CANONICAL SUBGROUPS VIA BREUIL-KISIN MODULESSHIN HATTORI
Abstract. Let p > 2 be a rational prime and K/Qp be an extensionof complete discrete valuation ﬁelds. Let G be a truncated BarsottiTate group of level n, height h and dimension d over OK with 0 < d <h. In this paper, we show that if the Hodge height of G is less than
1/(pn−2 (p + 1)), then there exists a ﬁnite ﬂat closed subgroup schemeof G of order pnd over OK with standard properties as the canonicalsubgroup.

1. Introduction
Let p be a rational prime and K/Qp be an extension of complete discrete
valuation ﬁelds. Let k be the residue ﬁeld of K and e = e(K) be the absolute
ramiﬁcation index of K. Let OK denote the ring of integers of K and put
ÕK = OK /pOK . We ﬁx an algebraic closure K̄ of K and let vp be thevaluation of K which is normalized as vp (p) = 1 and extended to K̄. Fora non-negative rational number i, we put m≥i= {a ∈ OK̄ | vp (a) ≥ i} andK̄
≥isimilarly for mK .
Let E be an elliptic curve over OK . By ﬁxing a formal parameter ofthe formal completion of E along the zero section, we identify the group of
p-torsion points E[p](OK̄ ) as a subset of mK̄ . If the group E[p](OK̄ ) has asubgroup C of order p whose elements have valuations greater than thoseof the elements of E[p](OK̄ ) \ C, then the subgroup C is called the (levelone) canonical subgroup of E. It was shown that the canonical subgroup of
E exists if the Hodge height of E, namely the p-adic valuation of the Hasse
invariant of E, is less than p/(p+1), and based on this result, Katz studied aspectral theory of the Up operator for overconvergent elliptic modular forms([14]).
For a similar investigation of p-adic modular forms for general reductive
algebraic groups, we need a generalization of the existence theorem of thecanonical subgroup to higher dimensional abelian schemes over OK . Sucha generalization was ﬁrst obtained by Abbes and Mokrane via a calculationof p-adic vanishing cycles of abelian schemes ([1]). Namely, for an abelianscheme A over OK with relative dimension g, they found a subgroup ofA[p](OK̄ ) of order pg in the upper ramiﬁcation ﬁltration {A[p]j+ (OK̄ )} if
the Hodge height of A ([1, Subsection 1.2]) is less than an explicit bound.
Date: October 3, 2012.
Supported by Grant-in-Aid for Young Scientists B-21740023.
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SHIN HATTORI

Their work was followed by many improvements and generalizations withvarious methods, such as [2], [7], [10], [11], [17], [19], [21] and especially [8]and [22].
In this paper, we prove an existence theorem of level n canonical subgroups for p > 2, using a classiﬁcation theory of ﬁnite ﬂat (commutative)
group schemes due to Breuil and Kisin ([4], [5], [15], [16]), and a ramiﬁcationtheoretic technique developed by the author ([12]).
Before stating the main theorem, we ﬁx some notation. For a ﬁnite ﬂatgroup scheme G over OK and its module of invariant diﬀerentials∑ωG overOK , write ωG ≃ ⊕i OK /(ai ) with some ai ∈ OK and put deg(G) = i vp (ai ).
We deﬁne the Hodge-Tate map
HTi : G(OK̄ ) → ωG ∨ ⊗ OK̄ /m≥iK̄by x 7→ (x∨ )∗ (dt/t), where x∨ : G ∨ × Spec(OK̄ ) → µp is the dual map ofx ∈ G(OK̄ ). Let G j and G j+ (resp. Gi and Gi+ ) denote the upper (resp.
lower) ramiﬁcation subgroup schemes of G (see [12]). Here we normalize theindices of the ﬁltrations by multiplying the usual ones by 1/e(K), so thatthese ramiﬁcation subgroup schemes are compatible with any base changeof complete discrete valuation rings. In particular, Gi is deﬁned by
Gi (OK̄ ) = Ker(G(OK̄ ) → G(OK̄ /m≥i)).
K̄For a truncated Barsotti-Tate group G of level n, height h and dimension
d < h over OK , let us consider its Cartier dual G ∨ and the p-torsion subgroupscheme G ∨ [p]. Then the Lie algebra Lie(G ∨ [p] × Spec(ÕK )) is a free ÕK module of rank h − d. We deﬁne the Hodge height Hdg(G) of G to be
the truncated valuation vp (det(VG ∨ [p] )) ∈ [0, 1] of the determinant of theaction of the Verschiebung VG ∨ [p] on this ÕK -module. Note the equality
Hdg(G) = Hdg(G ∨ ) (see for example [8, Proposition 2], where the Hodgeheight of G is denoted by Ha(G) and referred as the Hasse invariant of G).
Then the main theorem of this paper is the following.
Theorem 1.1. Let p > 2 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≥1−pn−1 w
canonical subgroup of G, such that Cn × Spec(OK /mK) coincides with the kernel of the n-th iterated Frobenius homomorphism
n−1 w
F n of G × Spec(OK /m≥1−p). Moreover, the group scheme Cn
K
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

CANONICAL SUBGROUPS VIA BREUIL-KISIN MODULES

3

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 .
(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+ forpw(pn − 1)/(p − 1)2 ≤ j < p(1 − w)/(p − 1).
Moreover, we show that such Cn is unique if w < p(p − 1)/(pn+1 − 1)
(Proposition 4.4). Since the upper ramiﬁcation subgroups can be patched
into a family (see Lemma 4.5), we also have the following corollary.
Corollary 1.2. Let K/Qp be an extension of complete discrete valuation
fields and j be a positive rational number. Let X be an admissible formal
scheme over Spf(OK ) which is quasi-compact and G be a truncated BarsottiTate group of level n over X of constant height h and dimension d with
0 < d < h. We let X and G denote the Raynaud generic fibers of theformal schemes X and G, respectively. For a finite extension L/K andx ∈ X(L), we put Gx = G ×X,x Spf(OL ), where we let x also denote themap Spf(OL ) → X obtained from x by taking the scheme-theoretic closureand 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.
Suppose p > 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), the fiber (Cn )x coincides with the generic fiber of thelevel n canonical subgroup of Gx .
Note that for a smaller range of w, Theorem 1.1 is also proved in [8] and[22]. The key idea of our approach is, ﬁrstly, to lift the conjugate Hodge
ﬁltration of G × Spec(ÕK ) to the Breuil-Kisin module associated to G. By
an induction, it suﬃces to consider the case of n = 1. We may assume
that the residue ﬁeld k is perfect and that the group scheme G is associatedto a φ-module M over the formal power series ring k[[u]] via the BreuilKisin classiﬁcation (see Section 2). The Lie algebra Lie(G ∨ ) is naturally
considered as a φ-stable direct summand of the φ-module M/ue M. Then
we show that the reduction modulo ue(1−w) of this direct summand lifts

