CANONICAL SUBGROUPS VIA BREUIL-KISIN MODULES
FOR p = 2
SHIN HATTORI
Abstract. Let p be a rational prime and K/Qp be an extension ofcomplete discrete valuation fields. Let G be a truncated Barsotti-Tategroup of level n, height h and dimension d over OK with 0 < d < h.
In this paper, we prove the existence of higher canonical subgroups
for G with standard properties if the Hodge height of G is less than
1/(pn−2 (p + 1)), including the case of p = 2.

1. Introduction
Let p be a rational prime and K/Qp be an extension of complete discrete
valuation ﬁelds. Let k be its residue ﬁeld, π be its uniformizer, e be its
absolute ramiﬁcation index, K̄ be its algebraic closure and vp be its valuation
extended to K̄ and normalized as vp (p) = 1. We let C denote the completionof K̄. For any valuation ﬁeld F (of height one) with valuation vF and≥ivaluation ring OF , put m≥i
F = {x ∈ F | vF (x) ≥ i} and OF,i = OF /mF
for any positive real number i. We also put ÕK = OK,1 , ÕK̄ = OK̄,1 and
Si = Spec(OK,i ).
One of the key ingredients of the theory of p-adic Siegel modular forms
is the existence theorem of canonical subgroups. Let X be the formal completion of the Siegel modular variety of genus g and level prime to p over
the Witt ring W (k) along the special ﬁber, X be its Raynaud generic ﬁber,
X ord be its ordinary locus considered as an admissible open subset of X and
A be the universal abelian scheme over X. Consider the unit component
A[pn ]0 of the pn -torsion of A and its Raynaud generic ﬁber (A[pn ]0 )rig . The
restriction (A[pn ]0 )rig |X ord is etale locally isomorphic to the constant group(Z/pn Z)g and it is a lift of the kernel of the n-th iterated Frobenius of thespecial ﬁber of A. Then the theorem asserts that this subgroup can be extended to a subgroup Cn with the same properties over a larger admissible
open subset of X containing X ord . In [9], the author proved the existence ofsuch a subgroup over the locus of the Hodge height less than 1/(2pn−1 ) forp ≥ 3. The aim of this paper is to generalize the result to the case of p = 2.
To state the main theorem, we introduce some notation. For any ﬁnite ﬂat(commutative) group scheme G (resp. Barsotti-Tate group Γ) over OK , welet ωG (resp. ωΓ ) denote its module of invariant diﬀerentials. Put deg(G) =Date: July 26, 2012.
Supported by Grant-in-Aid for Young Scientists B-23740025.
1

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SHIN HATTORI∑

i vp (ai ) by writing ωG ≃ ⊕i OK /(ai ). For any positive rational number i,the Hodge-Tate map for a ﬁnite ﬂat group scheme G over OK killed by pn
is deﬁned to be the natural homomorphism

HTi : G(OK̄ ) ≃ Hom(G ∨ × Spec(OK̄ ), µpn × Spec(OK̄ )) → ωG ∨ ⊗ OK̄,ideﬁned by g 7→ g ∗ (dT /T ), where ∨ means the Cartier dual and µpn =nSpec(OK [T ]/(T p − 1)) is the group scheme of pn -th roots of unity. We
normalize the upper and the lower ramiﬁcation subgroups of G to be adapted
to the valuation vp . Namely, writing the aﬃne algebra of G as
OK [T1 , . . . , Tr ]/(f1 , . . . , fs )r as x, we put
and an r-tuple (x1 , . . . , xr ) ∈ OK̄
r
G j (OK̄ ) = G(OK̄ ) ∩ {x ∈ OK̄| vp (fl (x)) ≥ j for any l}0 ,Gi (OK̄ ) = Ker(G(OK̄ ) → G(OK̄,i )),where (−)0 in the ﬁrst equality means the geometric connected component
as an aﬃnoid variety over K containing the zero section (see [1, Section′2]). We also put G j+ (OK̄ ) = ∪j ′ >j G j (OK̄ ) for any non-negative rational
number j. The scheme-theoretic closure of G j (OK̄ ) in G is denoted by G j
and deﬁne G j+ and Gi similarly. Finally, for any truncated Barsotti-Tategroup G ([10]) of level n, height h and dimension d over OK with d < h, wedeﬁne the Hodge height Hdg(G) to be the truncated valuation vp (det(V )) ∈[0, 1] of the determinant of the natural action of the Verschiebung V of
the group scheme G[p]∨ × Spec(ÕK ) on the free ÕK -module of ﬁnite rank
Lie(G[p]∨ × Spec(ÕK )). Then our main theorem is the following, which isproved in [9] except the case of p = 2.
Theorem 1.1. Let p be a rational prime and K/Qp be an extension ofcomplete discrete valuation fields. Let G be a truncated Barsotti-Tate groupof level n, height h and dimension d over OK with 0 < d < h and Hodgeheight w = Hdg(G).
(1) If w < 1/(pn−2 (p + 1)), then there exists a finite flat closed subgroupscheme Cn of G of order pnd over OK , which we call the level n canonical subgroup of G, such that Cn × S1−pn−1 w coincides with the kernelof the n-th iterated Frobenius homomorphism F n of G × S1−pn−1 w .
Moreover, the group scheme Cn has the following properties:(a) deg(G/Cn ) = w(pn − 1)/(p − 1).
(b) Put Cn′ to be the level n canonical subgroup of G ∨ . Then we havethe equality of subgroup schemes Cn′ = (G/Cn )∨ , or equivalently
Cn (OK̄ ) = Cn′ (OK̄ )⊥ , where ⊥ means the orthogonal subgroupwith respect to the Cartier pairing.
(c) If n = 1, then C1 = G(1−w)/(p−1) = G pw/(p−1)+ .
(2) If w < (p − 1)/(pn − 1), then the subgroup scheme Cn also satisfies
the following:(d) the group Cn (OK̄ ) is isomorphic to (Z/pn Z)d .

