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
with standard properties for G if the Hasse invariant of G is less than
1/(2pn−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 , we let
OF denote its valuation ring and put m≥i
F = {x ∈ F | vF (x) ≥ i} and≥i
OF,i = OF /mF for any positive real number i. We also put ÕK = OK,1
and ÕK̄ = OK̄,1 .
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 p-adic completion of the Siegel modular variety of genus g and level prime to p over
the Witt ring W (k), X be its Raynaud generic ﬁber, X ord be its ordinary
locus considered as an admissible open of X and A be the universal abelianscheme over X. Consider the unit component A[pn ]0 of the pn -torsion of
A. The rigid-analytic subgroup A[pn ]0 |X ord is etale locally isomorphic to theconstant group (Z/pn Z)g and it is a lift of the kernel of the n-th iterated
Frobenius of the special ﬁber of A. Then the theorem asserts that this subgroup can be extended to a subgroup Cn with the same properties over alarger rigid-analytic subspace X(r) of X containing X ord . In [9], the author
proved the existence of such a subgroup over the locus of the Hasse invariant
less than 1/(2pn−1 ) for p ≥ 3. The aim of this paper is to generalize theresult to the case of p = 2.
To state the main theorem, we introduce some notation. For any ﬁnite
torsion OK -module M which is isomorphic to OK /(π r1 ) ⊕ · · · ⊕ OK /(π rl ),we deﬁne its degree by deg(M ) = e−1 (r1 + · · · + rl ). For any ﬁnite ﬂat
Date: February 7, 2012.
Supported by Grant-in-Aid for Young Scientists B-23740025.
1

2

SHIN HATTORI(commutative) group scheme G (resp. Barsotti-Tate group Γ) over OK ,we let ωG (resp. ωΓ ) denote its module of invariant diﬀerentials and putdeg(G) = deg(ωG ). 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 thenatural 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 Hasse invariant Ha(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 of complete discrete valuation fields. Let G be a truncated Barsotti-Tate group oflevel n, height h and dimension d over OK with 0 < d < h and Hasse invariant w = Ha(G). If w < 1/(2pn−1 ), then the upper ramification subgroupscheme Cn = G j+ forpw(pn − 1)/(p − 1)2 ≤ j < p(1 − w)/(p − 1)satisfies Cn (OK̄ ) ≃ (Z/pn Z)d . Moreover, the group scheme Cn has the following properties:(a) deg(G/Cn ) = w(pn − 1)/(p − 1).
(b) Cn × Spec(OK,1−pn−1 w ) coincides with the kernel of the n-th iterated
Frobenius of G × Spec(OK,1−pn−1 w ).
(c) The scheme-theoretic closure of Cn (OK̄ )[pi ] in Cn coincides with the subgroup scheme Ci of G[pi ] for 1 ≤ i ≤ n − 1.
(d) The subgroup Cn (OK̄ ) coincides with the kernel of the Hodge-Tate map
HTn−w(pn −1)/(p−1) : G(OK̄ ) → ωG ∨ ⊗ OK̄,n−w(pn −1)/(p−1) .