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SHIN HATTORI

uniquely to a φ-stable direct summand of M. Our canonical subgroup is
deﬁned to be the ﬁnite ﬂat closed subgroup scheme of G associated to thequotient of M by this direct summand. Its properties stated in Theorem 1.1
follow easily from the construction, basically except the uniqueness and thecoincidence with ramiﬁcation subgroup schemes. The second key idea is toswitch to a complete discrete valuation ﬁeld of equal characteristic: Fromthe φ-module M, we can also construct a ﬁnite ﬂat group scheme H(M) overk[[u]] ([9]). Then, by the main theorem of [12], the ramiﬁcation subgroupsof G and H(M) are naturally isomorphic to each other. Moreover, it is also
proved that reductions of G and H(M) are isomorphic as pointed schemes([12, Corollary 4.6]). These reduce proofs of the remaining properties ofour canonical subgroup to an elementary calculation on the side of equalcharacteristic, which we can easily accomplish.
Acknowledgments. The author would like to thank the anonymous referee
for many helpful comments to improve the paper.
2. Review of the Breuil-Kisin classification of finite flatgroup schemes and their ramification theory
Let the notation be as in the previous section and suppose that the residue
ﬁeld k of K is perfect of characteristic p > 2. Let W = W (k) be the Witt ringof k and φ be the Frobenius endomorphism of W . Let us ﬁx once and for alla uniformizer π of K and a system {πn }n∈Z≥0 of its p-power roots in K̄ withpπ0 = π and πn+1
= πn . Put Kn = K(πn ) and K∞ = ∪Kn . Let E(u) ∈ W [u]be the Eisenstein polynomial of π over W and put c0 = p−1 E(0). In thissection, we brieﬂy recall a classiﬁcation theory of Breuil ([4], [5]) and Kisin([15], [16]) of ﬁnite ﬂat group schemes over OK and their ramiﬁcation theory,
while we concentrate mainly on the p-torsion case.
2.1. Breuil and Kisin modules. Put S = W [[u]] and S1 = k[[u]]. The φsemilinear continuous ring endomorphisms of these rings deﬁned by u 7→ up
are also denoted by φ. Then a Kisin module over S (of E-height ≤ 1) isan S-module M endowed with a φ-semilinear map φM : M → M such thatthe cokernel of the map 1 ⊗ φM : S ⊗φ,S M → M is killed by E(u). We
write φM also as φ if there is no risk of confusion. A morphism of Kisin
modules is an S-linear map which is compatible with φ’s of the source andthe target. Then the Kisin modules form a category and this category has
1,φan obvious notion of exact sequences. We let Mod1,φ/S1 (resp. Mod/S ) denoteits full subcategory consisting of the objects whose underlying S-modules
are free of ﬁnite rank over S1 (resp. S). For a k[[u]]-algebra B, we write φalso for the p-th power Frobenius endomorphism of B. Then we let Mod1,φ/B
denote the category of locally free B-modules M of ﬁnite rank endowedwith a φ-semilinear map φM : M → M such that the cokernel of the map
1 ⊗ φM : B ⊗φ,B M → M is killed by E(u).

CANONICAL SUBGROUPS VIA BREUIL-KISIN MODULES

51,φ
We also have categories Mod1,φ/S1 and Mod/S of Breuil modules deﬁned
as follows. Let S be the p-adic completion of the divided power envelope
W [u]DP of W [u] with respect to the ideal (E(u)) and the compatibility
condition with the canonical divided power structure on pW . The ring S
has a natural ﬁltration Fili S deﬁned as the closure in S of the ideal generated by E(u)j /j! for integers j ≥ i. The φ-semilinear continuous ringhomomorphism S → S deﬁned by u 7→ up is also denoted by φ. We haveφ(Fil1 S) ⊆ pS and put φ1 = p−1 φ|Fil1 S . This ﬁltration and the map φ1
induce a similar structure on the ring Sn = S/pn S. Note that we haveφ(Fil1 S1 ) = 0. Put c = φ1 (E(u)) ∈ S × . Then we let ′ Mod1,φ/S denotethe category of S-modules M endowed with an S-submodule Fil1 M containing (Fil1 S)M and a φ-semilinear map φ1,M : Fil1 M → M satisfying
φ1,M (s1 m) = c−1 φ1 (s1 )φ1,M (E(u)m) for any s1 ∈ Fil1 S and m ∈ M. A
morphism of this category is deﬁned to be a homomorphism of S-modules
compatible with Fil1 ’s and φ1 ’s. Note that this category also has an obvious
notion of exact sequences. We drop the subscript M of φ1,M if no confusion
1,φmay occur. We let Mod1,φ/S1 (resp. Mod/S ) denote the full subcategory of′ Mod1,φ consisting of the objects M such that M is free of ﬁnite rank over/S
S1 (resp. M is free of ﬁnite rank over S, M/Fil1 M is p-torsion free) andthe image φ1,M (Fil1 M) generates the S-module M.

The categories of Breuil and Kisin modules are in fact equivalent. By thenatural map S → S, we consider the ring S as an S-algebra. We deﬁne an
1,φexact functor MS (−) : Mod1,φ/S1 → Mod/S1 by putting MS (M) = S ⊗φ,S M
with
1⊗φ

Fil1 MS (M) = Ker(S ⊗φ,S M → (S1 /Fil1 S1 ) ⊗S M),
1⊗φ

φ1 ⊗1

φ1 : Fil1 MS (M) → Fil1 S1 ⊗S M → S1 ⊗φ,S1 M = MS (M).
Then the functor MS (−) is an equivalence of categories ([16, Proposition
1.1.11]). We also have a similar equivalence for the categories Mod1,φ/S andMod1,φ/S ([6, Theorem 2.2.1]).
2.2. Classification of finite flat group schemes and ramification. Set
ÕK̄ = OK̄ /pOK̄ and C to be the completion of K̄. Consider the ringR = lim(ÕK̄ ← ÕK̄ ← · · · ),←−where every transition map is the p-th power map. For an element x =n(x0 , x1 , . . .) ∈ R with xn ∈ ÕK̄ , we put x(0) = limn→∞ x̂pn ∈ OC , wherex̂n is a lift of xn in OK̄ . Then the ring R is a complete valuation ring ofcharacteristic p with valuation vR (x) = vp (x(0) ) whose fraction ﬁeld Frac(R)is algebraically closed, and the absolute Galois group GK = Gal(K̄/K)
naturally acts on this ring. We put m≥i
R = {x ∈ R | vR (x) ≥ i}. Deﬁne anelement π of R with vR (π) = 1/e by π = (π, π1 , π2 , . . .), where we abusively

6SHIN HATTORI

write πn also for its image in ÕK̄ . The ring R has a natural S-algebra
structure deﬁned by the continuous map S → R which sends u to theelement π. Then we have the following classiﬁcation theorem due to Breuil
and Kisin.
Theorem 2.1 ([4], [5], [15], [16]). There exists an anti-equivalence Gr(−)
1,φfrom the category Mod1,φ/S1 (resp. Mod/S ) to the category of finite flat groupschemes over OK killed by p (resp. Barsotti-Tate groups over OK ). Moreover, put G(−) = Gr(MS (−)). Then for any object M of the category
Mod1,φ/S1 , we have a natural isomorphism of GK∞ -modules
εM : G(M)(OK̄ )|GK∞ → TS∗ (M) = HomS,φ (M, R).
The anti-equivalence G(−) is compatible with Cartier duality in the following sense. For an object M of the category Mod1,φ/S1 , we can deﬁne a∨natural dual object M ([6], [18]), as follows. Put M∨ = HomS (M, S1 ).
∨Choose a basis e1 , . . . , eh of the free S1 -module M and let e∨
1 , . . . , eh denoteits dual basis. Deﬁne a matrix A ∈ Mh (S1 ) byφM (e1 , . . . , eh ) = (e1 , . . . , eh )A.
The φ-semilinear map φM∨ : M∨ → M∨ is deﬁned to be
∨
∨
∨
t −1
φM∨ (e∨1 , . . . , eh ) = (e1 , . . . , eh )(E(u)/c0 )( A) ,which is independent of the choice of the basis e1 , . . . , eh . Then we have anatural isomorphism of ﬁnite ﬂat group schemes over OK
G(M)∨ → G(M∨ ),where ∨ on the left-hand side means the Cartier dual (see [12, Proposition4.4]). This deﬁnes an isomorphism of GK∞ -modulesε∨MδM : G(M)∨ (OK̄ ) → G(M∨ )(OK̄ ) →TS∗ (M∨ ).

1,φ
Let S∨
1 be the object of the category Mod/S1 whose underlying S-moduleis S1 and Frobenius map is given by φS∨1 (1) = c−1
0 E(u). Then the naturalpairing⟨ , ⟩M : M × M∨ → S∨
1
induces a perfect pairing of GK∞ -modulesTS∗ (M) × TS∗ (M∨ ) → TS∗ (S∨1 ),which is denoted also by ⟨ , ⟩M . This pairing ﬁts into a commutative diagramof GK∞ -modules
G(M)(OK̄ ) × G(M)∨ (OK̄ )εM ≀/ Z/pZ(1)δM ≀

TS∗ (M) × TS∗ (M∨ )⟨ , ⟩M≀/ T ∗ (S∨ ),S

1where the top arrow is the Cartier pairing of G(M) ([12, Proposition 4.4]).

