_{2}}^{∘} ⊕ ℚ_{p}/ℤ_{p}] ∈ D_{𝒢_{\overline{K}_{0}}} (R̃′). It is clear that 𝒢_{R̃′,σ_{1}}^{∘} ≃ 𝒢_{R̃′,σ_{2}}^{∘}. Besides, the exact sequence (7.4.5) for σ_{1} splits; so we have C_{σ_{1}} = 0. It remains to prove the existence of σ_{2}. We note first that \overline{K}_{0} can be canonically imbedded into R̃′, since it is perfect. Since R′ is formally smooth over k and (t_{1}, ⋯, t_{n}) is a p-basis of R′ over k, by [EGA, 0_{IV} 21.2.7], there is a σ ∈ Σ such that σ(t_{i}) (1 ≤ i ≤ n − 1) form a system of regular parameters of R̃′ and σ(t_{n}) ∈ \overline{K}_{0} ⊂ R̃′. We claim that σ_{2} = σ answers the question. In fact, Lemma 7.8(ii) implies that 𝒢_{R̃′,σ}^{∘} is the universal deformation of 𝒢_{\overline{K}_{0}}^{∘}. It remains to

verify that \overline{C}_{σ} ≠ 0. Let A = \overline{K}_{0}[[π]] be a complete discrete valuation ring of characteristic p with residue field \overline{K}_{0}, T = Spec(A), ξ be the generic point of T, \overline{ξ} be a geometric over ξ, and I = Gal(\overline{ξ}/ξ) the Galois group. We define a homomorphism of \overline{K}_{0}-algebras f^{∗} : R̃′ → A by putting f^{∗}(σ(t_{1})) = π and f^{∗}(σ(t_{i})) = 0 for 2 ≤ i ≤ n − 1. This is possible, since (σ(t_{1}), ⋯, σ(t_{n−1})) is a system of regular parameters of R̃′. Let f : T → S̃′ be the homomorphism of schemes corresponding to f^{∗}, and 𝒢_{T} = 𝒢_{R̃′,σ} ×S̃′ T. By the functoriality of Hasse-Witt maps,

𝔥_{T} = (\begin{matrix}0 & 0 & ⋯ & 0 & −π \\ 1 & 0 & ⋯ & 0 & 0 \\ 0 & 1 & ⋯ & 0 & 0 \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & −f^{∗}(σ(t_{n})) \end{matrix}) ∈ M_{n×n}(R̃′)

is a matrix of φ_{𝒢_{T}} . By definition (5.4), the Hasse invariant of 𝒢_{T} is h(𝒢_{T}) = 1. In particular, 𝒢_{T} is generically ordinary. Let Ũ′_{σ} ⊂ S̃′ be the ordinary locus of 𝒢_{R̃′,σ}. We have f(ξ) ∈ Ũ′_{σ}. By the functoriality of fundamental groups, f induces a homomorphism of groups

π_{1}(f) : I = Gal(\overline{ξ}/ξ) → π_{1}(Ũ′_{σ}, f(\overline{ξ})) ≃ π_{1}(Ũ′_{σ}, \overline{x}).

Let 𝒢_{T}^{∘} be the connected part of 𝒢_{T}, and 𝒢_{T}^{ét} be the étale part of 𝒢_{T}. Then 𝒢_{T}^{ét} ≃ ℚ_{p}/ℤ_{p}. We have an exact sequence of 𝔽_{p}[I]-modules

0 → 𝒢_{T}^{∘}(1)(\overline{ξ}) → 𝒢_{T} (1)(\overline{ξ}) → 𝒢_{T}^{ét}(1)(\overline{ξ}) → 0,

which determines a cohomology class \overline{C}_{T} ∈ H^{1}(I, 𝒢_{T}^{∘}(1)(\overline{ξ})). We notice that 𝒢_{T} (1)(\overline{ξ}) is isomorphic to 𝒢_{R̃′,σ}(1)(\overline{x}) as an abelian group, and the action of I

on 𝒢_{T} (1)(\overline{ξ}) is induced by the action of π_{1}(Ũ′_{σ}, \overline{x}) on 𝒢_{R̃′,σ}(1)(\overline{x}). Therefore, \overline{C}_{T} is the image of \overline{C}_{σ} by the functorial map

H^{1}(π_{1}(Ũ′_{σ}, \overline{x}), 𝒢_{R̃′,σ}^{∘}(1)(\overline{x})) → H^{1}(I, 𝒢_{T}^{∘}(1)(\overline{ξ})).