CANONICAL SUBGROUPS FOR p = 23(e) Put Cn′ = (G ∨ )j+ for j as above. Then we have the equality Cn (OK̄ ) =Cn′ (OK̄ )⊥ , where ⊥ means the orthogonal subgroup with respect to theCartier pairing⟨ , ⟩G : G(OK̄ ) × G ∨ (OK̄ ) → µpn (OK̄ ).
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
the Raynaud generic fibers of the formal schemes X and G, respectively. Let
Gj+ be the family of j-th upper ramification subgroups as in [9, Lemma 4.3].
For any 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
the scheme-theoretic closure and the normalization. For any non-negative
rational number r, let X(r) be the rigid-analytic subspace of X defined byX(r)(K̄) = {x ∈ X(K̄) | Ha(Gx ) < r}.
Then the rigid-analytic group Gj+ |X(r) over X(r) is etale locally isomorphic to the constant group (Z/pn Z)d for r = 1/(2pn−1 ) and j = (2pn−1 −1)/(2pn−2 (p − 1)).
The basic strategy of the proof of the main theorem is the same as in[9]: we reduce to the case where k is perfect and then switch to a study of
ramiﬁcation of ﬁnite ﬂat group schemes over a complete discrete valuation
ring of equal characteristic p with residue ﬁeld k. In [9], we carried out
this switching by the ramiﬁcation correspondence theorem of the author([8]). The latter theorem is based on a classiﬁcation of ﬁnite ﬂat groupschemes over OK killed by p for p ≥ 3 in terms of ϕ-modules over the formal
power series ring k[[u]], which is due to Breuil ([3], [4]) and Kisin ([11],[12]). Instead, here we use a similar classiﬁcation of unipotent ﬁnite ﬂatgroup schemes also due to Kisin which holds for any p ([13]) to prove a
correspondence result of lower ramiﬁcation subgroups between mixed andequal characteristics. We seek the canonical subgroup Cn of the truncatedBarsotti-Tate group G in its upper ramiﬁcation subgroup schemes and they
depend only on its unit component G 0 . Moreover, we prove properties of
Cn by studying lower ramiﬁcation subgroups of the unipotent group scheme(G 0 )∨ via Cartier duality. Thus the classiﬁcation theorem of unipotent groupschemes suﬃces to prove our existence theorem of canonical subgroups.
2. Classification of unipotent finite flat group schemes
In this and the next section, we assume that the residue ﬁeld k of K isperfect. For p ≥ 3, we have a classiﬁcation theory of Barsotti-Tate groups
and ﬁnite ﬂat group schemes over OK due to Breuil ([3], [4]) and Kisin ([11],4SHIN HATTORI[12]) in terms of so-called Breuil-Kisin modules. Kisin ([13]) also extended
this classiﬁcation to the case of p = 2 and where groups are connected,using Zink’s classiﬁcation of formal Barsotti-Tate groups ([16], [17]). In thissection, 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 = πpand πn+1= πn . Put K∞ = ∪n≥0 K(πn ), S = W [[u]] and S1 = k[[u]]. 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 suchthat a ϕ-semilinear map ϕM : M → M is given. We write ϕM also as ϕ if
no confusion may occur. We follow the notation of [9, Subsection 2.1]. In1,ϕ1,ϕparticular, we have categories Mod1,ϕ/S , Mod/S1 , Mod/S∞ of Kisin modulesof E-height ≤ 1 and a category Mod1,ϕ/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 [13], respectively.
Let S be the p-adic completion of the divided power envelope of W [u] withrespect to the ideal (E(u)). The ring S has a natural ﬁltration Fil1 S inducedby the divided power structure, a ϕ-semilinear Frobenius endomorphism
denoted also by ϕ and a ϕ-semilinear map ϕ1 : Fil1 S → S. Then a Breuil
module is an S-module M such that an S-submodule Fil1 M containing(Fil1 S)M and a ϕ-semilinear map ϕ1,M : Fil1 M → M are given and thesesatisfy some compatibility conditions. The map ϕ1,M is also denoted by ϕ1
if there is no risk of confusion. We also have categories of Breuil modules
1,ϕ1,ϕMod1,ϕ/S , Mod/S1 and Mod/S∞ (for the deﬁnitions, see [8, Subsection 2.1]).
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 asϕ.

CANONICAL SUBGROUPS FOR p = 2

5∗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 [13] and [16]. The full subcategoryof topologically V -nilpotent objects is denoted by Mod1,ϕ,V/S . For any Kisin
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

Mod1,ϕ,V([13, Proposition 1.2.5]).
/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 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 )([6, Proposition B1.8.3] and [13, 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 ([11, 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 ([13, Proposition 1.2.7]). Similarly, for any object

6

SHIN HATTORI

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 ([13, 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 ([13, Subsection 1.3]).
Theorem 2.1.
(1) ([13], Theorem 1.2.8) There exists an anti-equivalenceof categories
G(−) : Mod1,ϕ,V→ (BT/OK )u/SCANONICAL SUBGROUPS FOR p = 2

7

with a natural isomorphism Mod(G(M)) → MS (M). Moreover, we
also have a natural isomorphism of GK∞ -modules εM : Tp (G(M)) →∗ (M (M)).
TcrysS(2) ([13], Theorem 1.3.9) There exists an anti-equivalence of categoriesuG(−) : Mod1,ϕ,V/S∞ → (p-Gr/OK )with a natural isomorphism of GK∞ -modules εM : G(M)(OK̄ ) →TS∗ (M).
□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
Let M be an object of the category Mod1,ϕ,V/S .