CANONICAL SUBGROUPS VIA BREUIL-KISIN MODULES

7

On the other hand, for a k[[u]]-algebra B, we also have an anti-equivalenceof categories H(−) from Mod1,φ/B to the category of ﬁnite locally free groupschemes H over B whose Verschiebung VH is zero such that the cokernel ofthe induced map VH∨ : Lie(H∨ )(p) → Lie(H∨ ) is killed by ue ([9, Théorème7.4]). Note that the anti-equivalence H(−) commutes with any base change.
Suppose that the B-module M is free of rank h. Let e1 , . . . , eh be its basis
and put φ(e1 , . . . , eh ) = (e1 , . . . , eh )A for some matrix A = (ai,j ) ∈ Mh (B).
Then the group scheme H(M) is by deﬁnition the additive group scheme
whose aﬃne algebra is
B[X1 , . . . , Xh ]/(X1p −h∑aj,1 Xj , . . . , Xhp −

j=1

h
∑

aj,h Xj ).

j=1

Moreover, for the case of B = k[[u]], we have a natural isomorphism of
GK∞ -modules
H(M)(R) → HomS,φ (M, R),
by which we identify both sides. Hence we obtain the isomorphism of GK∞ modules
G(M)(OK̄ )|GK∞ → H(M)(R).
We normalize the indices of the ramiﬁcation subgroup schemes of H(M) bymultiplying the usual ones by 1/e, so that
H(M)i (R) = Ker(H(M)(R) → H(M)(R/m≥i
R )).
Then we have the following correspondence of ramiﬁcation between the ﬁnite
ﬂat group schemes G(M) and H(M).
Theorem 2.2. Let M be an object of the category Mod1,φ/S1 .
(1) ([12, Theorem 1.1]) The natural isomorphism of GK∞ -modules
G(M)(OK̄ )|GK∞ → H(M)(R)
preserves the upper and the lower ramification subgroups of bothsides.
(2) ([12, Corollary 4.6]) By the k-algebra isomorphism k[[u1/p ]]/(ue ) →OK1 /pOK1 sending u1/p to π1 , we identify both sides of the isomorphism. Then there exists an isomorphism of schemes
G(M) × Spec(OK1 /pOK1 ) → H(M) × Spec(k[[u1/p ]]/(ue ))which preserves the zero section.
For a ﬁnite ﬂat group scheme G over OK which is killed by p, the upper andthe lower ramiﬁcation subgroups are in duality as in the following theorem.
Theorem 2.3 ([21], Theorem 1.6 or [8], Proposition 6). Let K/Qp be an
extension of complete discrete valuation fields and G be a finite flat groupscheme over OK killed by p. For j ≤ p/(p − 1), we have the equality
G j (OK̄ )⊥ = (G ∨ )l(j)+ (OK̄ )8

SHIN HATTORI

of subgroups of G ∨ (OK̄ ), where ⊥ means the orthogonal subgroup with respect
to the Cartier pairing and l(j) = 1/(p − 1) − j/p.
We insert here a lemma which gives an upper bound of the lower ramiﬁcation for ﬁnite ﬂat group schemes killed by p. For an extension K/k((u)) ofcomplete discrete valuation ﬁelds, let vu be the u-adic valuation on K normalized as vu (u) = 1. Let G be a ﬁnite ﬂat generically etale group scheme
over OK . Fix a positive integer m and we normalize the indices of the ramiﬁcation subgroup schemes of G by multiplying the usual ones by∑1/m. WriteωG ≃ ⊕i OK /(ai ) for some ai ∈ OK and put deg(G) = m−1 i vu (ai ). In
particular, for an object M of the category Mod1,φ/S1 , we deﬁne deg(H(M))
by putting m = e.
Lemma 2.4. Let K/Qp (resp. K/k((u))) be an extension of complete discrete valuation fields and G be a finite flat generically etale group scheme
over OK killed by p. Then Gi = 0 for any i > deg(G)/(p − 1).
Proof. Let Ksep be a separable closure of K. We may assume G(OKsep ) =G(OK ). For i as in the lemma, put x ∈ Gi (OK ) and let H be the schemetheoretic closure of the subgroup Fp x ⊆ G(OK ) in G. Then we have deg(H) ≤deg(G). By the Oort-Tate classiﬁcation ([20]), the aﬃne algebra of H isisomorphic to the ring OK [X]/(X p − aX) for some a ∈ OK with deg(H) =vp (a) (resp. deg(H) = m−1 vu (a)). Hence we obtain Hi = 0 and x = 0. □2.3. Hodge filtration and Breuil-Kisin modules. Finally, due to thelack of references, we explain how to decode the Hodge ﬁltration, the Hodgeheight and the Hodge-Tate map for a truncated Barsotti-Tate group oflevel one over OK from its corresponding Breuil-Kisin module. Put S1 =Spec(ÕK ) and E1 = Spec(S1 ). Consider the big crystalline site with thefppf topology (S1 /E1 )CRYS , as in [3]. For an fppf sheaf E over Spec(OK ),we let E1 denote its restriction to Spec(ÕK ) only in this subsection, and foran fppf sheaf F over Spec(ÕK ), let F denote the associated sheaf on thesite (S1 /E1 )CRYS ([3, (1.1.4.5)]). We also write the crystalline Dieudonné
functor as D∗ (−) = Ext 1S1 /E1 (−, OS1 /E1 ) ([3, Déﬁnition 3.1.5]). Let vu be
the u-adic valuation on the ring k[[u]] with vu (u) = 1, as before. By thek-algebra isomorphism k[[u]]/(ue ) → ÕK sending u to π, we identify bothsides of the isomorphism.
Lemma 2.5.
(1) Let G be a truncated Barsotti-Tate group of level one
over OK and M be the object of Mod1,φ/S1 which corresponds to G via
the anti-equivalence Gr(−). Then there exist natural isomorphismsof ÕK -modules
Fil1 M/(Fil1 S)M → ωG1 ,M/Fil1 M → Lie(G1∨ ).

(2) Let G be a finite flat group scheme over OK killed by p and M be
its corresponding object of the category Mod1,φ/S1 . Then we have anatural isomorphism of ÕK -modules Fil1 M/(Fil1 S)M → ωG1 .

CANONICAL SUBGROUPS VIA BREUIL-KISIN MODULES

9(3) Let M be an object of the category Mod1,φ/S1 . Then we have deg(G(M)) =−1
deg(H(M)) = e vu (det(φM )).
Proof. Let us consider the assertion (1). By [13, Théorème 4.4 (e)], we
can ﬁnd a Barsotti-Tate group Γ over OK such that its p-torsion subgroupscheme Γ[p] is isomorphic to G. Let N be the object of the category Mod1,φ/S
corresponding to Γ and put N1 = N /pN , which is naturally considered as an
object of the category Mod1,φ/S1 . By the construction of the anti-equivalence([15]), we have a natural isomorphism of exact sequences00/ Fil1 N1 /(Fil1 S)N1

/ N1 /(Fil1 S)N1

/ N1 /Fil1 N1

≀

≀

≀



/ ωΓ



/ D∗ (Γ1 )

ÕK

1


/ Lie(Γ∨ )
1

/0

/ 0,

where D∗ (Γ1 )ÕK is the set of sections on the divided power thickening ÕK →ÕK . Thus we also have the natural isomorphism00/ Fil1 M/(Fil1 S)M/ M/(Fil1 S)M/ M/Fil1 M

≀

≀

≀



/ ωG

1



/ D∗ (G1 )ÕK/ Lie(G ∨ )
1

/0

/ 0.

For the assertion (2), take a resolution
0 → G → Γ → Γ′ → 0
of G by Barsotti-Tate groups Γ and Γ′ over OK . Let N and N ′ be the objects′of the category Mod1,φ/S corresponding to Γ and Γ , respectively. Then we
have exact sequences
0 → ωΓ′ → ωΓ → ωG → 0,

0 → N ′ → N → M → 0.

By tensoring ÕK to the ﬁrst sequence and using the assertion (1), we obtainan exact sequence
Fil1 N ′ /(pFil1 N ′ + (Fil1 S)N ′ ) → Fil1 N /(pFil1 N + (Fil1 S)N ) → ωG1 → 0.
This and the second sequence induce the isomorphism Fil1 M/(Fil1 S)M →ωG1 .
For the assertion (3), choose a basis e1 , . . . , eh of M and put
φ(e1 , . . . , eh ) = (e1 , . . . , eh )Afor some A ∈ Mh (k[[u]]). Let us1 , . . . , ush be the elementary divisors ofA, which satisfy 0 ≤ si ≤ e for any i. Note that the matrix ue A−1 iscontained in Mh (k[[u]]) and its elementary divisors are ue−s1 , . . . , ue−sh . By
the deﬁnition of the functor MS (−), we have the equality
Fil1 MS (M)/(Fil1 S)MS (M) = SpanS1 ((1 ⊗ e1 , . . . , 1 ⊗ eh )ue A−1 ).

