RIMS Kôkyûroku Bessatsu
Bx (201x), 000–000

Ramification correspondence of finite flat groupschemes and canonical subgroups — a survey
By

Shin Hattori ∗Abstract
Let p > 2 be a rational prime, k be a perfect field of characteristic p and K be a finite
totally ramified extension of the fraction field of the Witt ring of k. Let G and H be finite
flat (commutative) group schemes killed by p over OK and k[[u]], respectively. In this survey
article, we explain the coincidence of ramification subgroups of G and H in the sense of AbbesMokrane when they are associated to the same Kisin module. We also give a survey of its
application to an existence theorem of canonical subgroups of truncated Barsotti-Tate groupsof higher dimension.

§ 1.

Introduction: canonical subgroups

This article is a survey of the author’s result on a correspondence of ramification
subgroups of finite flat group schemes over complete discrete valuation rings of mixed
and equal characteristics ([19]) and its application to an existence theorem of canonicalsubgroups of truncated Barsotti-Tate groups ([20]).
Let p be a rational prime and N ≥ 5 be an integer which is prime to p. Serre([29]) defined p-adic elliptic modular forms of level N as p-adic limits of q-expansionsof usual elliptic modular forms of level N , and Katz ([22]) introduced their modular
description as a function on the ordinary locus of the modular curve X1 (N ) over Qp .
Namely, a p-adic modular form f can be identified with a rule functorially associating
to any triplet (E, ω, ι) over a p-adically complete ring B an element f (E, ω, ι) of B,
where the triplet consists of an elliptic curve E over B such that Ē = E × Spec(B/pB)
Received April 20, 201x. Revised September 11, 201x.
2000 Mathematics Subject Classification(s): 11S23
Key Words: ramification, finite flat group schemes, Breuil-Kisin classification
Supported by Grant-in-Aid for Young Scientists B-21740023.
∗ Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan.
e-mail: shin-h@math.kyushu-u.ac.jp
c 201x Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

2

Shin Hattori

is ordinary, a nowhere vanishing diﬀerential form ω on E and a level Γ1 (N )-structureι : µN → E.
The space of p-adic modular forms admits actions of Hecke operators Tl (l - N )and Ul (l | N ) for l 6= p as usual, and the U operator which aﬀects the q-expansions as
f (q) =∑

an q n 7→ U f (q) =∑apn q n .

The definition of the U operator is as follows. Let f be a p-adic elliptic modular formand (E, ω, ι) be a triplet as above. Since Ē is ordinary, we have a unique subgroupscheme H of E which lifts the Frobenius kernel Ker(FĒ ) by Hensel’s lemma. Consider
the projection π : E → E/H and its dual map π ∨ . Then using the Frobenius operator(ϕf )(E, ω, ι) = f (E/H, (π ∨ )∗ ω, π(ι)),
the U operator is defined to be U = p−1 trϕ . However, the eigenspaces of the U operator
on the whole space of p-adic elliptic modular forms are all isomorphic to each other andinfinite dimensional ([17, Section II.3]).
To obtain a reasonable spectral theory of the U operator, we need to restrict U
to the subspace consisting of the p-adic modular forms which are also defined on someperipheral area outside the ordinary locus. The resulting class of p-adic modular forms
is called overconvergent. Katz ([22]) showed that the U operator acts complete continuously on the space of overconvergent modular forms and thus has meaningful eigenspaces
via the theory of the Fredholm determinant.
The key point for studying the action of the U operator on the subspace of overconvergent modular forms is an existence theorem of canonical subgroups. When E does
not have ordinary reduction, we do not have any Frobenius lift in general. Nevertheless,we do have a canonical Frobenius lift C when E has supersingular reduction but lies
suﬃciently near to the ordinary locus in X1 (N ). This subgroup scheme C is called thecanonical subgroup of E, and we can analyze the Frobenius operator and the U operator
by controlling C, instead of H in the ordinary case.
This whole story suits better for the rigid-analytic setting (see for example [9]).
We can realize the locus of elliptic curves “suﬃcient near to ordinary reduction” as anadmissible open of the associated rigid-analytic space X1 (N )rig and the overconvergent
modular forms can be identified with the sections of an invertible sheaf over this admissible open. Moreover, we can patch the canonical subgroup into a family. Namely,
there exists an admissible open subgroup of the p-torsion of the universal elliptic curve
over this locus which gives the canonical subgroup on each fiber.
Katz attributed the existence theorem of canonical subgroups to Lubin ([22, Theorem 3.1]), and the proof is accomplished by a calculation of formal power series of
one variable for the formal completion of E. In fact, when B is a complete discrete