D∗ (g) : D∗ (G(M))(Acrys → OC ) → D∗ (Qp /Zp )(Acrys → OC ) = Acrys .
A similar argument as in the proof of [12, Proposition 1.1.11] shows thatfor any 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.
This 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. 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 .
□

8SHIN HATTORI

3. Ramification correspondence
Let K be as in the previous section. Recall that, for any k[[u]]-algebra B,
we have an anti-equivalence H(−) from the category Mod1,ϕ/B to a category ofﬁnite locally free group schemes over B whose Verschiebung is the zero map([7, Théorème 7.4]. 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 thenatural isomorphism εM : G(M)(OK̄ ) → TS∗ (M) = H(M)(R) of Theorem
2.1. Consider the lower ramiﬁcation subgroups of the group scheme H(M)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. In this section, we prove that the isomorphism εM preserves the lower ramiﬁcation subgroups of both sides. In[8] where we treated the case of p ≥ 3, we proved this assertion for any ﬁnite
ﬂat group scheme killed by p by using an explicit description of the aﬃne
algebra of G(M) due to Breuil ([3]). For the case where group schemes areunipotent and p ≥ 2, a crystalline method as in [2] is enough to show thecorrespondence of lower ramiﬁcation subgroups, though it seems insuﬃcientfor obtaining a similar description of the aﬃne algebra.
Let i ≤ 1 be a positive rational number and put S¯i = Spec(OK̄,i ). We
let RiDP abusively denote the divided power envelope of the ring R withrespect to the kernel of the natural surjection pr0 : R → OK̄,i . The induced
surjection RiDP → OK̄,i is considered as an element of the crystalline site
CRYS(S1 /E1 ) and let this thickening be denoted by Ai . Note that, byﬁxing a generator x of the principal ideal m≥i
R , we have an isomorphism of
R-algebras
Rpi [Y1 , Y2 , . . .]/(Y1p , Y2p , . . .) → RiDP
which sends Yl to the pl -th divided power γpl (x). The ring RiDP is endowedwith the natural ﬁltration deﬁned by the divided power structure and wehave a natural isomorphism Rpi → RiDP /Filp RiDP . The ring R1DP is written
also as RDP and is naturally identiﬁed with the ring Acrys /pAcrys . We also
identify the module of sections D∗ (Z/pZ)(Ai ) with the ring RiDP .
Lemma 3.1. Let i ≤ 1 be a positive rational number and G be a unipotent
finite flat group scheme over OK,i killed by p. Then the natural homomorphism of abelian groups
G(O ) = Hom(Z/pZ, G × S¯i ) → Hom DP (D∗ (G)(Ai ), RDP /Filp RDP )
K̄,i

Ri

,ϕ

i

i

defined by g 7→ D∗ (g) and the identification D∗ (Z/pZ)(Ai ) ≃ RiDP above isan injection.
Proof. Note that, for any exact sequence of ﬁnite locally free group schemes
over OK̄,i killed by p
0 → H′ → H → H′′ → 0,

CANONICAL SUBGROUPS FOR p = 2

9

we have an exact sequence of abelian groups
0 → H′ (OK̄,i ) → H(OK̄,i ) → H′′ (OK̄,i )and an exact sequence of ϕ-modules over RiDP
0 → D∗ (H′′ )(Ai ) → D∗ (H)(Ai ) → D∗ (H′ )(Ai ) → 0
by [2, Proposition 4.3.1]. Thus, by a dévissage argument, we may assume
that the Verschiebung of the group scheme G is zero. In this case, by [2,Proposition 4.3.6] there exists a natural isomorphism
D∗ (G)(Ai ) → Lie(G ∨ × S¯i ) ⊗OK̄,i ,ϕ RiDP ,where the map ϕ : OK̄,i → RiDP is induced by ϕ : RiDP → RiDP . By thenatural isomorphism Rpi → RiDP /Filp RiDP , the group on the right-hand sideof the theorem is identiﬁed with
HomR,ϕ (Lie(G ∨ × S¯i ) ⊗OK̄,i ,ϕ Rpi , Lie((Z/pZ)∨ ) ⊗OK̄,i ,ϕ Rpi ).
Since the map pr0 : R → OK̄,i induces an isomorphism Ri → OK̄,i and themap ϕ : OK̄,i ≃ Ri → Rpi is also an isomorphism, the claim follows from [7,Théorème 7.4].
□
Theorem 3.2. For any object M of the category Mod1,ϕ,V/S1 , the natural
isomorphism εM : G(M)(OK̄ ) → H(M)(R) induces an isomorphism of lower
ramification subgroups
G(M)i (OK̄ ) → H(M)i (R)for any positive rational number i.
Proof. Put G = G(M). Since the i-th lower ramiﬁcation subgroups of G
and H(M) vanish for i > 1/(p − 1) ([8, Corollary 3.5 and Remark 3.6]), we
may assume i ≤ 1/(p − 1) ≤ 1. Consider a resolution of G by unipotent
Barsotti-Tate groups
0 → G → Γ1 → Γ2 → 0
and the associated exact sequence of Kisin modules
0 → N2 → N1 → M → 0.