10SHIN HATTORI

Thus, by the assertion (2), we obtain an isomorphism
ωG(M)1 ≃ ⊕i ue−si k[[u]]/ue k[[u]],∑which implies the equalities deg(G(M)) = i e−1 si = e−1 vu (det φ). Fromthe explicit description of the aﬃne algebra of H(M) given before, we seethat this is also equal to deg(H(M)).
□
Let G and M be as in Lemma 2.5 (1) and set M to be the object ofMod1,φ/S1 which corresponds to M via the equivalence MS (−). We let h andd denote the height and the dimension of G, respectively. Let e1 , . . . , eh bea basis of M and A be the element of Mh (k[[u]]) withφ(e1 , . . . , eh ) = (e1 , . . . , eh )A.
We also put M1 = M/ue M and Fil1 M1 = (1 ⊗ φ)(ÕK ⊗φ,ÕK M1 ) ⊆ M1 .
Then we have an isomorphism of ÕK -modules
M/(Fil1 S)M → ÕK ⊗φ,ÕK M1 .
From Lemma 2.5 (1) and the deﬁnition of the functor MS (−), we also havenatural isomorphisms of ÕK -modules
1⊗φLie(G1∨ ) → M/Fil1 M → Fil1 M1 .
Hence the ÕK -module Fil1 M1 is free of rank h − d and each elementary
divisor of the matrix A is either 1 or ue . This implies that the ÕK -module
M1 /Fil1 M1 is free of rank d.
Consider the diagram of ÕK -modulesÕK ⊗φ,ÕK (ÕK ⊗φ,ÕK M1 )
1⊗φ⊗φ

1⊗1⊗φ

/ ÕK ⊗



1
φ,ÕK Fil M1

ÕK ⊗φ,ÕK M1

1⊗φ/ Fil1 M1/ ÕK ⊗

φ,ÕK M1
1⊗φ



/ M1

whose left horizontal arrows are surjections and right horizontal arrows arenatural inclusions. The left vertical arrow is the map induced by the Frobenius FG1 via the natural isomorphism M ≃ D∗ (G1 )(S1 → ÕK ) and the middle vertical arrow is the map induced by the left vertical arrow. Lemma 2.5(1) implies that the truncated valuation of the determinant of the latter map
is equal to Hdg(G), since the natural isomorphism Hom(G1 , Ga ) → Lie(G1∨ )takes the action of the Frobenius FG1 on the left-hand side to the action of
the Verschiebung VG1∨ on the right-hand side. We see that the outer squareis commutative and thus the right square is also commutative. Hence wehave an exact sequence
0 → Fil1 M1 → M1 → M1 /Fil1 M1 → 0of the category Mod1,φwith vp (det(φFil1 M1 )) = Hdg(G).
/Õ
K

CANONICAL SUBGROUPS VIA BREUIL-KISIN MODULES

11

Let i ≤ 1 be a non-negative rational number. Then the zeroth projection≥ipr0 : R → ÕK̄ induces an isomorphism R/m≥i
R → OK̄ /mK̄ , by which we
identify both sides. Moreover, we also have natural isomorphisms≥i≥i∨
HomÕK (Fil1 M1 , R/m≥i
R ) → HomÕK (Lie(G1 ), OK̄ /mK̄ ) → ωG1∨ ⊗ OK̄ /mK̄ .

Lemma 2.6. Let G be a truncated Barsotti-Tate group of level one overOK and M be the object of Mod1,φ/S1 which corresponds to G via the antiequivalence G(−). Then the composite≥i
G(OK̄ ) → H(M)(R) → HomÕK (Fil1 M1 , R/m≥i
R ) → ωG1∨ ⊗ OK̄ /mK̄

coincides with the Hodge-Tate map HTi for any i ≤ 1.
Proof. For a sheaf E on the site (S1 /E1 )CRYS , we let EÕ

K̄

denote the set of

sections on the natural divided power thickening ÕK̄ → ÕK̄ . For a valued
point x : Spec(OK̄ ) → G, we let x∨ : G ∨ × Spec(OK̄ ) → µp denote its dual
map. Since G is a truncated Barsotti-Tate group of level one, we have acommutative diagram
Hom(D∗ (Z/pZ)Õ , D∗ (Z/pZ)Õ )K̄≀D∗ (x)∨/ Hom(D∗ (G1 )K̄∗ÕK̄ , D (Z/pZ)ÕK̄ )≀D∗ (µp )ÕD∗ (x∨ )K̄/ D∗ (G ∨ )1 ÕK̄

where the vertical arrows are isomorphisms compatible with Hodge ﬁltrations ([3, Proposition 5.3.6]). Put M = MS (M). In particular, by Lemma
2.5 (1) we also have a commutative diagram whose vertical arrows are isomorphisms
HomS (M/Fil1 M, ÕK̄ ) ≀ωG1∨ ⊗ ÕK̄ / HomS (M, Õ )K̄≀/ D∗ (G ∨ )1 ÕK̄ .

Then HT1 (x) coincides with the image of the identity map on the upper left
corner of the former diagram by the lower composite, which is contained in
the submodule ωG1∨ ⊗ ÕK̄ .

12

SHIN HATTORI

Now we consider a diagram
G(OK̄ )a a ` _ ^ ] ] \ [d c b/
HomS,φ (M, R)
HomS (M, RDP ) o

CC
CC
CC
CC

CC
CC
CC HomS (M, ÕK̄ ) o
O
CC
CC
CC
CC
!

HomS (M/Fil1 M, ÕK̄ ) o

HomS,φ (M, ÕK̄ )∼

HomÕ (Fil1 M1 , ÕK̄ ).
K̄

Here RDP is the divided power envelope of the ring R with respect to the∗ideal m≥1
R . The right-pointing solid arrows are deﬁned by x 7→ D (x) on
the sets of sections over the divided power thickenings RDP → ÕK̄ andÕK̄ → ÕK̄ . The top left-pointed horizontal arrow maps a homomorphism
f : M → R to the homomorphism M = S⊗φ,S M → RDP deﬁned by s⊗m 7→
sφ(f (m)) and the middle left-pointed horizontal arrow is deﬁned similarly.
The bottom one is induced by the isomorphism 1⊗φ : M/Fil1 M → Fil1 M1 .
The dotted arrow is the isomorphism εM : G(OK̄ ) → TS∗ (M). From the deﬁnition, we see that the bottom square is commutative and thus the diagram
is also commutative. Hence the lemma follows.
□
3. Level one canonical subgroup
In this section, we prove the following level one case of Theorem 1.1.
Theorem 3.1. Let the notation be as in Section 1 and suppose p > 2. Let
G be a truncated Barsotti-Tate group of level one, height h and dimension dover 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 subgroup≥1−w)scheme C of G of order pd over OK such that C × Spec(OK /mK
coincides with the kernel of the Frobenius homomorphism of G ×
Spec(OK /m≥1−w). We refer the subgroup scheme C as the canonicalK
subgroup of 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 offor w/(p−1) <the Hodge-Tate map HTb : G(OK̄ ) → ωG ∨ ⊗OK̄ /m≥bK̄b ≤ 1 − w.
(3) If w < 1/2, then C coincides both with the lower ramification subgroup scheme Gb for w/(p − 1) < b ≤ (1 − w)/(p − 1) and the