Ramification correspondence and canonical subgroups

3

valuation ring over Zp , an elliptic curve E over B has the canonical subgroup if andonly if the Newton polygon of the multiplication-by-p formula [p](X) of the formal completion of E has a vertex at x = p. Thus the proof of Katz-Lubin heavily relies on the
one-dimensionality of elliptic curves, and a similar consideration for higher dimensionalabelian schemes was too hard to carry out, at that time.
This had been one of the obstacles to establish the theory of overconvergent Siegel
modular forms, until Abbes-Mokrane ([1]) achieved a breakthrough. Let K be a complete discrete valuation field of residue characteristic p. They defined, for a finite flat
generically etale group scheme G over OK , a filtration {G j }j∈Q>0 by finite flat closed
subgroup schemes of G using a ramification theory of Abbes-Saito ([2], [3]), which iscalled the upper ramification filtration of G. The canonical subgroup of Katz-Lubin of
an elliptic curve E over OK appears in the upper ramification filtration of the p-torsion
subgroup scheme E[p]. Then they proved that, for an abelian scheme A of arbitrary
relative dimension over OK which is “suﬃciently close to ordinary reduction”, a subgroup scheme which appears in the filtration of A[p] satisfies similar properties to thecanonical subgroup of Katz-Lubin.
Since then several improvements of the result have been obtained, such as [4], [10],[15], [16], [25], [28], [31] and most recently [12] and [32]. Let us summarize some of main
points of these improvements. First, the canonical subgroup theorem is generalized fortruncated Barsotti-Tate groups ([21]) over OK instead of abelian schemes. In particular,a higher analogue of the canonical subgroup theorem, namely the existence of a similarcanonical subgroup in A[pn ] instead of A[p], is also known. Moreover, this improvementmeans that we can construct the canonical subgroup for an abelian scheme A without
deep geometric techniques such as p-adic vanishing cycles but just from the finite flatgroup scheme A[pn ]. Second, the original condition on “suﬃcient closeness to ordinarity”
in Abbes-Mokrane’s work is much relaxed.
One of the two main theorems of this survey article is also an improvement of thecanonical subgroup theorem along these two lines. To state the result, we fix somenotations. Let K/Qp be an extension of complete discrete valuation fields. For a finite
flat group scheme G over OK and its module of invariant diﬀerentials ωG over OK ,∑write ωG ' ⊕i OK /(ai ) with some ai ∈ OK and put deg(G) = i vp (ai ), where vp is thenormalized p-adic valuation. Put ÕK = OK /pOK . For 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 subgroup scheme G ∨ [p]. Then the Lie algebra Lie(G ∨ [p] × Spec(ÕK ))over ÕK is a free ÕK -module of rank h − d. We define the Hasse invariant Ha(G) of G tobe the truncated p-adic valuation vp (det(VG ∨ [p] )) ∈ [0, 1] of the determinant of the actionof the Verschiebung VG ∨ [p] on this ÕK -module. This invariant measures the distance ofG from ordinary reduction. In fact, G is an extension of a finite etale group scheme by4