To verify that \overline{C}_{σ} ≠ 0, it suffices to check that \overline{C}_{T} ≠ 0. We consider the polynomial P (X) = X^{p^{n}} + f^{∗}(σ(t_{n}))X^{p^{n−1}} + πX ∈ A[X]. According to 5.12, it suffices to find a α ∈ \overline{K}_{0} ⊂ A such that P (α) is a uniformizer of A. But by the choice of σ, we have σ(t_{n}) ∈ \overline{K}_{0} and σ(t_{n}) ≠ 0; so f^{∗}(σ(t_{n})) ≠ 0 lies in \overline{K}_{0}. Let α be a p^{n−1}(p − 1)-th root of −f^{∗}(σ(t_{n})) in \overline{K}_{0}. Then we have α ∈ \overline{K}_{0}^{×}, and P (α) = απ is a uniformizer of A. This completes the proof of 7.5.

8. End of the Proof of Theorem 1.3

In this section, k denotes an algebraically closed field of characteristic p > 0.

8.1. First, we recall some preliminaries on Newton stratification due to F. Oort. Let G be an arbitrary BT-group over k, S be the local moduli of G in characteristic p, and G be the universal deformation of G over S (3.8). Put d = dim(G) and c = dim(G^{∨}). We denote by N (G) the Newton polygon of G which has endpoints (0, 0) and (c + d, d). Here we use the normalization of Newton polygons such that slope 0 corresponds to étale BT- groups and slope 1 corresponds to groups of multiplicative type. Let N P(c + d, d) be the set of Newton polygons with endpoints (0, 0) and (c + d, d) and slopes in (0, 1). For α, β ∈ N P(c + d, d), we say that α ⪯ β if no point of α lies below β; then “⪯” is a partial order on N P(c + d, d). For each β ∈ N P(c + d, d), we denote by V_{β} the subset of S consisting of points x with N (G_{x}) ⪯ β, and by V_{β}^{∘} the subset of S consisting of points x with N (G_{x}) = β. By Grothendieck-Katz’s specialization theorem of Newton polygons, V_{β} is closed in S, and V_{β}^{∘} is open (maybe empty) in V_{β}. We put

♢(β) = {(x, y) ∈ ℤ×ℤ | 0 ≤ y < d, y < x < c+d, (x, y)lies on or above the polygon β},

and dim(β) = #(♢(β)).

Theorem 8.2 ([Oo2] Theorem 2.11). Under the above assumptions, for each β ∈ N P(c + d, d), the subset V_{β}^{∘} is non-empty if and only if N (G) ⪯ β. In that case, V_{β} is the closure of V_{β}^{∘} and all irreducible components of V_{β} have dimension dim(β).

8.3. Let G be a connected and HW-cyclic BT-group over k of dimension d = dim(G) ≥ 2. Let β ∈ N P(c + d, d) be the Newton polygon given by the following slope sequence:

β = (\underbrace{1/(c + 1), ⋯, 1/(c + 1)}_{c+1}, \underbrace{1, ⋯, 1}_{d−1}).

We have N (G) ⪯ β since G is supposed to be connected. By Oort’s Theorem 8.2, V_{β} is a equal dimensional closed subset of the local moduli S of dimension c(d − 1). We endow V_{β} with the structure of a reduced closed subscheme of S.

Lemma 8.4. Under the above assumptions, let R be the ring of S, and

(\begin{matrix}0 & 0 & ⋯ & 0 & −a_{1} \\ 1 & 0 & ⋯ & 0 & −a_{2} \\ 0 & 1 & ⋯ & 0 & −a_{3} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & −a_{c} \end{matrix}) ∈ M_{c×c}(R)

be a matrix of the Hasse-Witt map φ_{G}. Then the closed reduced subscheme V_{β} of S is defined by the prime ideal (a_{1}, ⋯, a_{c}). In particular, V_{β} is irreducible.

Proof. Note first that {a_{1}, ⋯, a_{c}} is a subset of a system of regular parameters of R by 4.11(i). Let I be the ideal of R defining V_{β}. Let x be an arbitrary point of V_{β}, we denote by 𝔭_{x} the prime ideal of R corresponding to x. Since the Newton polygon of the fibre G_{x} lies above β, G_{x} is connected. By Lemma 4.4, we have a_{i} ∈ 𝔭_{x} for 1 ≤ i ≤ c. Since V_{β} is reduced, we have a_{i} ∈ I. Let 𝔓 = (a_{1}, ⋯, a_{c}), and V (𝔓) the closed subscheme of S defined by 𝔓. Then V (𝔓) is an integral scheme of dimension c(d − 1) and V_{β} ⊂ V (𝔓). Since Theorem 8.2 implies that dim V_{β} = c(d−1), we have necessarily V_{β} = V (𝔓). □