CANONICAL SUBGROUPS VIA BREUIL-KISIN MODULES

13

upper ramification subgroup scheme G j+ for pw/(p − 1) ≤ j <p(1 − w)/(p − 1).
Proof. Let W (k̄) be the Witt ring of an algebraic closure k̄ of k. Choose
a Cohen ring C(k) with residue ﬁeld k and inclusions C(k) → W (k̄) andC(k) → OK . Then the OK -algebra OK ′ = OK ⊗C(k) W (k̄) is a complete
discrete valuation ring with residue ﬁeld k̄ and relative ramiﬁcation index
one over OK . Note that ﬁnite ﬂat closed subgroup schemes D and D′ of G
of the same order over OK coincide if and only if the map D → G/D′ is zeroand the latter can be checked after the base change to OK ′ . Hence if weshow the theorem for G × Spec(OK ′ ), then the upper ramiﬁcation subgroupscheme C = G pw/(p−1)+ satisﬁes all the properties in the theorem. Therefore,
replacing OK with OK ′ , we may assume that k is perfect.
We adopt the notation of the previous section. Let M be the object of
the category Mod1,φ/S1 corresponding to G via the anti-equivalence G(−). Put
M1 = M/ue M and Fil1 M1 = (1 ⊗ φ)(ÕK ⊗φ,ÕK M1 ). We identify thek-algebras k[[u]]/(ue ) and ÕK by u 7→ π, as before. As we have seen in
Subsection 2.3, we have the exact sequence
0 → Fil1 M1 → M1 → M1 /Fil1 M1 → 0, where the ÕK -module Fil1 M1 (resp. M1 /Fil1 M1 )of the category Mod1,φ/Õ
K

is free of rank h − d (resp. rank d) and the equality vp (det(φFil1 M1 )) = w
holds.
We choose once and for all a basis e1 . . . , eh of M such that e1 , . . . , eh−d
is a lift of a basis of the ÕK -module Fil1 M1 to the k[[u]]-submodule (1 ⊗φ)(S1 ⊗φ,S1 M) and eh−d+1 , . . . , eh is a lift of a basis of the ÕK -module
M1 /Fil1 M1 . Then the elements e1 , . . . , eh−d , ue eh−d+1 , . . . , ue eh form a basisof the k[[u]]-module (1 ⊗ φ)(S1 ⊗φ,S1 M) and we have
)
(
P1
P2
φ(e1 , . . . , eh ) = (e1 , . . . , eh ) e
u P3 ue P4
for some matrices Pi with entries in the ring k[[u]], where P4 is a d × dmatrix. Since the elements on the right-hand side also form a basis of(1 ⊗ φ)(S1 ⊗φ,S1 M), the matrix
(
)
P1 P2
P3 P4

is contained in GLh (k[[u]]). Moreover, since w < 1, we also have the equalityvu (det(P1 )) = ew and thus there exists a matrix P̂1 ∈ Mh−d (k[[u]]) satisfying
P1 P̂1 = uew Ih−d , where Ih−d is the identity matrix. Here we note that the
number ew is a non-negative integer.
For the uniqueness assertion in (1), we ﬁrst show the following lemma.
Lemma 3.2. Let D be a finite flat closed subgroup scheme of G over OK
and L be the subobject of M in the category Mod1,φ/S1 corresponding to the14

SHIN HATTORI

quotient G/D via the anti-equivalence G(−). Then, for any i ∈ 1e Z with
0 < i ≤ 1, the following conditions are equivalent:(1) The subgroup scheme D × Spec(OK /m≥iK ) coincides with the kernelof the Frobenius homomorphism of G × Spec(OK /m≥iK ).
(2) L/uei L = Fil1 M1 /uei Fil1 M1 .
Proof. Put N = M/L, which is the object of the category Mod1,φ/S1 corresponding to the ﬁnite ﬂat group scheme D. Note that the Frobenius
map is compatible with any morphism between schemes of characteristic p,
and that the kernel of the Frobenius of the truncated Barsotti-Tate groupd
G × Spec(OK /m≥i
K ) is a ﬁnite ﬂat closed subgroup scheme of order p . By
Theorem 2.2 (2), we see that the kernel of the Frobenius of the group scheme
H(M) × Spec(k[[u]]/(uei )) is also ﬁnite ﬂat of rank pd . Now the condition(1) in the lemma is equivalent to saying that D × Spec(OK /m≥i
K ) is killed bydthe Frobenius and of order p . By Theorem 2.2 (2), this holds if and only if
H(N) × Spec(k[[u]]/(uei )) is killed by the Frobenius and of order pd , namely
if the latter subgroup scheme coincides with the kernel of the Frobenius ofH(M) × Spec(k[[u]]/(uei )).
From the deﬁnition of the anti-equivalence H(−), we see that the Frobenius of H(M/uei M) corresponds to the natural map
1 ⊗ φ : k[[u]]/(uei ) ⊗φ,k[[u]] M/uei M → M/uei M.
Since Coker(1 ⊗ φ) = M1 /Fil1 M1 ⊗ k[[u]]/(uei ) is free of ﬁnite rank overk[[u]]/(uei ), the kernel of the Frobenius coincides with H(Coker(1 ⊗ φ)).
Thus the subgroup scheme H(N/uei N) coincides with the kernel of the
Frobenius of H(M/uei M) if and only if L/uei L = Im(1⊗φ) = Fil1 M1 /uei Fil1 M1 .
□By this lemma, the unique existence of C as in Theorem 3.1 (1) follows
from the lemma below, by putting N = M/L and C = G(N).
Lemma 3.3. There exists a unique direct summand L of the k[[u]]-module
M which is free of rank h − d such that L is stable under φ = φM and L satisfies the condition (2) of Lemma 3.2 for i = 1 − w. Moreover, the φ-module
L defines a subobject of M in the category Mod1,φ/S1 with vu (det(φL )) = ew.
Proof. Let L be a direct summand of M satisfying the condition (2) ofLemma 3.2 for i = 1 − w. Let δ1 , . . . , δh−d be a basis of L. Then we have)(
Ih−d + ue(1−w) B ′(δ1 , . . . , δh−d ) = (e1 , . . . , eh )ue(1−w) B
with some B ∈ Md,h−d (k[[u]]) and B ′ ∈ Mh−d (k[[u]]). By multiplying theinverse of the invertible matrix Ih−d + ue(1−w) B ′ , we may assume B ′ = 0.
It is enough to show that there exists B uniquely such that the resulting
L is stable under φ = φM , and that L deﬁnes an element of the category
Mod1,φ/S1 satisfying vu (det φL ) = ew.

CANONICAL SUBGROUPS VIA BREUIL-KISIN MODULES

Note that we have(P
P2
φ(δ1 , . . . , δh−d ) = (e1 , . . . , eh ) e 1
u P3 ue P4

15

)(

)
Ih−d
.
uep(1−w) φ(B)Consider the equation
φ(δ1 , . . . , δh−d ) = (δ1 , . . . , δh−d )Dfor D ∈ Mh−d (k[[u]]). This is equivalent to the following equations:{
P1 + uep(1−w) P2 φ(B) = D,ue P3 + ue+ep(1−w) P4 φ(B) = ue(1−w) BD.
From this we obtain the equation for B
BP1 = uew P3 − uep(1−w) BP2 φ(B) + uew+ep(1−w) P4 φ(B).
By multiplying P̂1 , we have
B = P3 P̂1 − uep(1−w)−ew BP2 φ(B)P̂1 + uep(1−w) P4 φ(B)P̂1 .
The assumption w < p/(p + 1) implies the inequalities ep(1 − w) − ew >0 and ep(1 − w) > 0. Therefore we can solve this equation uniquely byrecursion to obtain B and D satisfying the above equations. Moreover,
we also have D = P1 (Ih−d + uep(1−w)−ew P̂1 P2 φ(B)) and the matrix Ih−d +uep(1−w)−ew P̂1 P2 φ(B) is invertible. Hence we see that the module L deﬁnes
an object of the category Mod1,φ/S1 and vu (det(φL )) = ew. This concludes
the proof of the lemma.
□
By Lemma 2.5 (3), we obtain the equalities deg(G/G(N)) = deg(G(L)) =e−1 vu (det(φL )) = w and the part (a) of Theorem 3.1 follows.
≥1−w). Since Ḡ is a truncated
For the part (b), put Ḡ = G × Spec(OK /mK
Barsotti-Tate group of level one, we have the equalities of ﬁnite ﬂat closed
subgroup schemes(Im(FḠ ))∨ = Im(VḠ ∨ ) = Ker(FḠ ∨ ).
≥1−w
Thus the subgroup scheme (G/C)∨ × Spec(OK /mK) coincides with the∨kernel of the Frobenius of Ḡ . By the uniqueness of the canonical subgroup
we have just proved, we obtain the equality C ′ = (G/C)∨ .
Let us prove the part (c). By Theorem 2.3 and the part (b), it suﬃces
to show the equality C = G(1−w)/(p−1) . By Theorem 2.2 (1), we are reduced
to show the equality H(N) = H(M)(1−w)/(p−1) . For this, we identify anelement of the group H(M)(R) = HomS1 ,φ (M, R) deﬁned by ei 7→ xi for1 ≤ i ≤ h − d and eh−d+i 7→ yi for 1 ≤ i ≤ d with a solution (x, y) =(x1 , . . . , xh−d , y1 , . . . , yd ) in Rh of the equation()
PP2(xp , y p ) = (x, y) e 1,
u P3 ue P4