CANONICAL SUBGROUPS FOR p = 23(e) The scheme-theoretic closure of Cn (OK̄ )[pi ] in Cn coincides withthe subgroup scheme Ci of G[pi ] for 1 ≤ i ≤ n − 1.
(3) If w < (p − 1)/pn , then the subgroup Cn (OK̄ ) coincides with thekernel of the Hodge-Tate map HTn−w(pn −1)/(p−1) .
(4) If w < 1/(2pn−1 ), then the subgroup scheme Cn coincides with theupper ramification subgroup scheme G j+ for any j satisfying
pw(pn − 1)/(p − 1)2 ≤ j < p(1 − w)/(p − 1).
We also show the uniqueness of the canonical subgroup Cn for w <p(p − 1)/(pn+1 − 1) (Proposition 3.8). From Theorem 1.1, we can show
the following corollary just as in the proof of [9, Corollary 1.2].
Corollary 1.2. Let K/Qp be an extension of complete discrete valuation
fields. Let X be an admissible formal scheme over Spf(OK ) which is quasicompact and G be a truncated Barsotti-Tate group of level n over X of constant height h and dimension d with 0 < d < h. We let X and G denote theRaynaud generic fibers of the formal schemes X and G, respectively. For a
finite extension L/K and x ∈ X(L), we put Gx = G ×X,x Spf(OL ), where
we let x also denote the map Spf(OL ) → X obtained from x by taking thescheme-theoretic closure and the normalization. For a non-negative rational
number r, let X(r) be the admissible open subset of X defined byX(r)(K̄) = {x ∈ X(K̄) | Hdg(Gx ) < r}.
Put r1 = p/(p + 1) and rn = 1/(2pn−1 ) for n ≥ 2.
Then there exists an admissible open subgroup Cn of G|X(rn ) such that,
etale locally on X(rn ), the rigid-analytic group Cn is isomorphic to the constant group (Z/pn Z)d and, for any finite extension L/K and x ∈ X(L), thefiber (Cn )x coincides with the generic fiber of the level n canonical subgroupof Gx .
The basic strategy of the proof of the main theorem is the same as in[9]: we construct the level one canonical subgroup by lifting the conjugate
Hodge ﬁltration of a reduction of G to the subobject of the associated BreuilKisin module of G. Since the canonical subgroup is required to be a lift of
the Frobenius kernel of a reduction of G, it should be connected. Thus
what we need is just a classiﬁcation of connected ﬁnite ﬂat group schemes
allowing the case of p = 2, which is due to Kisin ([14]). Then the propertiesof our canonical subgroup follow easily from the construction, except thecoincidence with ramiﬁcation subgroups.
In [9], to show that the canonical subgroups appear in the ramiﬁcation ﬁltrations of G, we use the ramiﬁcation correspondence theorem ([8, Theorem1.1]), which is shown only for p ≥ 3. Instead, we prove the compatibility of
the canonical subgroups with any ﬁnite base extension and reduce ourselvesto the case of G(OK̄ ) = G(OK ), where we can easily show that the lower
ramiﬁcation subgroups of G are computed on the side of equal characteristic.

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SHIN HATTORI

2. Classification of unipotent finite flat group schemes
In this section, we assume that the residue ﬁeld k of K is perfect. For p ≥3, we have a classiﬁcation theory of Barsotti-Tate groups and ﬁnite ﬂat groupschemes over OK due to Breuil ([3], [4]) and Kisin ([12], [13]) in terms of socalled Breuil-Kisin modules. Kisin ([14]) also extended this classiﬁcation to
the case of p = 2 and where groups are connected, using Zink’s classiﬁcationof formal Barsotti-Tate groups ([20], [21]). In this section, we brieﬂy recall
this result of Kisin. Since we adopt a contravariant notation contrary to his,what we describe here is a classiﬁcation of unipotent Barsotti-Tate groups
and unipotent ﬁnite ﬂat group schemes.
Let W = W (k) be the Witt ring of k and φ be its natural Frobenius
endomorphism which lifts the p-th power map of k. Natural φ-semilinear
Frobenius endomorphisms of various W -algebras are denoted also by φ. Let
E(u) ∈ W [u] be the Eisenstein polynomial of π over W . Let us ﬁx once
and for all a system {πn }n≥0 of p-power roots of π in K̄ with π0 = πp= πn . Put K∞ = ∪n≥0 K(πn ), S = W [[u]] and S1 = k[[u]].
and πn+1
We write the φ-semilinear continuous ring endomorphisms of the latter two
rings deﬁned by u 7→ up also as φ. Then a Kisin module is an S-module
M endowed with a φ-semilinear map φM : M → M. We write φM also as
φ if no confusion may occur. We follow the notation of [8, Subsection 2.1]
1,φand [9, Subsection 2.1]. In particular, we have categories Mod1,φ/S , Mod/S1 ,1,φMod1,φ/S∞ of Kisin modules of E-height ≤ 1 and a category Mod/B for any
k[[u]]-algebra B.
∗
Let M be an object of the category Mod1,φ/S∞ and put φ M = S ⊗φ,S M.
Then the map 1 ⊗ φ : φ∗ M → M is injective and we have a unique map
ψM : M → φ∗ M satisfying (1 ⊗ φ) ◦ ψM = E(u). We say M is V -nilpotent
if the composite

φn−1∗ (ψM ) ◦ φn−2∗ (ψM ) ◦ · · · ◦ ψM : M → φn∗ M
factors through the submodule (p, u)φn∗ M for any suﬃciently large n. Similarly, we say an object M of the category Mod1,φ/S is topologically V -nilpotent
if the same condition holds. The full subcategories of V -nilpotent (resp.
1,φ,V
topologically V -nilpotent) objects are denoted by Mod1,φ,V/S1 and Mod/S∞(resp. Mod1,φ,V/S ). Note that these notions are called connected and formalin [14], respectively.
Let S be the p-adic completion of the divided power envelope of W [u]with respect to the ideal (E(u)). The ring S has a natural ﬁltration Fil1 S
induced by the divided power structure, a φ-semilinear Frobenius endomorphism denoted also by φ and a φ-semilinear map φ1 : Fil1 S → S, since theinclusion φ(Fil1 S) ⊆ pS holds for any p. Then a Breuil module is an Smodule M endowed with an S-submodule Fil1 M containing (Fil1 S)M anda φ-semilinear map φ1,M : Fil1 M → M satisfying some compatibility conditions. The map φ1,M is also denoted by φ1 if there is no risk of confusion.