10SHIN HATTORIPut M = D∗ (G)(S → OK ) and Ni = D∗ (Γi )(S → OK ). Then we have adiagram∼Tp (Γ1 )∼

Tp (Γ2 )πG/ T ∗ (N1 ) ocrys/ T ∗ (N2 ) o
crys
πM





TS∗ (N1 )∼TS∗ (N2 )



πM

εM



,

/ HomS,ϕ (M, RDP ) o

G(OK̄ )
G(OK̄,i ) ∼/ HomS,ϕ (M, RDP /Filp RDP ) o
i

∼

i



TS∗ (M)


HomS,ϕ (M, Ri ),where the left horizontal arrows are induced by g 7→ D∗ (g) and the righthorizontal arrows are the maps sending f to 1 ⊗ (ϕ ◦ f ). The middle left
vertical arrow πG : Tp (Γ2 ) → G(OK̄ ) is deﬁned as follows: For g ∈ Tp (Γ2 ),the element pg is contained in the image of Tp (Γ1 ) = limn Γ1 [pn ](OK̄ ) and←−put pg = h = (hn )n>0 . Then the element h1 ∈ Γ1 [p](OK̄ ) is contained in thesubgroup G(OK̄ ) and the map πG is deﬁned by g 7→ h1 . We deﬁne the map∗ (N ) → HomDP ) similarly: For any map f : N → A
πM : Tcrys
22crys ,S,ϕ (M, R
the map pf induces a map N1 → Acrys . Its composite with the natural map
Acrys → RDP factors through M and deﬁnes the map πM (f ) : M → RDP .
The map πM is deﬁned in the same way. From these deﬁnitions, we see thatthe diagram is commutative. Note that the bottom right horizontal arrowis an isomorphism since ϕ : Ri → Rpi is an isomorphism and the bottom left
horizontal arrow is an injection by Lemma 3.1. The upper squares induce
the isomorphism εM : G(OK̄ ) → TS∗ (M) = H(M)(R) and we also havethe equality H(M)(Ri ) = HomS,ϕ (M, Ri ). Thus we obtain a commutative
diagram of abelian groups∼εM

G(OK̄ )


G(OK̄,i ) / H(M)(R)/ H(M)(Ri ),where the upper horizontal arrow is a bijection and the lower horizontal
arrow is an injection. Hence the theorem follows.
□
4. 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. A key step isthe following theorem treating the case of level one.

CANONICAL SUBGROUPS FOR p = 2

11

Theorem 4.1. Let K/Qp be an extension of complete discrete valuation
fields. Let G be a truncated Barsotti-Tate group of level one, height h anddimension d over OK with 0 < d < h and Hasse invariant w = Ha(G).
(1) Suppose w < (p2 − p − 1)/(p2 − 1) for p ≥ 3 and w < 1/2 for p = 2.
Then the upper ramification subgroup scheme C = G pw/(p−1)+ is oforder pd . Moreover, the group scheme C has the following properties:(a) deg(G/C) = w.
(b) C × Spec(OK,1−w ) coincides with the kernel of the Frobenius of
G × Spec(OK,1−w ).
(2) If w < 1/2, then C coincides with the upper ramification subgroupscheme G j+ for pw/(p−1) ≤ j < p(1−w)/(p−1). The group scheme
C also has the following properties:(c) The subgroup C(OK̄ ) coincides with the kernel of the HodgeTate map HTb : G(OK̄ ) → ωG ∨ ⊗ OK̄,b for w/(p − 1) < b ≤(1 − w)/(p − 1).
(d) Put C ′ = (G ∨ )j+ for j as above. Then we have the equality
C(OK̄ ) = C ′ (OK̄ )⊥ , where ⊥ means the orthogonal subgroupwith respect to the Cartier pairing ⟨ , ⟩G .
Proof. By a base change argument as in the proof of [9, Theorem 3.2], 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.
Let M be the object of the category Mod1,ϕ,V/S1 corresponding to the unipo0∨
tent group scheme (G ) via the anti-equivalence G(−). Put M1 = M/ue M
and AM1 = (1⊗ϕ)(ϕ∗ M1 ). By the k-algebra isomorphism k[[u]]/(ue ) → ÕK
deﬁned by u 7→ π, we identify both sides of the isomorphism. Then themodules M1 and AM1 are naturally considered as objects of the category
Mod1,ϕand by Lemma 2.2 we can show that there exists a natural iso/ÕK