Shin Hattoria finite flat group scheme of multiplicative type if and only if Ha(G) = 0. Finally, we0
put G j+ to be the scheme-theoretic closure in G of the subgroup ∪j 0 >j G j (OK̄ ). Thenour canonical subgroup theorem is as follows.
Theorem 1.1 ([20], Theorem 1.1).
Let p > 2 be a rational prime, K/Qp be an
extension of complete discrete valuation fields and e be the absolute ramification index ofK. Put m≥i
K = {x ∈ K | evp (x) ≥ i}. Let G be a truncated Barsotti-Tate group of leveln, 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 subgroup scheme Cn = G j+ forpew(pn − 1)/(p − 1)2 ≤ j < pe(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).
≥e(1−pn−1 w)(b) Cn ×Spec(OK /mK
homomorphism Fn) coincides with the kernel of the n-th iterated Frobenius≥e(1−pn−1 w)of G × Spec(OK /mK).

(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.
Since it is known that the upper ramification filtration can be patched into a family,
we have the following corollary.
Corollary 1.2 ([20], 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 Barsotti-Tate groupof level n over X of constant height h and dimension d with 0 < d < h. We let G and X
denote the Raynaud generic fibers of the formal schemes X and G, respectively. For a
finite extension L/K and x ∈ X(L), we put Gx = G ×X,x Spf(OL ), where we let x also
denote the map Spf(OL ) → X induced from x by taking the scheme-theoretic closureand the normalization. Let Gj+ be the admissible open subgroup of G over X such thatfor any x ∈ X(L) as above, the fiber Gj+
x coincides with the upper ramification subgroupje(L/K)+(K̄). For a non-negative rational number r, let X(r) be the admissible openGxof the rigid-analytic space X defined byX(r)(K̄) = {x ∈ X(K̄) | Ha(Gx ) < r}.
Suppose p > 2. Then the finite etale 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)).

Ramification correspondence and canonical subgroups

5

In particular, using this corollary, we can generalize results of [1] to the case ofHa(A) < 1/2.
The idea of the proof of Theorem 1.1 is as follows. By an elementary argument
as in [12] and [32], it suﬃces to treat the case of level one. Moreover, by a basechange, we may assume that the residue field k of K is perfect. Then, the key pointis that we can reduce ourselves to showing a similar statement to Theorem 1.1 forthe lower ramification filtration of a finite flat generically etale group scheme over acomplete discrete valuation ring of equal characteristic with residue field k. This equalcharacteristic counterpart can be shown by an easy calculation in a spirit of the Elkik
approximation ([11, Section I]). The reduction to the equal characteristic case is aconsequence of a ramification correspondence theorem between finite flat group schemes
over complete discrete valuation rings of mixed and equal characteristics, which is theother main theorem of this survey article (Theorem 3.2).
§ 2.

Breuil-Kisin classification

In this section, we recall classification theories of finite flat generically etale (commutative) group schemes over a complete discrete valuation ring with perfect residue
field k of characteristic p. The case of equal characteristic is classical: Let p be a rational prime and T be a scheme with pOT = 0. We let φ denote the absolute Frobenius
morphism of T . We define a φ-module over T to be a pair (M, φM ) of an OT -module
M and a φ-semilinear homomorphism φM : M → M. A φ-module (M, φM ) is said tobe finite locally free if M is a locally free OT -module of finite rank. Then we have thefollowing classification theorem for finite locally free group schemes over T .
Theorem 2.1 ([14], Théorème 7.4).
Let T be a scheme with pOT = 0. For a
finite locally free group scheme H over T , we regard the Lie algebra of the Cartierdual Lie(H∨ ) as a φ-module over T via the map Lie(VH∨ ) induced by the Verschiebung
VH∨ . Then we have an anti-equivalence H(−) from the category of finite locally freeφ-modules over T to the category of finite locally free group schemes over T killed bytheir Verschiebung. Its quasi-inverse is given by the functor H 7→ Lie(H∨ ), and theseanti-equivalences are compatible with any base change.
Let us specialize to the case of T = Spec(k[[u]]). Put S1 = k[[u]] and we also let
φ denote the absolute Frobenius endomorphism of this ring. Let (M, φM ) be a finite(locally) free φ-module over k[[u]]. Namely, this is a pair of a free S1 -module of finite
rank with a φ-semilinear map φM : M → M. Put φ∗ M = S1 ⊗φ,S1 M. We say (M, φM )is a Kisin module of u-height ≤ r if the cokernel of the map 1 ⊗ φM : φ∗ M → M is killed
by ur , though it was introduced by Breuil ([5]). Then by Theorem 2.1 we also havean anti-equivalence H(−) from the category of Kisin modules of u-height ≤ r to the6Shin Hattori