CANONICAL SUBGROUPS FOR p = 2

5

1,φ
1,φ
We also have categories of Breuil modules Mod1,φ/S , Mod/S1 and Mod/S∞(for the deﬁnitions, see [8, Subsection 2.1]. Though there these are deﬁned
only for p ≥ 3, the deﬁnitions are valid also for the case of p = 2). For any
object M of these categories, we deﬁne a φ-semilinear map φM : M → M
by φM (x) = φ1 (E(u))−1 φ1 (E(u)x), which we abusively write as φ.
∗
Let M be an object of the category Mod1,φ/S and put φ M = S ⊗φ,S M.
Then the map 1 ⊗ φ : φ∗ M → M is injective and we have a unique map
ψM : M → φ∗ M satisfying (1⊗φ)◦ψM = p. Then we say M is topologicallyV -nilpotent if the composite

φn−1∗ (ψM ) ◦ φn−2∗ (ψM ) ◦ · · · ◦ ψM : M → φn∗ M
factors through the submodule (p, Fil1 S)φn∗ M for any suﬃciently large n.
This notion is called S-window over OK in [14] and [20]. The full subcategoryof topologically V -nilpotent objects is denoted by Mod1,φ,V. For any Kisin/S
module M, deﬁne a Breuil module MS (M) = S ⊗φ,S M by putting1⊗φFil1 MS (M) = Ker(S ⊗φ,S M → S/Fil1 S ⊗S M),
1⊗φ

φ1 ⊗1

φ1 : Fil1 MS (M) → Fil1 S ⊗S M → MS (M).
1,φ
1,φ
1,φ
This gives exact functors Mod1,φ/S∞ → Mod/S∞ and Mod/S → Mod/S , which

are both denoted by MS (−), and the latter induces a functor Mod1,φ,V→/S([14, Proposition 1.2.5]).
Mod1,φ,V/S
We can associate Galois representations to Kisin and Breuil modules.
Consider the ring R = lim(ÕK̄ ← ÕK̄ ← · · · ), where the transition maps←−
are p-th power maps. An element r ∈ R is written as r = (rn )n≥0 withnrn ∈ ÕK̄ , and deﬁne r(0) ∈ OC by r(0) = limn→∞ r̂np , where r̂n is a lift of rn
in OK̄ . Then the ring R is a complete valuation ring of characteristic p with
its valuation deﬁned by vR (r) = vp (r(0) ) whose fraction ﬁeld is algebraically≥iclosed, and we put m≥i
R = {r ∈ R | vR (r) ≥ i} and Ri = R/mR . We
have a natural ring surjection W (R) → OC which lifts the zeroth projection
pr0 : R → ÕK̄ . The ring Acrys is the p-adic completion of the divided powerenvelope of W (R) with respect to the kernel of this surjection. Thus wehave the induced surjection Acrys → OC . Put π = (π, π1 , π2 , . . .) ∈ R andconsider the rings W (R) and Acrys as S-algebras by the map u 7→ [π]. In
particular, we consider the ring k[[u]] as a subring of R by the map u 7→ π
and let vR also denote the induced valuation on the former ring, which
satisﬁes vR (u) = 1/e.
1,φFor any objects M of the category Mod1,φ/S and M of Mod/S , we associate
to them GK∞ -modulesTS∗ (M) = HomS,φ (M, W (R)),∗
Tcrys(M) = HomS,Fil1 ,φ (M, Acrys )6SHIN HATTORI([6, Proposition B1.8.3] and [14, Subsection 1.2.6]). If the S-module M is
free of rank h, then the Zp -module TS∗ (M) is also free of rank h ([12, Corollary 2.1.4]). We also have a natural injection of GK∞ -modules TS∗ (M) →∗ (M (M)) deﬁned by f 7→ 1 ⊗ (φ ◦ f ) and this is a bijection if M is
Tcrys
S
topologically V -nilpotent ([14, Proposition 1.2.7]). Similarly, for any object
M of the category Mod1,φ/S∞ , we have the associated GK∞ -moduleTS∗ (M) = HomS,φ (M, W (R) ⊗ Qp /Zp ).
Put Sn = Spec(OK,n ), Sn = S/pn S and En = Spec(Sn ). Let us consider
the big crystalline site CRYS(Sn /En ) with the fppf topology and its topos(Sn /En )CRYS . For any Barsotti-Tate group Γ over OK , we have the contravariant Dieudonné crystal D∗ (Γ × Sn ) = Ext 1Sn /En (Γ × Sn , OSn /En ) (forthe notation, see [2]). We put
D∗ (Γ)(S → OK ) = lim D∗ (Γ × Sn )(Sn → OK,n ).
←−nThis module is considered as an object Mod(Γ) of the category Mod1,φ/S withthe natural φ-semilinear Frobenius map induced by the Frobenius of Γ × S1
and the ﬁltration deﬁned as the inverse image of the natural inclusionωΓ ⊆ lim D∗ (Γ × Sn )(OK,n → OK,n ).
←−n