morphism of ÕK -modules Lie(G 0 ) → AM1 as in [9, Subsection 2.3]. Since
Ha(G) = Ha(G ∨ ) and Lie(G) = Lie(G 0 ), we obtain an exact sequence of
ϕ-modules over ÕK
0 → AM1 → M1 → M1 /AM1 → 0

which splits as a sequence of ÕK -modules and the equality of truncatedvaluation vp (det(ϕAM1 )) = w. We choose a basis e1 , . . . , eh0 of M such thatthe images of e1 , . . . , ed form a basis of AM1 and the images of ed+1 , . . . , eh0form a basis of M1 /AM1 . Deﬁne matrices P1 , P2 , P3 , P4 with coeﬃcients in12SHIN HATTORI

the ring k[[u]] by P1 ∈ Md (k[[u]]) and

(

)
P1
P2
ϕM (e1 , . . . , eh0 ) = (e1 , . . . , eh0 ) e.
u P3 ue P4

We have the equality vR (det(P1 )) = w and thus there exists an elementP̂1 of Md (k[[u]]) satisfying P1 P̂1 = P̂1 P1 = uew Id , where Id is the identity
matrix. Put L = k[[u]]e1 ⊕ · · · ⊕ k[[u]]ed , which we consider as an object of
the category Mod1,ϕ/S1 by puttingϕL (e1 , . . . , ed ) = (e1 , . . . , ed )P1 .
We can lift the modulo ue(1−w) of the above sequence to an exact sequenceof the category Mod1,ϕ/S10→L→M→N→0
as in the proof of [9, Lemma 3.4] and the objects L and N are also containedin the full subcategory Mod1,ϕ,V/S1 .
Lemma 4.2. The subgroup H(N)(R) coincides with the lower ramification
subgroup H(M)(1−w)/(p−1) (R).
Proof. Let x be an element of the subgroup H(N)(R) ⊆ H(M)(R) andidentify this with a map M → R deﬁned by ei 7→ xi for 1 ≤ i ≤ d anded+i 7→ zi for 1 ≤ i ≤ h0 − d. The h0 -tuple (x1 , . . . , xd , z1 , . . . , zh0 −d ) is asolution in R with vR (xi ) ≥ 1 − w of the equation
π ew (x1 , . . . , xd ) = (xp1 , . . . , xpd )P̂1 − π e (z1 , . . . , zh0 −d )P3 P̂1 ,(z1p , . . . , zhp0 −d ) = (x1 , . . . , xd )P2 + π e (z1 , . . . , zh0 −d )P4 .
Then the proof of [9, Lemma 3.5] works verbatim and the lemma follows forp ≥ 3.
We need a special care for the case of p = 2, as follows. Put ε = 1/2 − w.
As in loc. cit., we ﬁrst show the inequality vR (xi ) ≥ 1 for any i. Consider
the sequences {ξn }n and {ζn }n deﬁned by ξ0 = 1/2 + ε, ζ0 = 1/4 + ε/2 andξn+1 = min(2ξn , 1 + ζn ) − w,ζn+1 = 2−1 min(ξn , 1 + ζn ),which satisfy ξn ≥ 0 and ζn ≥ 0. It is enough to show ξn ≥ 1 for some n.
Suppose on the contrary ξn < 1 for any n. Then we have ζn+1 = ξn /2 andξn+1 = min(2ξn − 1/2 + ε, ξn−1 /2 + 1/2 + ε).
Each ξn is an increasing function of ε and we may assume ε < 1/6. Then we
have ξ1 = 1/2+3ε. Suppose that the minimum on the right-hand side of theabove equality is obtained by the ﬁrst term 2ξn − 1/2 + ε for any n. Then we
have ξn+1 = 1/2 + (2n+2 − 1)ε, which contradicts to the assumption ξn < 1.
Let N be the smallest integer such that the minimum is obtained by the