category of finite flat group schemes H over k[[u]] killed by VH such that the φ-module
Lie(H∨ ) is of u-height ≤ r. For a Kisin module M of u-height ≤ r, we can explicitly
write defining equations of the associated group scheme H(M). Indeed, choose a basis
e1 , . . . , ed of the S1 -module M and define a matrix A = (ai,j ) byφM (e1 , . . . , ed ) = (e1 , . . . , ed )A.
Then the group scheme H(M) is naturally isomorphic to the additive group scheme
defined by the equations
Xip −

d
∑

aj,i Xj(i = 1, . . . , d).

j=1

Next we suppose that K is of mixed characteristic (0, p) with p > 2 and perfectresidue field k. In this case, the classification theorem for finite flat group schemes is due
to Breuil ([5], [7], [8]) and Kisin ([23]). For simplicity, we concentrate on the classificationof finite flat group schemes killed by p. Let e be the absolute ramification index of K.
ˆ be its completion and put G = Gal(K̄/K).
Let K̄ be an algebraic closure of K, K̄
K
1,φWe let Mod/S1 denote the category of Kisin modules of u-height ≤ e. Then we havethe following theorem.
Theorem 2.2 ([8], Theorem 3.3.2).
Let p > 2 and K as above. Then there
exists an anti-equivalence G(−) from the category Mod1,φ/S1 to the category of finite flatgroup schemes over OK killed by p.
Remark. Breuil ([7]) proved a similar classification in terms of slightly diﬀerent
linear algebraic data, which are now called Breuil modules, and he showed Theorem 2.2
in [8] by constructing an equivalence of categories between those of Breuil and Kisin
modules. Kisin ([23]) gave a much simpler proof independent of Breuil’s earlier classification: he first proved directly a classification of Barsotti-Tate groups over OK via
Breuil modules by an elementary argument using the deformation theory of Messing([27]) and then derived the classification of finite flat group schemes by taking a resolution by Barsotti-Tate groups, the idea which he attributed to Beilinson. These twoconstruction of the anti-equivalence G(−) are naturally isomorphic to each other.
From a Kisin module M ∈ Mod1,φ/S1 , we can decode the action of GK on the finite
module G(M)(OK̄ ) associated to the finite flat group scheme G(M) corresponding to M
as follows. Let us fix a uniformizer π of K and a system of its p-power roots {πn }n∈Z≥0pin K̄ such that π0 = π and πn = πn+1
. Put Kn = K(πn ) and K∞ = ∪n∈Z≥0 Kn . We
define a ring R to be
R = lim(OK̄ /pOK̄ ← OK̄ /pOK̄ ← OK̄ /pOK̄ ← · · · ),←−nRamification correspondence and canonical subgroups

7

where the arrows are the p-th power map, and also an element π of R to be π =(π, π1 , π2 , . . .). The ring R is a complete valuation ring whose valuation vR is given
as follows: for x = (x0 , x1 , . . .) ∈ R, choose a lift x̂n of xn in OK̄ and put x(0) =nlimn→∞ x̂pn ∈ OK̄ˆ . Then the valuation of the element x is defined as vR (x) = vK (x(0) ).
The ring R admits a natural action of the Galois group GK and an S1 -algebra
structure defined by u 7→ π which is compatible with the action of GK∞ = Gal(K̄/K∞ ).
Moreover, the ring R is considered as a φ-module over S1 via the absolute Frobenius
endomorphism φ. For a Kisin module M ∈ Mod1,φ/S1 , we put∗
TS(M) = HomS1 ,φ (M, R),