The Acrys -moduleD∗ (Γ)(Acrys → OC ) = lim D∗ (Γ × Sn )(Acrys /pn Acrys → OC,n )←−nalso has a φ-semilinear Frobenius map and a ﬁltration deﬁned in the sameway. Similarly, for any ﬁnite ﬂat group scheme G over OK , the S-moduleD∗ (G)(S → OK ) = lim D∗ (G × Sn )(Sn → OK,n )←−nis endowed with a natural φ-semilinear Frobenius map which is induced by
the Frobenius of the group scheme G × S1 and is also denoted by φ, and aﬁltration deﬁned by the submoduleFil1 D∗ (G)(S → OK ) = lim Ext 1Sn /En (G × Sn , JSn /En )(Sn → OK,n ),←−nwhere JSn /En is the canonical divided power ideal sheaf of the structure
sheaf OSn /En .
We say a Barsotti-Tate group or a ﬁnite locally free group scheme is unipotent if its Cartier dual is connected. We let (BT/OK )u (resp. (p-Gr/OK )u )denote the category of unipotent Barsotti-Tate groups (resp. the categoryof unipotent ﬁnite ﬂat group schemes killed by some p-power) over OK . If a
Barsotti-Tate group Γ over OK is unipotent, then the object Mod(Γ) is topologically V -nilpotent ([14, Lemma 1.1.3]). Moreover, we have the followingclassiﬁcation theorem of unipotent Barsotti-Tate groups and unipotent ﬁnite
ﬂat group schemes, whose second assertion follows from the ﬁrst assertionby an argument of taking a resolution ([14, Subsection 1.3]).

CANONICAL SUBGROUPS FOR p = 2

7

Theorem 2.1.
(1) ([14], Theorem 1.2.8) There exists an anti-equivalenceof exact categories
G(−) : Mod1,φ,V→ (BT/OK )u/S
with a natural isomorphism Mod(G(M)) → MS (M). Moreover, we
also have a natural isomorphism of GK∞ -modules∗
εM : Tp (G(M)) → Tcrys(MS (M)).

(2) ([14], Theorem 1.3.9) There exists an anti-equivalence of exact categoriesuG(−) : Mod1,φ,V/S∞ → (p-Gr/OK )with a natural isomorphism of GK∞ -modules
εM : G(M)(OK̄ ) → TS∗ (M).
On the other hand, for any k[[u]]-algebra B, we have an exact antiequivalence H(−) from the category Mod1,φ/B to a category of ﬁnite locally freegroup schemes over B whose Verschiebung is the zero map ([7, Théorème7.4]. See also [8, Subsection 3.2]). Moreover, for B = k[[u]] and M ∈Mod1,φ,V/S1 , the anti-equivalences G(−) and H(−) are related by the natural
isomorphismεM : G(M)(OK̄ ) → TS∗ (M) = H(M)(R)of Theorem 2.1 (2). We deﬁne the lower ramiﬁcation subgroups of the groupscheme H(M) to be adapted to the valuation vR . Namely, we deﬁne
H(M)i (R) = Ker(H(M)(R) → H(M)(Ri ))for any positive rational number i. We also deﬁne∑ deg(H(M)) by writing
ωH(M) ≃ ⊕i k[[u]]/(ai ) and putting deg(H(M)) = i vR (ai ).
Let M be an object of the category Mod1,φ,V/S . If we identify an elementg ∈ Tp (G(M)) with a homomorphism of Barsotti-Tate groups from Qp /Zp
to G(M) over OC , then by the natural isomorphism Mod(G(M)) → MS (M)the element εM (g) is identiﬁed with the induced map
D∗ (g) : D∗ (G(M))(Acrys → OC ) → D∗ (Qp /Zp )(Acrys → OC ) = Acrys .
A similar argument as in the proof of [13, Proposition 1.1.11] shows that forany exact sequence
0 → N → N′ → M → 0
of Kisin modules such that N and N′ are objects of the category of Mod1,φ/S
and M is of Mod1,φ/S∞ , the functor MS (−) induces an exact sequence ofBreuil modules
0 → MS (N) → MS (N′ ) → MS (M) → 0.

8SHIN HATTORIThis and [2, Lemme 4.2.5 (ii)] imply that, for any object M of the category
Mod1,φ,V/S∞ , there exists a natural isomorphism of S-modules
D∗ (G(M))(S → OK ) → MS (M)which is compatible with Fil1 and φ.
Let us consider the k-algebra isomorphism k[[u]]/(ue ) → ÕK deﬁned by
u 7→ π. Using this isomorphism, we identify the k-algebras of both sides.
Then we can show the following lemma just as in the proof of [9, Lemma
2.4].
Lemma 2.2.
(1) Let G be a unipotent truncated Barsotti-Tate group oflevel one over OK and M be the corresponding object of Mod1,φ,V/S1 via
the anti-equivalence G(−). Then there exist natural isomorphisms ofÕK -modules
Fil1 MS (M)/(Fil1 S)MS (M) → ωG ,MS (M)/Fil1 MS (M) → Lie(G ∨ ).

(2) Let G be a unipotent finite flat group scheme over OK killed by p
and M be the corresponding object of the category Mod1,φ,V/S1 via theanti-equivalence G(−). Then there exists a natural isomorphism ofÕK -modules Fil1 MS (M)/(Fil1 S)MS (M) → ωG .
(3) Let M be an object of the category Mod1,φ,V/S1 . Then we have theequalities
deg(G(M)) = deg(H(M)) = vR (det(φM )).
Let G be a unipotent truncated Barsotti-Tate group of level one, heighth and dimension d over OK with 0 < d < h and Hodge height Hdg(G) = w.
Let M be the object of the category Mod1,φ,V/S1 corresponding to G via theeanti-equivalence G(−). Put M1 = M/u M andFil1 M1 = Im(1 ⊗ φ : ÕK ⊗φ,ÕK M1 → M1 ).
Then the modules M1 and Fil1 M1 are naturally considered as objects ofthe category Mod1,φ. By Lemma 2.2 (1), we can show that there exists a/ÕKnatural isomorphism of ÕK -modules Lie(G ∨ ) → Fil1 M1 as in [9, Subsection2.3]. Thus the ÕK -module Fil1 M1 is free of rank h − d. Moreover, we obtainan exact sequence of φ-modules over ÕK
0 → Fil1 M1 → M1 → M1 /Fil1 M1 → 0

which splits as a sequence of ÕK -modules and the equality of truncatedvaluation vp (det(φFil1 M1 )) = w. We can also prove the following lemma asin the proof of [9, Lemma 2.5].
Lemma 2.3. Let G be a unipotent truncated Barsotti-Tate group of level
one over OK and M be the object of Mod1,φ,V/S1 which corresponds to G via