CANONICAL SUBGROUPS FOR p = 2

13second term ξN −1 /2 + 1/2 + ε. Then we have ξN +1 = 3/4 + (2N −1 + 1/2)ε,ξN +2 = 3/4 + (2N + 1/2)ε andξN +2l+1 = (2l+2 − 1)/2l+2 + (2N −l−1 + 2 − 3/2l+1 )ε,ξN +2l+2 = (2l+2 − 1)/2l+2 + (2N −l + 2 − 3/2l+1 )εfor any non-negative integer l. However, this sequence satisﬁes ξN +2l+1 ≥ 1
for suﬃciently large l, which is a contradiction and we obtain the inequalityvR (xi ) ≥ 1 for any i. Then the rest of the argument of loc. cit. works
verbatim and the lemma follows also for p = 2.
□
Hence we see that if w is as in the theorem, the equalities
H(N)(R) = Ker(H(M)(R) → H(AM1 )(Ri )) = H(M)(1−w)/(p−1) (R)hold for any rational number i satisfying w/(p − 1) < i ≤ 1 − w and thesesubgroups are of order ph0 −d . In this case, we put b = (1 − w)/(p − 1).
Moreover, we also see that if w < 1/2, then these subgroups coincide withthe subgroup H(M)b (R) for w/(p − 1) < b ≤ (1 − w)/(p − 1). By Theorem
3.2, the subgroup scheme G(N) of (G 0 )∨ is equal to the lower ramiﬁcation
subgroup scheme ((G 0 )∨ )b for both cases.
Now we deﬁne the subgroup scheme C of G 0 to be the scheme-theoretic closure of the orthogonal subgroup ((G 0 )∨ )b (OK̄ )⊥ with respect to the Cartierpairing ⟨ , ⟩G 0 . The group scheme C is of order pd . We insert here thefollowing lemma due to the lack of references.
Lemma 4.3. 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 .
Then 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], thefunctor 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.

14

SHIN HATTORI

By this lemma and a theorem of Tian and Fargues ([15, Theorem 1.6]or [5, Proposition 6]), we have the equalities C = (G 0 )j+ = G j+ for j =p/(p − 1) − pb and the ﬁrst assertions of (1) and (2) hold.
Let us show the assertion (a). From the deﬁnition, we see that the groupscheme C is isomorphic to ((G 0 )∨ /G(N))∨ and thus we have the equalities
deg(G/C) = deg((G 0 )∨ /G(N)) = deg(G(L))= deg(Fil1 MS (L)/(Fil1 S)MS (L)),
where the last equality follows from Lemma 2.2. The ÕK -module of the lastterm is equal to
SpanÕK ((1 ⊗ e1 , . . . , 1 ⊗ ed )P̂1 ue(1−w) )and we see that its degree is equal to w.
For the assertion (b), we need the following lemma.
Lemma 4.4. Let i ≤ 1 be a positive rational number and H be a finite
locally free group scheme over OK̄,i killed by p and of order ph . Then the
Verschiebung VH of the group scheme H is zero if and only if the OK̄,i module νH∨ = Ext 1S¯ (H, Ga )S¯i is locally free of rank h.
iProof. By [2, Proposition 4.3.1], we have an exact sequence of locally free
RiDP -modules
D∗ (H)(Ai ) → D∗ (H(p) )(Ai ) → D∗ (Ker(VH ))(Ai ) → 0,
V

where V denotes the induced map D∗ (VH ). On the other hand, by [2, Proposition 4.3.9], we also have a natural isomorphism of RiDP -modules
D∗ (H(p) )(Ai )/V D∗ (H)(Ai ) → νH∨ ⊗OK̄,i ,ϕ RiDP .
Thus the map VH is zero if and only if the natural surjectionD∗ (H(p) )(Ai ) → D∗ (H(p) )(Ai )/V D∗ (H)(Ai )is an isomorphism, and it is equivalent to saying that the RiDP -moduleνH∨ ⊗OK̄,i ,ϕ RiDP is locally free of rank h. Since we have isomorphisms
ϕOK̄,i → Rpi → RiDP /Filp RiDP , this holds if and only if the OK̄,i -module νH∨is locally free of rank h.
□
Now we put H = G(L) × S¯i . We have an exact sequence of OK̄,i -modules
ωH → D∗ (H)(OK̄,i → OK̄,i ) → νH∨ → 0
for any positive rational number i ≤ 1. Note that the OK̄,i -module νH∨is the ﬁrst homology group of the Lie complex of the Cartier dual H∨ ([2,Subsection 3.2.1]) and thus its formation is compatible with any base change.
Therefore we obtain a natural isomorphism of OK̄,i -modules(MS (L)/Fil1 MS (L)) ⊗ÕK OK̄,i → νH∨ .