16

SHIN HATTORI

where we put xp = (xp1 , . . . , xph−d ) and similarly for y p . We use the notationin the proof of Lemma 3.3. Then we can also identify the group H(L)(R)with the set of solutions z = (z1 , . . . , zh−d ) in Rh−d of the equationz p = zD
and the natural map H(M) → H(L) is identiﬁed with the map (x, y) 7→ z =x + ue(1−w) yB.
Let (x, y) be an element of H(N)(R) = Ker(H(M)(R) → H(L)(R)). Then
we have x + ue(1−w) yB = 0 and thus, by eliminating x from the equationfor (x, y), we obtainy p = ue(1−w) y(uew P4 − BP2 ).
Hence the inequality
p min vR (yi ) ≥ 1 − w + min vR (yi )
1≤i≤d

1≤i≤d

follows, which implies vR (yi ) ≥ (1 − w)/(p − 1) and thus vR (xi ) ≥ p(1 −w)/(p−1) for any i. This shows the inclusion H(N)(R) ⊆ H(M)(1−w)/(p−1) (R).
Conversely, let (x, y) be an element of H(M)(1−w)/(p−1) (R). Then we canwrite (x, y) = ue(1−w)/(p−1) (a, b) with some tuples a, b in R and a (p − 1)-stroot ue(1−w)/(p−1) of ue(1−w) in R. These tuples satisfy the equality
)
(
P1
P2
e(1−w) p p.
u(a , b ) = (a, b) e
u P3 ue P4
)
)
(
(
P1 P2
Q1 Q2
. By multiplying this,be the inverse of the matrix
Let
P3 P4
Q3 Q4
we obtaina = ue(1−w) (ap Q1 + bp Q3 )and thus vR (xi ) ≥ p(1 − w)/(p − 1) for any i. Let z be the image of theelement (x, y) by the natural map H(M)(R) → H(L)(R). Then we havevR (zi ) ≥ p(1 − w)/(p − 1) > w/(p − 1). By Lemma 2.5 (3), the equalitydeg(H(L)) = w holds and Lemma 2.4 implies z = 0. Hence the inclusion
H(M)(1−w)/(p−1) (R) ⊆ H(N)(R) follows and we conclude the proof of thepart (c).
For the assertion (2) of Theorem 3.1, let us assume w < (p − 1)/p. Let b
be a rational number with w/(p − 1) < b ≤ 1 − w. By the deﬁnition of thesubmodule L, we have a commutative diagram
H(M)(R)/ H(L)(R)/ H(L/ue(1−w) L)(R/m≥b )H(M/ue(1−w) M)(R/m≥b
R )

WWWWW
WWWWW
WWWWW
WWWWW
+

R

H(Fil1 M1 /ue(1−w) Fil1 M1 )(R/m≥b
R )CANONICAL SUBGROUPS VIA BREUIL-KISIN MODULES

17

whose upper right vertical arrow is an injection by Lemma 2.4. Therefore
the assertion (2) follows from Lemma 2.6.
Finally, let us prove the assertion (3). Assume w < 1/2. Take i >w/(p − 1) and consider the commutative diagram with exact upper row
0/ C(O )K̄/ G(O )K̄/ (G/C)(O )K̄/ (G/C)(O /m≥i ).
K̄
K̄

)G(OK̄ /m≥iK̄/0

By Lemma 2.4, we have (G/C)i = 0 and the right vertical arrow in the
diagram is an injection. This implies the inclusion Gw/(p−1)+ ⊆ C. Thus we
obtain the inclusions
C = G(1−w)/(p−1) ⊆ Gw/(p−1)+ ⊆ C
and the equality C = Gb holds for any b satisfying w/(p − 1) < b ≤ (1 −w)/(p − 1). The assertion for upper ramiﬁcation subgroups follows from thepart (b) and Theorem 2.3.
□
4. Higher canonical subgroups
In this section, we prove Theorem 1.1 and Corollary 1.2. Though this canbe done basically by repeating arguments in [1], [8] and [22], we give a proof
here with necessary modiﬁcations for the convenience of the reader. First we
recall the following two lemmas in [8] whose proofs depend only on elementary arguments and are independent of the theory of Harder-Narasimhan
ﬁltrations or Hodge-Tate maps developed there. Note that for Lemma 4.1,
the proof of [8, Théorème 5] remains valid also for our subgroup scheme C
by Theorem 3.1 (1).
Lemma 4.1 ([8], Théorème 5). Let G be a truncated Barsotti-Tate group oflevel two, height h and dimension d < h over OK with Hdg(G) < 1/(p + 1).
Let C be the level one canonical subgroup of G[p] constructed in Theorem 3.1.
Then the group scheme p−1 C/C is a truncated Barsotti-Tate group of levelone, height h and dimension d with Hdg(p−1 C/C) = pHdg(G).
Lemma 4.2 ([8], Proposition 12). Let G be a truncated Barsotti-Tate groupof level n and dimension d over OK . Let C be a finite flat closed subgroupscheme of G over OK of order pnd . Suppose that we have the inequality
deg(C) > nd − 1. Then the group C(OK̄ ) is isomorphic to (Z/pn Z)d .
Proof of Theorem 1.1. We proceed by induction on n. The case of n = 1
is Theorem 3.1. For n ≥ 2, suppose that the theorem holds for any level
less than n. By assumption, we have the level one canonical subgroup C
of G[p] as in Theorem 3.1. Then Lemma 4.1 implies that, for n ≥ 2, thegroup scheme p−(n−1) C/C is a truncated Barsotti-Tate group of level n − 1over OK with Hdg(p−(n−1) C/C) = pw. Using the induction hypothesis forp−(n−1) C/C, we deﬁne the level n canonical subgroup Cn of G to be the18

SHIN HATTORI

unique ﬁnite ﬂat closed subgroup scheme of G over OK containing C suchthat the quotient Cn /C is the level n − 1 canonical subgroup of p−(n−1) C/C.
Then Cn is of order pnd .
By the assertion on the Frobenius kernel for p−(n−1) C/C, we have the
n−1 w
n−1
equality Cn = (F n−1 )−1 (C (p ) ) of fppf sheaves over Spec(OK /m≥1−p
).
K
≥1−p
Since the subgroup scheme C × Spec(OK /mK

n−1 w) also coincides with

≥1−pn−1 w
the Frobenius kernel of G × Spec(OK /mK), we obtain the assertion

on the Frobenius kernel for G.
Since the multiplication by pn−1 induces an isomorphism G/p−(n−1) C →G[p]/C, we have the equality
deg(G/Cn ) = deg(G[p]/C) + deg((p−(n−1) C/C)/(Cn /C))and the part (a) of the theorem follows from the induction hypothesis.
For the part (b), it is enough to show the Cartier pairing ⟨ , ⟩G kills thesubset Cn (OK̄ ) × Cn′ (OK̄ ). Let C ′ be the canonical subgroup of G ∨ [p]. By
the construction of Cn′ , we have C ′ ⊆ Cn′ and Cn′ /C ′ ⊆ p−(n−1) C ′ /C ′ . Forx ∈ C(OK̄ ) and y ∈ Cn′ (OK̄ ), we have pn−1 y ∈ C ′ (OK̄ ) and ⟨x, y⟩G =⟨x, pn−1 y⟩G[p] = 0 by Theorem 3.1 (b). Thus the subgroup Cn′ (OK̄ ) is
contained in C(OK̄ )⊥ = (G/C)∨ (OK̄ ) ⊆ G ∨ (OK̄ ) and we have ⟨x, y⟩G =⟨x̄, y⟩G/C for any x ∈ Cn (OK̄ ) and y ∈ Cn′ (OK̄ ), where x̄ is the image of x in(G/C)(OK̄ ). Since we have an exact sequencepn−1

0 → p−(n−1) C/C → G/C → G[p]/C → 0
and an isomorphism C ′ ≃ (G[p]/C)∨ of subgroup schemes of G ∨ [p], a similarargument on the side of Cn′ implies the equality ⟨x̄, y⟩G/C = ⟨x̄, ȳ⟩p−(n−1) C/C ,where ȳ is the image of y by the surjection (G/C)∨ (OK̄ ) → (p−(n−1) C/C)∨ (OK̄ ).
Moreover, we have an isomorphism (G/C)∨ ≃ p−(n−1) C ′ of subgroup schemesof G ∨ ﬁtting into the commutative diagram
00/ (G[p]/C)∨/ (G/C)∨

≀

≀



/ C′


/ p−(n−1) C ′/ (p−(n−1) C/C)∨/0/ p−(n−1) C ′ /C ′/ 0.