CANONICAL SUBGROUPS FOR p = 2

9the anti-equivalence G(−). Then the composite
M
G(OK̄ ) →H(M)(R) → HomÕK (Fil1 M1 , R/m≥i
R ) → ωG ∨ ⊗ OK̄,iεcoincides with the Hodge-Tate map HTi for any i ≤ 1.
Remark 2.4. A similar classiﬁcation for ﬁnite ﬂat group schemes over OK
via Breuil-Kisin modules allowing the case of p = 2 and with non-trivial etale
part is obtained independently by Kim ([11]), Lau ([15]) and Liu ([17]). By
using [11, Corollary 4.3], we can generalize Lemma 2.2 and Lemma 2.3 tothis case.
3. Canonical subgroups
In this section, we prove Theorem 1.1. The proof is a modiﬁcation of theargument in [9], where we had to exclude the case of p = 2. We begin witha consideration on a dual situation, as below. By this passage to the dual,
we can replace the use of the congruence of the deﬁning equations of G(M)and H(M) ([8, Corollary 4.6]) in [9] to show the uniqueness of the level one
canonical subgroup, by that of [5, Proposition 1] which is valid for any p.
This uniqueness in turn replaces the use of the ramiﬁcation correspondence
theorem ([8, Theorem 1.1]) to show that the canonical subgroup coincides
with a ramiﬁcation subgroup.
Suppose that the residue ﬁeld k of K is perfect. Let G be a unipotent
truncated Barsotti-Tate group of level one, height h and dimension d overOK with 0 < d < h and Hodge height Hdg(G) = w. Let M be the objectof the category Mod1,φ,V/S1 corresponding to G via the anti-equivalence G(−).
Consider the objects M1 and Fil1 M1 of the category Mod1,φ/Õ

as in theK

previous section. Then, a verbatim argument as in the proof of [9, Lemma
3.3] shows the following proposition.
Proposition 3.1. Let the notation be as above. For any finite flat closed
subgroup scheme D of G over OK , let L be the subobject of M in the category
Mod1,φ,V/S1 corresponding to the quotient G/D. Suppose w < p/(p + 1). Then
there exists a unique D satisfying L/ue(1−w) L = Fil1 M1 /ue(1−w) Fil1 M1 .
Moreover, for this unique D, we have the equality vR (det(φL )) = w.
We temporarily refer to the unique D in Proposition 3.1 as the conjugate
Hodge subgroup of G, which will be shown to be equal to the canonicalsubgroup of G.
Lemma 3.2. Let G be as above and D be the conjugate Hodge subgroup of
G. Then D is the unique finite flat closed subgroup scheme of G over OK
such that (G/D)∨ × S1−w coincides with the Frobenius kernel of G ∨ × S1−w .
In particular, the construction of the conjugate Hodge subgroup is compatible
with any finite extension of K.

10

SHIN HATTORI

Proof. Put i = 1 − w. First let us show that the group scheme (G/D)∨ × Si
coincides with the Frobenius kernel of G ∨ × Si . By comparing orders, it
is enough to show that the group scheme (G/D)∨ × Si is contained in the
Frobenius kernel of G ∨ × Si . By [5, Proposition 1], it is equivalent to sayingthat the natural map
ωG/D ⊗ OK,i → ωG ⊗ OK,iis zero. By Lemma 2.2 (2), this map can be identiﬁed with the top horizontal
arrow of the commutative diagram(Fil1 MS (L)/(Fil1 S)MS (L)) ⊗ OK,i/ (Fil1 MS (M)/(Fil1 S)MS (L)) ⊗ OK,i_/ (MS (M)/(Fil1 S)MS (M)) ⊗ OK,i ,(MS (L)/(Fil S)MS (L)) ⊗ OK,i1where the bottom horizontal arrow and the right vertical arrow are injective.
Let δ1 , . . . , δh−d be a basis of the S-module L and D be the element ofMh−d (k[[u]]) satisfyingφ(δ1 , . . . , δh−d ) = (δ1 , . . . , δh−d )D.
From the deﬁnition of the functor MS (−), we see that the module on the
top left corner of the diagram is equal to the OK,i -module
SpanÕK ((1 ⊗ δ1 , . . . , 1 ⊗ δh−d )ue D−1 ) ⊗ OK,i .
The equality vR (det(D)) = w implies that the entries of the matrix ue D−1
are divisible by uei and the left vertical arrow of the diagram is zero. Hencethe assertion follows.
On the other hand, let D′ be a ﬁnite ﬂat closed subgroup scheme of G overOK such that (G/D′ )∨ × Si coincides with the Frobenius kernel of G ∨ × Si .
Let L′ be the subobject of M corresponding to the quotient G/D′ . Since G isa truncated Barsotti-Tate group of level one, the group scheme D′ × Si also
coincides with the Frobenius kernel of G × Si . Note that the principal idealm≥i
R of the ring Rpi has a unique divided power structure satisfying γn (x) = 0
for any x ∈ m≥i
R and n ≥ p. We can consider the surjection pr0 : Rpi → OK̄,i
as a divided power thickening over the surjection S → OK . Evaluating theexact sequence
0 → D∗ (Im(F )) → D∗ (G × Si ) → D∗ (Ker(F )) → 0
on this divided power thickening, we obtain the equality
1⊗φ