where the module on the right-hand side consists of the S1 -linear homomorphisms∗(M).
compatible with φ’s. The Galois group GK∞ acts naturally on the module TS∗
Then there is a natural isomorphism of GK∞ -modules εM : G(M) → TS (M) ([8, loc.
cit.]).
The field Frac(R) can be identified with the completion of an algebraic closure of
the subfield X = k((u)). This identification induces the “field-of-norms” isomorphismof Galois groups GK∞ ' GX which is compatible with the upper ramification subgroupsof both sides up to a shift by the Herbrand function of K∞ /K ([6, Subsection 4.2], [33]).
On the other hand, we also have a natural isomorphism of GX -modules H(M)(R) →∗TS(M), by which we identify both sides. Thus we have a natural isomorphism of GK∞ modules
εM : G(M)(OK̄ ) → H(M)(R).
We can show that the greatest upper ramification jumps of the Galois modules in theclassical sense ([30]) of both sides are no more than pe/(p − 1), where we follow thenormalization of the upper ramification subgroups in [13]. Since the greatest upper
ramification jump of the extension K1 /K is equal to 1 + pe/(p − 1), we see by using
the isomorphism εM that both sides of the isomorphism have exactly the same greatest
upper ramification jump.
§ 3.

Ramification correspondence

In this section, we briefly recall the ramification theory of finite flat group schemes([1]) and state the ramification correspondence theorem mentioned before. For a while,
let p be a rational prime which may be two and K be a complete discrete valuation
field with residue field of characteristic p which may be imperfect. We fix a uniformizer
π and a separable closure K sep of K, and put GK = Gal(K sep /K). We let vK denotethe valuation on K sep with vK (π) = 1 and put m≥i
K sep = {x ∈ OK sep | vK (x) ≥ i} fori ∈ Q≥0 .

8Shin Hattori

Let B be a finite flat OK -algebra locally of complete intersection which is genericallyetale, that is, B ⊗OK K is etale over K. Put F (B) = HomOK -alg. (B, OK sep ). This is a
finite GK -set, namely a finite set where the Galois group GK acts continuously. Fix apresentation
B ' OK [X1 , . . . , Xr ]/(f1 , . . . , fs )and consider the K-aﬃnoid varietyjrXK(B) = {(x1 , . . . , xr ) ∈ OK
sep | vK (fi (x1 , . . . , xr )) ≥ j(i = 1, . . . , s)}j
for j ∈ Q>0 . The set of geometric connected components of XK(B) turns out to be
independent of the choice of a presentation in an appropriate sense and is denoted by
F j (B). This set is also a finite GK -set and, since the set F (B) is identified with theset of zeros of the equations f1 , . . . , fs , we have a GK -equivariant functorial surjection
F (B) → F j (B).
Let G = Spec(B) be a finite flat generically etale group scheme over OK . Then theaﬃne algebra B is locally of complete intersection ([7, Proposition 2.2.2]) and thus wecan apply the above formalism to G. Put F (G) = F (B) = G(OK sep ) and F j (G) = F j (B).
Then, by a functoriality, the set F j (G) is shown to have a structure of a GK -module,
namely a module where the Galois group GK acts continuously and compatibly withthe module structure. Moreover, the natural map G(OK sep ) = F (G) → F j (G) turns out
to be a surjective homomorphism. Its kernel is denoted by G j (OK sep ) and called the0j-th upper ramification subgroup of G. We also put G j+ (OK sep ) = ∪j 0 >j G j (OK sep ) and
let G j and G j+ denote their scheme-theoretic closures in G.
As in the classical ramification theory of local fields, we also have a “lower” variant of the upper ramification filtration. Consider the reduction map G(OK sep ) →G(OK sep /m≥i
K sep ) for i ∈ Q≥0 and let Gi (OK sep ) denote its kernel, which is called the i-th
lower ramification subgroup of G. We define Gi+ (OK sep ), Gi and Gi+ similarly. These
two filtrations are compatible with base extensions: for an extension L/K of complete
discrete valuation fields with relative ramification index e(L/K), we have natural isomorphisms