CANONICAL SUBGROUPS FOR p = 2

15

Let us1 , . . . , usd be the elementary divisors of the matrix P1 and put L1 =L/ue L. Then we have isomorphisms
1⊗ϕ

MS (L)/Fil1 MS (L) → (1 ⊗ ϕ)(ϕ∗ L1 ) → ⊕dl=1 k[[u]]/(ue−sl ).
The equality s1 + · · · + sd = ew implies the inequality e(1 − w) ≤ e − sl for
any l and we see that the OK̄,i -module νH∨ is free of rank d for i = 1 − w.
By Lemma 4.4, the Verschiebung of the group scheme G(L) × S¯1−w is zeroand its Cartier dual C × S¯1−w is killed by the Frobenius. Since the groupscheme C is of order pd , we obtain the assertion (b).
Next we prove the assertion (d). We ﬁrst show the following lemmas.
Lemma 4.5. Let H be a finite flat group scheme over OK killed by pn .
Then the Cartier pairing⟨ , ⟩H : H(OK̄ ) × H∨ (OK̄ ) → µpn (OK̄ )sends the subset Hi (OK̄ ) × (H∨ )i′ (OK̄ ) into (µpn )i+i′ (OK̄ ) for any positive
rational numbers i and i′ .
Proof. Let B and J (resp. B ∨ and J ∨ ) be the aﬃne algebra and theaugmentation ideal of H (resp. of H∨ ). Let (h, h∨ ) be an element of thesubset in the lemma. Note that we have a natural decomposition B =OK ⊕ J as an OK -module. We identify the element h with an OK -algebra
homomorphism h∗ : B → OK̄ which sends the ideal J into m≥i. On the otherK̄
hand, the element h∨ can be identiﬁed with an OK̄ -algebra homomorphismn(h∨ )∗ : OK̄ [T ]/(T p − 1) → B ⊗ OK̄ which sends the element T − 1 into′J ⊗ m≥i. The element ⟨h, h∨ ⟩H of the group µpn (OK̄ ) is deﬁned by theK̄ncomposite h∗ ◦(h∨ )∗ : OK̄ [T ]/(T p −1) → OK̄ . Hence the lemma follows. □Lemma 4.6. Let H be a finite flat group scheme over OK . Then the i-th
lower ramification subgroup scheme Hi is connected for any positive rational
number i.
Proof. We have a commutative diagram0/ H0 (O )K̄/ H(O )K̄

/ Het (O )
K̄





H(OK̄,i )/0≀/ Het (OK̄,i ),

where the upper row is exact and the right vertical arrow is an isomorphism.
This implies that the subgroup Hi (OK̄ ) is contained in H0 (OK̄ ) and thelemma follows.
□
Put D = G(N) = ((G 0 )∨ )b and set D̃ to be the inverse image of D by
the natural epimorphism f ∨ : G ∨ → (G 0 )∨ . Then, for any x ∈ C(OK̄ ) andx∨ ∈ G ∨ (OK̄ ), we have the equality of Cartier pairings⟨x, x∨ ⟩G = ⟨x, f ∨ (x∨ )⟩G 0

16

SHIN HATTORI

and thus the orthogonal subgroup of C(OK̄ ) with respect to the Cartierpairing ⟨ , ⟩G is D̃(OK̄ ). Moreover, since the group scheme D is connectedby Lemma 4.6, the group scheme D̃, which is an extension of D by theconnected group scheme T = (G et )∨ , is also connected and thus it is a
closed subgroup scheme of (G ∨ )0 .
Put D′ = (((G ∨ )0 )∨ )b . By deﬁnition, the subgroup C ′ (OK̄ ) is the orthogonal subgroup of D′ (OK̄ ) with respect to the Cartier pairing ⟨ , ⟩(G ∨ )0 .
To prove the assertion (d), it is enough to show that the Cartier pairing⟨ , ⟩(G ∨ )0 kills the subset D̃(OK̄ ) × D′ (OK̄ ), since both of the group schemes
C ′ and D̃ are of order ph−d . For this, ﬁrst note that the group scheme T isof multiplicative type and thus it is a closed subgroup scheme of the lower
ramiﬁcation subgroup scheme ((G ∨ )0 )1/(p−1) . Since b > 0, Lemma 4.5 implies that the Cartier pairing ⟨ , ⟩(G ∨ )0 kills the subset T (OK̄ ) × D′ (OK̄ ).
Thus we reduce ourselves to showing that the Cartier pairing ⟨ , ⟩(G ∨ )0 /T
kills the subset D(OK̄ ) × D′ (OK̄ ). Since the lower ramiﬁcation subgroupis compatible with subgroups, we have the equalities D = ((G ∨ )0 /T )b andD′ = (((G ∨ )0 /T )∨ )b . Putting b = (1 − w)/(p − 1) and observing the inequality 2(1 − w)/(p − 1) > 1/(p − 1), the assertion (d) follows from Lemma
4.5.
Finally we show the assertion (c). For simplicity, after replacing G by G ∨we show the assertion for C ′ . Note that we have shown the equality C ′ = D̃.
Now we have a commutative diagram with exact rows0