Rpi ⊗S MS (L′ ) = Im(Rpi ⊗φ,S MS (M) → Rpi ⊗S MS (M)).
Since the ÕK -module Fil1 M1 is a direct summand of M1 , the Rpi -module
Rpi ⊗φ,ÕK Fil1 M1 is a submodule of Rpi ⊗φ,ÕK M1 and is equal to the image
on the right-hand side of the above equality. Thus we obtain the equality
Rpi ⊗φ,OK,i (L′ /uei L′ ) = Rpi ⊗φ,OK,i (Fil1 M1 /uei Fil1 M1 )

CANONICAL SUBGROUPS FOR p = 2

11

and the natural map
L′ /uei L′ → (M1 /uei M1 )/(Fil1 M1 /uei Fil1 M1 )is zero after tensoring the injection φ : OK,i → Rpi . Since the OK,i modules of the both sides of this natural map is free, we obtain the inclusion
L′ /uei L′ ⊆ Fil1 M1 /uei Fil1 M1 . Since the OK,i -module L′ /uei L′ is also a direct summand of M1 /uei M1 , the reverse inclusion follows similarly and theequality
L′ /uei L′ = Fil1 M1 /uei Fil1 M1holds. Then the uniqueness assertion in Proposition 3.1 implies the equality
D = D′ .
For the last assertion, let L/K be a ﬁnite extension. Put GOL = G ×
Spec(OL ) and similarly for DOL . The subgroup scheme (GOL /DOL )∨ ×
S1−w also coincides with the Frobenius kernel of (GOL )∨ × S1−w . By theuniqueness we have just proved, the subgroup scheme DOL coincides withthe conjugate Hodge subgroup of GOL . This concludes the proof of thelemma.
□
Lemma 3.3. Let G be as above and D be the conjugate Hodge subgroup ofG.
(1) D = G(1−w)/(p−1) .
(2) If w < (p − 1)/p, then the subgroup D(OK̄ ) coincides with the kernelof the Hodge-Tate map
HTb : G(OK̄ ) → ωG ∨ ⊗ OK̄,bfor any b satisfying w/(p − 1) < b ≤ 1 − w.
(3) If w < 1/2, then we have D = Gb for any b satisfying w/(p − 1) <b ≤ (1 − w)/(p − 1).
Proof. First we consider the assertion (1). It is enough to show the equality
DOL = (GOL )(1−w)/(p−1) for a ﬁnite extension L/K. By Lemma 3.2, thesubgroup scheme DOL is the conjugate Hodge subgroup of GOL . Thus we
may assume G(OK̄ ) = G(OK ). Let M and N be the objects of the category
Mod1,φ,V/S1 corresponding to G and D, respectively. Then we can show theequality
H(N) = H(M)(1−w)/(p−1)
as in the proof of [9, Theroem 3.1 (c)]. Take x ∈ G(M). Let G ′ be thescheme-theoretic closure in G of the subgroup Fp x ⊆ G(OK̄ ) and M′ be thequotient of M corresponding to G ′ . By the Oort-Tate classiﬁcation ([18]),
we have the following equivalences:
G(M′ )(1−w)/(p−1) = G(M′ ) ⇔ vR (det(φM′ )) ≥ 1 − w⇔ H(M′ )(1−w)/(p−1) = H(M′ ).

12

SHIN HATTORI

Note the commutative diagram
_

εM′
∼

/ H(M′ )(R)_

G(M)(OK̄ )
O

εM
∼


/ H(M)(R)O

G(N)(OK̄ )

εN
∼

?
/ H(N)(R),

G(M′ )(O K̄ )
?

where the horizontal arrows are isomorphisms and the vertical arrows are
injections. Then we havex ∈ D(OK̄ ) = G(N)(OK̄ ) ⇔ εM (x) ∈ H(N)(R) = H(M)(1−w)/(p−1) (R)⇔ H(M′ )(1−w)/(p−1) = H(M′ )⇔ G(M′ )(1−w)/(p−1) = G(M′ )⇔ x ∈ G(M)(1−w)/(p−1) (OK̄ )and the assertion (1) follows. The assertions (2) and (3) can be shown by averbatim argument as in the proof of [9, Theorem 3.1 (2), (3)].
□Remark 3.4. By using results of [11] and Remark 2.4, we can drop theassumption that G is unipotent from the arguments above, though we do no
use this fact in what follows.
Now we proceed to prove the following theorem, which is the level one
case of Theorem 1.1.
Theorem 3.5. Let the notation be as in Section 1. Let G be a truncatedBarsotti-Tate group of level one, height h and dimension d over OK with
0 < d < h and Hodge height w = Hdg(G).
(1) If w < p/(p+1), then there exists a unique finite flat closed subgroupscheme C of G of order pd over OK such that C × S1−w coincides
with the kernel of the Frobenius homomorphism of G × S1−w . We
refer to the subgroup scheme C as the (level one) canonical subgroupof G. Moreover, the subgroup scheme C has the following properties:(a) deg(G/C) = w.
(b) Let C ′ be the canonical subgroup of G ∨ . Then we have the equality of subgroup schemes C ′ = (G/C)∨ , or equivalently C(OK̄ ) =C ′ (OK̄ )⊥ , where ⊥ means the orthogonal subgroup with respect
to the Cartier pairing.
(c) C = G(1−w)/(p−1) = G pw/(p−1)+ .
(2) If w < (p−1)/p, then the subgroup C(OK̄ ) coincides with the kernel ofthe Hodge-Tate map HTb : G(OK̄ ) → ωG ∨ ⊗ OK̄,b for any b satisfying
w/(p − 1) < b ≤ 1 − w.
(3) If w < 1/2, then C coincides both with the lower ramification subgroup scheme Gb for any b satisfying w/(p − 1) < b ≤ (1 − w)/(p − 1)