G j × Spec(OL ) → (G × Spec(OL ))je(L/K) ,Gi × Spec(OL ) → (G × Spec(OL ))ie(L/K) .
Example 3.1.
If the aﬃne algebra of G is generated by one element, we caneasily calculate the ramification filtrations ([18, Corollary 5]). For example, supposethat K is of mixed characteristic (0, p) with absolute ramification index e and let E be
an elliptic curve over OK . Consider its formal completion Ê along the zero section withformal parameter X and write the multiplication-by-p formula as[p](X) = pX + · · · + cX p + · · · .

Ramification correspondence and canonical subgroups9

Put f = vK (c) and define a subgroup scheme C of E[p] to be the scheme-theoretic
closure of the subgroup {x ∈ Ê[p](mK̄ ) | vK (x) ≥ (e − f )/(p − 1)} of E[p](OK̄ ). Then,for f < pe/(p + 1), we have
 E[p] (0 < j ≤ pf /(p − 1))jE[p] = C(pf /(p − 1) < j ≤ (pe − f )/(p − 1))0((pe − f )/(p − 1) < j),
2 E[p] (0 ≤ i ≤ f /(p − p))E[p]i = C
(f /(p2 − p) < i ≤ (e − f )/(p − 1))0((e − f )/(p − 1) < i).
On the other hand, for f ≥ pe/(p + 1), we have{E[p] (0 < j ≤ p2 e/(p2 − 1))
E[p]j =
0(p2 e/(p2 − 1) < j),{E[p] (0 ≤ i ≤ e/(p2 − 1))
E[p]i =
0(e/(p2 − 1) < i).
The subgroup scheme C coincides with the canonical subgroup of E in the sense ofKatz-Lubin.
Now we can state the other main theorem of this survey article, which establishes
a correspondence of ramification filtrations between finite flat group schemes over complete discrete valuation rings of mixed and equal characteristics.
Theorem 3.2 ([19], Theorem 1.1).
Let p > 2 be a rational prime and K be acomplete discrete valuation field of mixed characteristic (0, p) with perfect residue fieldk and absolute ramification index e. Consider the category Mod1,φ/S1 of Kisin modules
as before and let M be its object. Then the natural isomorphism εM : G(M)(OK̄ ) →H(M)(R) induces isomorphisms of ramification subgroups
G(M)j (OK̄ ) → H(M)j (R),G(M)i (OK̄ ) → H(M)i (R)for any j ∈ Q>0 and i ∈ Q≥0 .
This can be seen as a generalization of the coincidence of the classical greatest
upper ramification jumps of both sides mentioned before.
§ 4.

Sketch of the proofs

In this section, we give a brief sketch of the proofs of the main theorems. First weshow compatibilities of the Breuil-Kisin classification with base extensions inside K∞ /K

10Shin Hattori

and Cartier duality. For the compatibility with the base change, put S01 = k[[v]] andlet φ also denote the absolute Frobenius endomorphism of the ring S01 . Fix a positive0ninteger n and consider the category Mod1,φ/S0 of Kisin modules over S1 of v-height ≤ ep .
1

Using the k-algebra homomorphism S01 → R defined by v 7→ π 1/p , we define a similar∗∗0pnfunctor TS
0 to TS . The homomorphism of k-algebras S1 → S1 defined by u 7→ v
1,φinduces a natural functor (−)0 : Mod1,φ/S1 → Mod/S0 byn1

M 7→ M0 = S01 ⊗S1 M1 ,φM0 = φ ⊗ φM .

0∗∗
On theThen we have a natural isomorphism of GK∞ -modules TS(M) → TS0 (M ).
1,φother hand, by Theorem 2.2 the category Mod/S0 classifies finite flat group schemes
1over OKn killed by p, and we let G 0 (−) denote the anti-equivalences of the theorem overOKn . Then we can show the following proposition.