Thus we obtain an isomorphism (p−(n−1) C/C)∨ → p−(n−1) C ′ /C ′ and theequality ⟨x̄, ȳ⟩p−(n−1) C/C = 0 holds by induction hypothesis.
Moreover, for w < (p − 1)/(pn − 1), the inequality deg(Cn ) = nd − w(pn −1)/(p − 1) > nd − 1 holds by the part (a) and Lemma 4.2 implies the part(d).
Next we show the part (e). By induction hypothesis, we have the subgroupscheme Ci of G[pi ] as in the theorem. Note that, by the part (d), a subgroupof Cn (OK̄ ) which is isomorphic to (Z/pn−1 Z)d is equal to Cn (OK̄ )[pn−1 ].
Hence, by induction hypothesis, it is enough to check that Cn−1 is containedin Cn . The case of n = 2 follows from the deﬁnition of Cn . Suppose n > 2.

CANONICAL SUBGROUPS VIA BREUIL-KISIN MODULES

19

For any i with 2 ≤ i ≤ n, the group scheme Ci /C is the level i − 1 canonicalsubgroup of the truncated Barsotti-Tate group p−(i−1) C/C of Hodge heightpw. The inequality p(p − 1)/(pn − 1) < (p − 1)/(pn−1 − 1) and the induction
hypothesis imply that Cn−1 /C is contained in Cn /C. Thus the part (e) follows.
Let us show the assertion (3). Assume w < (p − 1)/pn and set Kn to be
the scheme-theoretic closure in G of the subgroup Ker(HTn−w(pn −1)/(p−1) ).
We have the inequality w/(p − 1) < 1 − ϵ ≤ 1 − w for ϵ = w(pn − 1)/(p − 1)
and Theorem 3.1 (2) implies that the kernel of the map HT1−ϵ : G[p](OK̄ ) →is of order pd . Using this, we can show that the orderωG[p]∨ ⊗ OK̄ /m≥1−ϵK̄
of the group scheme Kn is no more than pnd , by an elementary argument
just as in [8, Proposition 13]. On the other hand, we have a commutative
diagram with exact rows00

/ C(O )K̄/ G(O )
K̄

HT

HT



/ ωC ∨ ⊗ O

K̄

/0

HT



/ ωG ∨ ⊗ O/ (G/C)(O )
K̄

K̄


/ ω(G/C)∨ ⊗ O

K̄

/ 0,

where we put HT(x) = (x∨ )∗ (dt/t) as before. Take x ∈ Cn (OK̄ ) andlet x̄ denote its image in (G/C)(OK̄ ). By the construction of Cn and induction hypothesis, we see that the element HT(x̄) is killed by the ideal≥w(pn −p)/(p−1)
mK̄
. Since deg(C ∨ ) = w, we have m≥w(ωC ∨ ⊗ OK̄ ) = 0. HenceK̄≥w(pn −1)/(p−1)

the element HT(x) is killed by the ideal mK̄
and we obtain the
inclusion Cn ⊆ Kn , which implies the assertion (3) by comparing orders.
Finally, the assertion (4) follows just as in the proof of [22, Theorem 2.5],
using induction hypothesis and assertions of the theorem we have alreadyproved. This concludes the proof of Theorem 1.1.
□
We can also prove the following result on anti-canonical isogenies, slightly
generalizing [8, Proposition 16].
Proposition 4.3. Let G be a truncated Barsotti-Tate group over OK of level
two, height h, dimension d 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
3.1. Let D be a finite flat closed subgroup scheme of G[p] over OK such thatthe 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. Then, by [13, Théorème 4.4 (e)], there exists a
Barsotti-Tate group Γ over OK satisfying G ≃ Γ[p2 ].
Note that the truncated Barsotti-Tate group p−1 D/D is also of heighth and dimension d. The natural homomorphism C → G[p]/D induces an20

SHIN HATTORI

isomorphism between the generic ﬁbers of both sides. Now we claim that≥1−w
the group scheme (G[p]/D) × Spec(OK /mK) is killed by the Frobenius.
′
Indeed, let N and N be the objects of the category Mod1,φ/S1 corresponding
to the ﬁnite ﬂat group schemes C and G[p]/D via the anti-equivalence G(−),
respectively. By [18, Corollary 2.2.2], the generic isomorphism (G[p]/D)∨ →C ∨ corresponds to an injection N∨ → (N′ )∨ . Then the S1 -modules ∧d N∨
and ∧d (N′ )∨ are free of rank one and we also have an injection ∧d N∨ →∧d (N′ )∨ . Hence we obtain the inequality vu (det φ(N′ )∨ ) ≤ vu (det φN∨ ). By
Theorem 3.1 (b), the object N∨ corresponds to the quotient G[p]∨ /C ′ of the
Cartier dual G[p]∨ by its canonical subgroup C ′ . Thus Lemma 2.5 (3) andTheorem 3.1 (a) imply the equality vu (det φN∨ ) = ew. From this and thedeﬁnition of the dual object of the category Mod1,φ/S1 , we see that the map
φN′ is zero modulo ue(1−w) and the claim follows from Theorem 2.2 (2).
Since the group scheme p−1 D/D is a truncated Barsotti-Tate group oflevel one and dimension d, the group scheme in the claim coincides with the≥1−wkernel of the Frobenius of the group scheme (p−1 D/D) × Spec(OK /mK).
Note the equality p−1 (G[p]/D) = G/D of closed subgroup schemes of thetruncated Barsotti-Tate group p−2 D/D. Therefore the Frobenius induces≥1−w)
an isomorphism of group schemes over Spec(OK /mK
G[p] ≃ (G/D)/(G[p]/D) = p−1 (G[p]/D)/(G[p]/D) → (p−1 D/D)(p) .
F

Considering the Hodge heights of both sides, we have the equalitymin{w, 1 − w} = min{pHdg(p−1 D/D), 1 − w},from which the assertion (1) follows. The uniqueness of the canonical subgroup in Theorem 3.1 (1) implies the assertion (2). The last assertion follows
from the isomorphism (p−1 D/D)/(G[p]/D) ≃ D and Theorem 3.1 (a).
□We give here remarks on the uniqueness of the canonical subgroup Cn .
Let G be a truncated Barsotti-Tate group over OK of level n, height h,dimension d and Hodge height w < 1/(pn−2 (p + 1)). Let Dn be a ﬁnite ﬂat
closed subgroup scheme of G over OK . Then an induction and the uniquenessassertion in Theorem 3.1 (1) show the following uniqueness of Cn : Supposethat there exists a ﬁltration 0 = D0 ⊆ D1 ⊆ D2 ⊆ · · · ⊆ Dn−1 ⊆ Dn ofﬁnite ﬂat closed subgroup schemes over OK such that Di /Di−1 is killed by
i−1 w
p, of order pd and its modulo m≥1−p
is killed by the Frobenius for any
Ki. Then we have Cn = Dn .
On the other hand, the following stronger uniqueness also holds for w <p(p − 1)/(pn+1 − 1). Note the inequalities(p − 1)/pn < p(p − 1)/(pn+1 − 1) < (p − 1)/(pn − 1).
Proposition 4.4. Let G be a truncated Barsotti-Tate group over OK of leveln, height h, dimension d and Hodge height w < p(p − 1)/(pn+1 − 1). Let Dnbe a finite flat closed subgroup scheme of G over OK such that Dn (OK̄ ) ≃CANONICAL SUBGROUPS VIA BREUIL-KISIN MODULES≥1−p(Z/pn Z)d and the group scheme Dn × Spec(OK /mK
n-th iterated Frobenius F n . Then we have Cn = Dn .