CANONICAL SUBGROUPS FOR p = 2

13

and the upper ramification subgroup scheme G j+ for any j satisfying
pw/(p − 1) ≤ j < p(1 − w)/(p − 1).
Proof. By a base change argument as in the proof of [9, Theorem 3.1], we
may assume that the residue ﬁeld k of K is perfect. Let G 0 (resp. G et ) be
the unit component (resp. the maximal etale quotient) of the group schemeG, and consider their Cartier duals (G 0 )∨ and (G et )∨ . These four groupschemes are all truncated Barsotti-Tate groups of level one over OK and let
h0 be the height of G 0 . Then (G 0 )∨ is a unipotent truncated Barsotti-Tategroup of level one, height h0 and dimension h0 −d. If h0 = d, then the groupscheme G is ordinary (namely, (G 0 )∨ is etale) and the assertions are clear.
Thus we may assume h0 > d.
Since Hdg(G) = Hdg(G ∨ ) and Lie(G) = Lie(G 0 ), the truncated BarsottiTate group (G 0 )∨ satisﬁes the assumption on the Hodge height in Proposition
3.1. Let D be the conjugate Hodge subgroup of (G 0 )∨ , which is of order
ph0 −d . We deﬁne the canonical subgroup C by C = ((G 0 )∨ /D)∨ , which is a
ﬁnite ﬂat closed subgroup scheme of G 0 of order pd over OK . By Lemma
3.2, the group scheme C × S1−w coincides with the Frobenius kernel of
G 0 × S1−w , which is equal to the Frobenius kernel of G × S1−w . If a ﬁnite
ﬂat closed subgroup scheme E of G has the reduction E × S1−w equal to thisFrobenius kernel, then E is connected and Lemma 3.2 implies C = E. Thus
the uniqueness assertion of Theorem 3.5 (1) follows.
By Lemma 2.2 (3) and Proposition 3.1, we have deg((G 0 )∨ /D) = w anddeg(G/C) = deg(G 0 /C) = deg(D∨ ) = h0 − d − deg(D)= h0 − d − deg((G 0 )∨ ) + deg((G 0 )∨ /D) = w.
Thus the part (a) of the theorem follows. Cartier duality and the uniquenessof the canonical subgroup we have just proved imply the part (b).
For the part (c), we insert here the following lemma due to the lack ofreferences.
Lemma 3.6. Let H be a finite flat group scheme over OK and H0 be its
unit component. Then we have Hj = (H0 )j for any positive rational numberj.
Proof. Replacing K by a ﬁnite extension, we may assume that the maximal
etale quotient Het is a constant group scheme M for some abelian group M
and that we have an isomorphism of schemes over OK
H0 × M → H
which induces the natural isomorphism of group schemes H0 × {0} → H0 .
Let F j be the functor of the set of geometric connected components of
the j-th tubular neighborhood as in [1, Section 2]. By [1, Lemme 2.1.1], the14SHIN HATTORI

functor F j is compatible with products and we have a commutative diagram
H0 (OK̄ ) × M (OK̄ )/ H(O )K̄
/ F j (H),F j (H0 ) × F j (M )where the vertical arrows are homomorphisms and the horizontal arrows arebijections preserving zero elements. For j > 0, the natural map M (OK̄ ) →F j (M ) is an isomorphism and the kernel of the left vertical arrow is the
subgroup (H0 )j (OK̄ ) × {0}. Thus the kernel of the right vertical arrow is
the subgroup (H0 )j (OK̄ ) and the lemma follows.
□
Now Lemma 3.3 (1) implies the equality D = ((G 0 )∨ )b for b = (1 −w)/(p−1). By Lemma 3.6 and a theorem of Tian and Fargues ([19, Theorem
1.6] or [5, Proposition 6]), we have the equalities C = (G 0 )j+ = G j+ for
j = p/(p − 1) − pb = pw/(p − 1). From this and the part (b), we also obtain
the equality C = Gb and the part (c) follows. The part (3) can be shown
similarly, by using Lemma 3.3 (3).
Finally we show the assertion (2). By replacing G by G ∨ , it is enough toshow the assertion for the canonical subgroup C ′ of G ∨ . Put T = (G et )∨ . Let
D be the conjugate Hodge subgroup of (G 0 )∨ as above and D̃ be the inverse
image of D by the natural epimorphism ι∨ : G ∨ → (G 0 )∨ . We claim theequality C ′ = D̃. Indeed, by Lemma 3.3 (1), we have D = ((G 0 )∨ )(1−w)/(p−1) .
By the part (c) of the theorem, this implies that D is the canonical subgroupof (G 0 )∨ and also coincides with ((G 0 )∨ )pw/(p−1)+ . Since C ′ = (G ∨ )pw/(p−1)+ ,the natural map ι∨ induces the surjection C ′ (OK̄ ) → D(OK̄ ). In particular,
the subgroup C ′ (OK̄ ) is contained in D̃(OK̄ ). On the other hand, since thegroup scheme T is isomorphic to a direct sum of µp after a ﬁnite extension,
we have the inclusions
T = T1/(p−1) ⊆ (G ∨ )1/(p−1) ⊆ (G ∨ )(1−w)/(p−1) = C ′ ,from which the claim follows.
Therefore, we have the commutative diagram with exact rows0

/ T (O )K̄/ C ′ (O )K̄

/ D(O )K̄

/00

/ T (O )K̄/ G ∨ (O )K̄
/ (G 0 )∨ (O )
K̄

/0



ωG ⊗ OK̄,b∼/ ω 0 ⊗O
G

K̄,b ,

where the lowest vertical arrows are the Hodge-Tate maps HTb and thelowest horizontal arrow is an isomorphism. By Lemma 3.3 (2), the subgroup
D(OK̄ ) coincides with the kernel of the lowest right vertical arrow for any b