/ T (O )K̄/ D̃(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. The proof of [9, Lemma 2.5] isvalid also for p = 2 and the subgroup D(OK̄ ) coincides with the kernel ofthe lowest right vertical arrow. This implies that the subgroup D̃(OK̄ ) is
the kernel of the lowest left vertical arrow and the assertion (c) follows. □Since the arguments in [9, Section 4] work verbatim also for p = 2, Theorem 1.1 follows from Theorem 4.1.
We can also prove the following result on anti-canonical isogenies for any
p, generalizing [5, Proposition 16]. For a Barsotti-Tate group Γ over OK ,we deﬁne its Hasse invariant Ha(Γ) by Ha(Γ) = Ha(Γ[p]).
Proposition 4.7. Let Γ be a Barsotti-Tate group over OK of height h,dimension d with 0 < d < h and Hasse invariant w = Ha(Γ). Suppose

CANONICAL SUBGROUPS FOR p = 2

17

w < 1/2 and let C be the canonical subgroup of Γ[p] as in Theorem 4.1. Let
E be a finite flat closed subgroup scheme of Γ[p] such that the natural map
C(OK̄ ) ⊕ E(OK̄ ) → Γ[p](OK̄ ) is an isomorphism. Then we have the equality Ha(Γ/E) = p−1 Ha(Γ) and the subgroup scheme Γ[p]/E is the canonicalsubgroup of (Γ/E)[p].
Proof. Note that the Barsotti-Tate group Γ/E is also of height h and dimension d. The natural homomorphism C → Γ[p]/E 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 Γ[p]/E is also
connected. Now we claim that the group scheme (Γ[p]/E) × Spec(OK,1−w )is killed by the Frobenius. For this, by replacing K as in the proof of [9,Theorem 3.2], we may assume that the residue ﬁeld k is perfect. Let L and
L′ be the objects of the category Mod1,ϕ,V
corresponding to the ﬁnite ﬂat/S∨∨
unipotent group schemes C and (Γ[p]/E) via the anti-equivalence G(−), respectively. By [14, Corollary 2.2.2], the generic isomorphism (Γ[p]/E)∨ → 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 vu (det ϕL′ ) ≤ vu (det ϕL ) = ew. As in the
proof of Theorem 4.1 (b), this implies the claim and the group scheme inthe claim coincides with the kernel of the Frobenius of the group scheme(p−1 E/E) × Spec(OK,1−w ). Considering the Hasse invariant of the BarsottiTate group (Γ/E)/(Γ[p]/E) ≃ Γ, we have the equalitymin{w, 1 − w} = min{pHa(Γ/E), 1 − w},from which the equality of the proposition follows.
On the other hand, for any rational number j with pw/(p − 1) < j <p(1 − w)/(p − 1), we have the commutative diagram
C(OK̄ )

OOO
OOO
OOO
OOO

'
/ (Γ[p]/E)(O )Γ[p](OK̄ )K̄

/ (p−1 E/E)(O ),K̄

where the oblique arrow is an isomorphism. Since C = Γ[p]j , the functoriality of j-th upper ramiﬁcation subgroups implies that the closed subgroup scheme Γ[p]/E of p−1 E/E is contained in (p−1 E/E)j . By the inequalityw/(p − 1) < j < p(1 − w/p)/(p − 1), the latter group scheme is the canonicalsubgroup of p−1 E/E and its height is equal to d. Thus the group scheme
coincides with the subgroup scheme Γ[p]/E and we conclude the proof of the
proposition.
□
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E-mail address: shin-h@math.kyushu-u.ac.jp
Faculty of Mathematics, Kyushu University