n−1 w

21

) is killed by the

Proof. We proceed by induction on n. The case of n = 1 follows from
Theorem 3.1 (1). Suppose that n ≥ 2 and the assertion holds for n − 1.
Let C1 be the scheme-theoretic closure of Cn (OK̄ )[p] in Cn and deﬁne D1
similarly. By Theorem 1.1 (e), the subgroup scheme C1 is the canonicalsubgroup of G[p]. First we claim C1 = D1 . For this, by a base change
argument as before, we may assume that the 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 the scheme-theoretic closure of thelatter intersection in G[p], which is a closed subgroup scheme of E. Let L, E
and F be the objects of the category Mod1,φ/S1 corresponding to G[p]/C1 , E andF, respectively. Since we have the inequality w < p(p − 1)/(p3 − 1) < 1/2,
Proposition 4.3 (3) implies vu (det φF ) ≤ vu (det φE ) = p−1 ew. On the other
hand, since F is also a closed subgroup scheme of Dn , the n-th iterated
n−1 w
Frobenius of F × Spec(OK /m≥1−p) is zero. By Theorem 2.2 (2), we seeK
that the entries of a representing matrix of the map
1⊗φF

1⊗φF

1⊗φF

1⊗φF

S1 ⊗φn ,S1 F → S1 ⊗φn−1 ,S1 F → · · · → S1 ⊗φ,S1 F → F
have u-adic valuations no less than e(1 − pn−1 w). Taking the valuation of
the determinant of this map, we obtain the inequalities
1 − pn−1 w ≤ e−1 vu (det φF )(pn − 1)/(p − 1) ≤ w(pn − 1)/(p2 − p),
which contradict the assumption on w and the equality C1 = D1 follows.
Now the group scheme p−(n−1) C1 /C1 is a truncated Barsotti-Tate group oflevel n − 1, height h, dimension d and Hodge height pw with the level n − 1canonical subgroup Cn /C1 . The ﬁnite ﬂat closed subgroup scheme Dn /C1≥1−pn−2 pw
satisﬁes (Dn /C1 )(OK̄ ) ≃ (Z/pn−1 Z)d and its modulo mK
is killed byn−1
F. Thus the induction hypothesis implies the equality Cn /C1 = Dn /C1 .
This concludes the proof of the proposition.
□
To show Corollary 1.2, we need a patching lemma for upper ramiﬁcation
subgroups, which is a slight generalization of [1, Proposition 8.2.2]. Let
K/Qp be an extension of complete discrete valuation ﬁelds and j be a positive rational number. Let X be an admissible formal scheme over Spf(OK )which is quasi-compact and X be its Raynaud generic ﬁber. For a ﬁnite
extension L/K, an OL -valued point x : Spf(OL ) → X and an ideal I of
OX , we let I (x) denote the ideal of OL generated by the image of I . The
p-adic valuation of a generator of this ideal is denoted by vp (I (x)). For any
x ∈ X(L), we let x also denote the map Spf(OL ) → X obtained from x bytaking the scheme-theoretic closure and the normalization.
Lemma 4.5. Let the notation be as above. Let G be a finite locally freeformal group scheme over X, G be its Raynaud generic fiber and H be a

22

SHIN HATTORI

coherent open ideal of OX . Then there exists an admissible open subgroup
GjH + of G such that for any finite extension L/K and x ∈ X(L), the fiber+
GjH
coincides with the generic fiber of the upper ramification subgroupx
jv (H (x))+
scheme Gx pof the finite flat (formal) group scheme Gx = G ×X,xSpf(OL ) over OL .
Proof. By replacing X by the admissible blow-up of X along the ideal H, we
may assume that the ideal H is invertible. It suﬃces to show the existence ofan admissible open subgroup as in the lemma locally. Thus we may assume
that X = Spf(B) is aﬃne, the ideal H is generated by h ∈ B and there exists
a closed immersion of G to a projective abelian scheme A over Spec(B), by atheorem of Raynaud ([3, Théorème 3.1.1]). Let P be the formal completionof A along the special ﬁber, P be its Raynaud generic ﬁber and I be thedeﬁning ideal of the closed immersion G → P. For positive rational numbers
j and j ′ , write j = m/n and j ′ = m′ /n with positive integers m, m′ , n and′
put J = I n + pm hm OP . Let B be the admissible blow-up of P along′′
the ideal J and Zm,m ,n,h be the formal open subscheme of B where pm hm
generates the ideal J OB .
′The Raynaud generic ﬁber of Zm,m ,n,h is the admissible open subset of
P whose set of K̄-valued points is given by{y ∈ P (K̄)|vp (I (y)) ≥ jvp (h(y)) + j ′ }and it is independent of the choice of m, m′ , n. We write this Raynaud′′
generic ﬁber as Z jh+j . We claim that the admissible open subset Z jh+j
is a rigid-analytic subgroup of P . For this, take a ﬁnite extension L/K′
and x ∈ X(L). It is enough to show that the ﬁber (Z jh+j )x is a rigidanalytic subgroup of Px . This is the admissible open subset of Px consistingof y ∈ Px (L̄) such that the inequality vL (I (y)) ≥ e(L/Qp )(jvp (h(x)) + j ′ )holds, where vL is the normalized valuation of L. Taking a suﬃciently large
L, we may assume that the constant e(L/Qp )(jvp (h(x)) + j ′ ) is an integer.
Then the universality of the dilatation implies the claim, as in the proof of[1, Proposition 8.2.2].
′
By [1, Proposition A.1.2], there exists an admissible open subgroup Z jh+j ,0′′of Z jh+j such that for any x ∈ X(K̄), the ﬁber (Z jh+j ,0 )x coincides with′′the unit component (Zxjh+j )0 of the rigid-analytic group Zxjh+j . Put∪∪′′
Z jh+ =
Z jh+j , Z jh+,0 =Z jh+j ,0
j ′ >0j ′ >0and GjH + = G ∩ Z jh+,0 . These are admissible open subgroups of P and
G, respectively. To show that GjH + satisﬁes the property as in the lemma,
it is enough to show that, for any ﬁnite extension L/K and x ∈ X(L),′the ﬁber (Z jh+j )x is naturally isomorphic to the tubular neighborhood′
X jvp (h(x))+j (Gx → Z) of a closed immersion from Gx to a formal aﬃnescheme Z which is an object of the category SFAOL ([1, Notation 1.5]).
Take x ∈ X(L). Since A is projective and Gx is ﬁnite over OL , there exists

CANONICAL SUBGROUPS VIA BREUIL-KISIN MODULES

23

an aﬃne open subscheme Ux of Ax such that the closed immersion Gx → Ax
factors through Ux . Let Ûx be its formal completion along the special ﬁber
and Ux be its Raynaud generic ﬁber. Then Ûx is an object of the category
SFAOL . Moreover, Ûx is a formal aﬃne open subscheme of Px through which′
the closed immersion Gx → Px factors and the restriction (Z jh+j )x |Ux co′
incides with the tubular neighborhood X jvp (h(x))+j (Gx → Ûx ). Therefore′′
we are reduced to show the equality (Z jh+j )x |Ux = (Z jh+j )x . For this,′take y ∈ (Z jh+j )x (L̄). Then we have vp (I (y)) ≥ jvp (h(x)) + j ′ > 0. Put
Px \Ûx = V (Ix′ ) with some ideal Ix′ ⊆ OPx . Let P̄x and ȳ denote the special
ﬁbers of Px and y, respectively. Consider the inverse image I¯x (resp. I¯x′ )of Ix (resp. Ix′ ) in P̄x . Since V (I¯x ) ∩ V (I¯x′ ) = ∅, we have I¯x + I¯x′ = (1)in any open aﬃne neighborhood of the image of ȳ in P̄x . Thus the image is′
not contained in V (I¯x′ ) and we obtain y ∈ (Z jh+j )x |Ux .
□Proof of Corollary 1.2. Deﬁne a positive rational number j and an open
coherent ideal H of OX by j = p(pn − 1)/(p − 1)2 and H = HaI (G[p]) ([8,2.2.2]). We deﬁne Cn to be the admissible open subgroup GjH + of G as inLemma 4.5. Then, by Theorem 1.1 (c) and (4), each ﬁber (Cn )x coincides
with the generic ﬁber of the level n canonical subgroup of Gx . Since G is
etale over X, the admissible open subgroup Cn of G is also etale over X.
Since its ﬁbers over X(rn ) are isomorphic to the group (Z/pn Z)d by Theorem
1.1 (d), [1, Lemme A.1.1] implies that Cn is ﬁnite over X(rn ). Hence therigid-analytic group Cn over X(rn ) is etale locally constant, and thus etale
locally on X(rn ) this is isomorphic to the constant group (Z/pn Z)d .
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E-mail address: shin-h@math.kyushu-u.ac.jp
Faculty of Mathematics, Kyushu University