We keep the assumptions above. Let (t_{i,j})_{1≤i≤c,1≤j≤d} be a regular system of parameters of R such that t_{i,d} = a_{i} for all 1 ≤ i ≤ c. Let x be the generic point of the Newton strata V_{β}, k′ = κ(x), and R′ = 𝒪̂_{S,x} . Since R is noetherian and integral, the canonical ring homomorphism R → 𝒪_{S,x} → R′ is injective. The image in R′ of an element a ∈ R will be denoted also by a. By choosing a k-section k′ → R′ of the canonical projection R′ → k′, we get a (non-canonical) isomorphism of k-algebras R′ ≃ k′[[t_{1,d}, ⋯, t_{c,d}]]. Let k′′ be an algebraic closure of k′, and R′′ = k′′[[t_{1,d}, ⋯, t_{c,d}]]. Then we have a natural injective homomorphism of k-algebras R′ → R′′ mapping t_{i,d} to t_{i,d} for 1 ≤ i ≤ c. Let S′′ = Spec(R′′), \overline{x} be its closed point. By the construction of S′′, we have a morphism of k-schemes

(8.4.1)  f : S′′ → S

sending \overline{x} to x. We put 𝒢 = G ×_{S} S′′. By the choice of the Newton polygon β, the closed fibre 𝒢\overline{x} has a BT-subgroup ℋ\overline{x} of multiplicative type of height d−1. Since S′′ is henselian, ℋ\overline{x} lifts uniquely to a BT-subgroup ℋ of 𝒢. We put 𝒢′′ = 𝒢 /ℋ . It is a connected BT-group over S′′ of dimension 1 and height c+1.

Lemma 8.5. Under the above assumptions, 𝒢′′ is the universal deformation in equal characteristic of its special fiber.

This lemma is a particular case of [Lau, Lemma 3.1]. Here, we use 4.11(ii) to give a simpler proof.

Proof. We have an exact sequence of BT-groups over S′′

0 → ℋ → 𝒢 → 𝒢′′ → 0,

which induces an exact sequence of Lie algebras 0 → Lie(𝒢^{′′∨}) → Lie(𝒢^{∨}) → Lie(ℋ ^{∨}) → 0 compatible with Hasse-Witt maps. Since ℋ is of multiplicative type, we get Lie(ℋ ^{∨}) = 0 and an isomorphism of Lie algebras Lie(𝒢^{′′∨}) ≃ Lie(𝒢^{∨}). By the choice of the regular system (t_{i,j})_{1≤i≤c,1≤j≤d}, there is a basis (v_{1}, ⋯, v_{c}) of Lie(𝒢^{′′∨}) over 𝒪_{S′′} such that φ_{𝒢′′} is given by the matrix

𝔥 = (\begin{matrix}0 & 0 & ⋯ & 0 & −t_{1,d} \\ 1 & 0 & ⋯ & 0 & −t_{2,d} \\ 0 & 1 & ⋯ & 0 & −t_{3,d} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & −t_{c,d} \end{matrix}) .

Now the lemma results from Proposition 4.11(ii). □

8.6. Proof of Theorem 1.3. The one-dimensional case is treated in 7.3. If dim(G) ≥ 2, we apply the preceding discussion to obtain the morphism f: S′′ → S and the BT-groups 𝒢 = G ×_{S} S′′ and 𝒢′′, which is the quotient of 𝒢 by the maximal subgroup of 𝒢 of multiplicative type. Let U′′ be the common ordinary locus of 𝒢 and 𝒢′′ over S′′, and \overline{ξ} be a geometric point of U′′. Then f maps U′′ into the ordinary locus U of G. We denote by

ρ_{𝒢} : π_{1}(U′′, \overline{ξ}) → Aut_{ℤ_{p}}(T_{p}(𝒢 , \overline{ξ}))

the monodromy representation associated to 𝒢, and the same notation for ρ_{𝒢′′} . By the functoriality of monodromy, we have Im(ρ_{𝒢} ) ⊂ Im(ρ_{G}). On the other hand, the canonical map 𝒢 → 𝒢′′ induces an isomorphism of Tate modules T_{p}(𝒢 , \overline{η}) ⟶^{∼} T_{p}(𝒢′′, \overline{η}) compatible with the action of π_{1}(U′′, \overline{η}). Therefore, the group Im(ρ_{𝒢} ) is identified with Im(ρ_{𝒢′′} ). Since 𝒢′′ is one-dimensional, we conclude the proof by Lemma 8.5 and Theorem 7.3.

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Yichao Tian

Department of Mathematics Princeton University Princeton New Jersey 08544 USA yichaot@princeton.edu