Proposition 4.1 ([19], Proposition 4.3).

Let M be an object of the category Mod1,φ/S1and M0 be the associated object of the category Mod1,φ/S01 . Then there exists a natural
isomorphism
G 0 (M0 ) → G(M) × Spec(OKn )of finite flat group schemes over OKn which makes the following diagram commutative:
G(M)(OK̄ )|GK∞

∼

εM o


∗
TS
(M)

/ G 0 (M0 )(OK̄ )|GK

∞

εM0 o

∼


/ T ∗ 0 (M0 ).
S

Remark. The author does not know if a similar base change compatibility of theBreuil-Kisin classification holds for wildly ramified extensions in general. Thus it seems
diﬃcult to prove Theorem 3.2 by killing the Galois action, taking scheme-theoretic
closures on both sides of the isomorphism εM and reducing to the case of rank one.
Instead, by using the restricted compatibility as in Theorem 4.1, we reduce ourselves tocomparing the defining equations of both sides, as we explain below.
For the compatibility with Cartier duality, we recall a duality of the category
Mod1,φ/S1 ([26, Section 3]). Let W = W (k) be the Witt ring and E(u) be the Eisenstein polynomial over W of the uniformizer π of K. Put c0 = p−1 E(0). We also fix
a system of p-power roots of unity {ζpn }n∈Z≥0 in K̄ with ζp 6= 1 and ζpn = ζppn+1 .
Then we have an element t̄ of the ring R associated to the system on which the Galois
group GK∞ acts via the modulo p cyclotomic character ([26, loc. cit.]). Let M be an∨
object of the category Mod1,φ/S1 and put M = HomS1 (M, S1 ). Consider the naturalpairing h , iM : M × M∨ → S1 . Then we can give M∨ a natural structure of an object

Ramification correspondence and canonical subgroups

11

−1 e
∨
∨
∨
of Mod1,φ/S1 satisfying hφM (m), φM (m )iM = c0 u φ(hm, m iM ) for any m ∈ M andm∨ ∈ M∨ ([26, Proposition 3.1.7]). This induces a natural perfect pairing of GK∞ ∗∗modules TS(M) × TS(M∨ ) → (Z/pZ)t̄, which is also denoted by h , iM . Then we canalso show the following compatibility with Cartier duality.

Let M be an object of the category Mod1,φProposition 4.2 ([19], Proposition 4.4).
/S1
and M∨ be its dual object. Then there exists a natural isomorphism G(M)∨ → G(M∨ )of finite flat group schemes over OK such that the induced map
ε

∨

M
∗
δM : G(M)∨ (OK̄ ) → G(M∨ )(OK̄ ) →
(M∨ )
TS

fits into the commutative diagram of GK∞ -modules
G(M)(OK̄ ) × G(M)∨ (OK̄ )
εM o

o

δM o



∗
∗
TS (M) × TS (M∨ )/ Z/pZ(1)h , iM/ (Z/pZ)t̄,

where the top arrow is the Cartier pairing of G(M) and the right vertical arrow is theisomorphism defined by ζp 7→ t̄.
We also need the following duality result for upper and lower ramification subgroups. For G, this is due to Tian and Fargues ([31, Theorem 1.6], [12, Proposition 6]),
while the case of H(M) is [19, Theorem 3.3].
Let K be a complete discrete valuation field of mixed charProposition 4.3.
acteristic (0, p) with residue field k and absolute ramification index e. Put l(j) =e/(p − 1) − j/p.
1. Let G be a finite flat group scheme over OK killed by p. Then we have the equality
G j (OK̄ )⊥ = (G ∨ )l(j)+ (OK̄ ) for j ≤ pe/(p − 1), where ⊥ means the orthogonal
subgroup with respect to the Cartier pairing.
2. Assume that the residue field k is perfect and let M be an object of the catej⊥
gory Mod1,φ= (H(M∨ ))l(j)+ (R) for/S1 . Then we have the equality H(M) (R)j ≤ pe/(p − 1), where ⊥ means the orthogonal subgroup with respect to the pairing
h , iM .
To prove Theorem 3.2, it is enough by Proposition 4.2 and Proposition 4.3 to show
the assertion on lower ramification filtrations. By Proposition 4.1, we may replace K
with K1 and assume that the entries of a representing matrix of φM for some basis
are contained in the subring k[[up ]] of S1 . In this case, we can write down defining