CANONICAL SUBGROUPS FOR p = 2

15satisfying w/(p − 1) < b ≤ 1 − w. This implies that the subgroup D̃(OK̄ ) =C ′ (OK̄ ) is the kernel of the lowest left vertical arrow for any such b and theassertion (2) follows. This concludes the proof of Theorem 3.5.
□
Since the arguments in the proof of [9, Theorem 1.1] work verbatim also
for p = 2, Theorem 1.1 follows from Theorem 3.5.
Moreover, we can show the following proposition on anti-canonical subgroups, by modifying the proof of [9, Proposition 4.3].
Proposition 3.7. Let G be a truncated Barsotti-Tate group of level two,height h and dimension d over OK with 0 < d < h and Hodge height w =Hdg(G). Suppose w < 1/2 and let C be the canonical subgroup of G[p] as in
Theorem 3.5. Let D be a finite flat closed subgroup scheme of G[p] over OK
such that the natural map C(OK̄ ) ⊕ D(OK̄ ) → G[p](OK̄ ) is an isomorphism.
(1) The truncated Barsotti-Tate group p−1 D/D of level one has Hodgeheight Hdg(p−1 D/D) = p−1 w.
(2) The subgroup scheme G[p]/D is the canonical subgroup of p−1 D/D.
(3) deg(D) = p−1 w.
Proof. By a base change argument as before, we may assume that the residue
ﬁeld k of K is perfect. Note that the truncated Barsotti-Tate group p−1 D/Dof level one is also of height h and dimension d. The natural homomorphism
C → G[p]/D induces an isomorphism between the generic ﬁbers of both sides.
Since the group scheme C is connected, the connected-etale sequence implies
that the group scheme G[p]/D is also connected. Now we claim that thegroup scheme (G[p]/D)×S1−w is killed by the Frobenius. For this, let L andcorresponding to the unipotent
L′ be the objects of the category Mod1,φ,V/S∨∨
ﬁnite ﬂat group schemes C and (G[p]/D) via the anti-equivalence G(−),
respectively. By [16, Corollary 2.2.2], the generic isomorphism (G[p]/D)∨ →C ∨ corresponds to an injection L → L′ . Then the S1 -modules ∧d L and ∧d L′
are free of rank one and this injection induces an injection ∧d L → ∧d L′ .
Hence we obtain the inequality vR (det φL′ ) ≤ vR (det φL ) = w. Since thegroup scheme G[p]/D is a ﬁnite ﬂat closed subgroup scheme of the truncatedBarsotti-Tate group (p−1 D/D)0 of level one over OK , a similar argument to
the proof of Lemma 3.2 implies the claim. Then the proposition follows asin the proof of [9, Proposition 4.3].
□
We also have the following generalization of [9, Proposition 4.4] to thecase of p = 2.
Proposition 3.8. Let G be a truncated Barsotti-Tate group of level n, heighth and dimension d over OK with 0 < d < h and Hodge height w < p(p −1)/(pn+1 − 1). Let Cn be the level n canonical subgroup of G, which is definedby Theorem 1.1. Let Dn be a finite flat closed subgroup scheme of G overOK such that Dn (OK̄ ) ≃ (Z/pn Z)d and the group scheme Dn × S1−pn−1 w is
killed by the n-th iterated Frobenius F n . Then we have Cn = Dn .

16

SHIN HATTORI

Proof. Suppose n ≥ 2. Let C1 be the scheme-theoretic closure of Cn (OK̄ )[p]in Cn and deﬁne D1 similarly. The group scheme C1 coincides with thecanonical subgroup of G[p] by Theorem 1.1 (e). We claim C1 = D1 , which
implies the proposition by an induction as in the proof of [9, Proposition
4.4]. For this, by a base change argument as before, we may assume thatthe residue ﬁeld k of K is perfect and G(OK̄ ) = G(OK ). Suppose C1 ̸= D1 .
Then we can ﬁnd a ﬁnite ﬂat closed subgroup scheme E of G[p] such that
C1 (OK̄ ) ⊕ E(OK̄ ) = G[p](OK̄ ) and D1 (OK̄ ) ∩ E(OK̄ ) ̸= 0. Let F be thescheme-theoretic closure of the latter intersection in G[p]. As in the proof of
Theorem 3.5 (2), we see that the group scheme C1 contains the multiplicative
part of G[p]. Thus the quotient G[p]/C is unipotent. Since the natural map
E → G[p]/C is a generic isomorphism, the connected-etale sequence showsthat E, and thus also F, are unipotent. Let F be the object of the category
Mod1,φ,V/S1 corresponding to F. Then Lemma 2.2 (3) and Proposition 3.7 (3)impliesvR (det(φF )) = deg(F) ≤ deg(E) = p−1 w.
On the other hand, the group scheme F × S1−pn−1 w is killed by the n-th
iterated Frobenius. Put i = 1 − pn−1 w. Evaluating D∗ (F) on the divided
power thickening Rpi → OK̄,i as before, we see that the map
1 ⊗ φ ⊗ φF : Rpi ⊗φ,Rpi (Rpi ⊗φ,S F) → Rpi ⊗φ,S F
is the one induced by the FrobeniusD∗ (F ) : D∗ ((F × Si )(p) ) → D∗ (F × Si )and that the composite of its pull-backsφn∗ (Rpi ⊗φ,S F) → φn−1∗ (Rpi ⊗φ,S F) → · · · → Rpi ⊗φ,S F
is the map induced by the n-th iterated Frobenius. Taking the valuation of
the determinant of this map, we obtain the inequalitiesp(1 − pn−1 w) ≤ vR (det φF )p(pn − 1)/(p − 1) ≤ w(pn − 1)/(p − 1),which contradict the assumption on w and the equality C1 = D1 follows. □References[1] A. Abbes and A. Mokrane: Sous-groupes canoniques et cycles évanescents p-adiquespour les variétés abéliennes, Publ. Math. Inst. Hautes Etudes Sci. 99 (2004), 117–162.
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E-mail address: shin-h@math.kyushu-u.ac.jp
Faculty of Mathematics, Kyushu University