12

Shin Hattori

equations of the group scheme G(M) explicitly in terms of M, by using [7, Proposition
3.1.2]. Let us consider the isomorphism of k-algebras k[[u]]/(ue ) → ÕK = OK /pOK
sending u to π, by which we identify both sides. Then we can check that the defining
equations of the group schemes G(M) and H(M) over ÕK ' k[[u]]/(ue ) coincide witheach other and this coincidence preserves zero sections. In other words, we can construct
an isomorphism of schemesηM : G(M) × Spec(ÕK ) → H(M) × Spec(k[[u]]/(ue ))which preserves zero sections. Though this is is not compatible with group structuresin general, we can show that this induces the commutative diagram of sets
G(M)(OK̄ )
G(M)(OK̄ /m≥i
)
K̄

εM
∼

/ H(M)(R)ηM
/ H(M)(R/m≥i )R

for any i ≤ e, where we put m≥i
R = {x ∈ R | vR (x) ≥ i}. In the diagram, the arrows arehomomorphism of groups except the bottom one, which is at least a bijection compatible
with zero elements. Moreover, the i-th lower ramification subgroups on both sides ofεM are the inverse images of the zero elements by the vertical arrows. Hence theisomorphism εM is compatible with the i-th lower ramification filtrations for i ≤ e.
Since we can easily show that the i-th lower ramification subgroups of both sides vanish
for i > e/(p − 1), we can conclude the proof of Theorem 3.2.
e
Put M = S1 e for a basis e with φM (e) = c−1
Example 4.4.
0 u e. Then we
have G(M) = µp = Spec(OK [X]/(X p − 1)) ([5, Exemple 2.2.3]). On the other hand, theeadditive group scheme H(M) is isomorphic to Spec(k[[u]][Y ]/(Y p −c−1
0 u Y )). Reducing
modulo p and modulo ue respectively, we have an isomorphism of schemes over ÕK 'k[[u]]/(up )ÕK [X]/(X p − 1) → (k[[u]]/(ue ))[Y ]/(Y p )defined by X 7→ 1 + Y . This isomorphism is compatible with zero sections, but not withgroup structures. Nevertheless, the j-th upper (resp. i-th lower) ramification subgroupsof G(M) and H(M) are zero if and only if j > pe/(p − 1) (resp. i > e/(p − 1)).
As for Theorem 1.1, we may assume n = 1, as mentioned before. By Proposition
4.3, it suﬃces to show the following.
Proposition 4.5 ([19], Theorem 3.2).
Let p > 2 and K be as in Theorem 1.1.
Let G be a truncated Barsotti-Tate group of level one, height h and dimension d overOK with d < h and Hasse invariant w = Ha(G).

Ramification correspondence and canonical subgroups13

1. If w < (p − 1)/p, then the lower ramification subgroup scheme D = Ge(1−w)/(p−1) isof order pd . The group scheme D has the following properties:(a) deg(G/D) = w.
≥e(1−w)(b) The reduction modulo mKof the closed subgroup scheme (G/D)∨ of G ∨coincides with the kernel of the Frobenius homomorphism of the reduction G ∨ ×≥e(1−w)
Spec(OK /mK).
≥e(1−w)(c) D × Spec(OK /mK) also coincides with the kernel of the Frobenius ho≥e(1−w)
momorphism of G × Spec(OK /mK).
2. If w < 1/2, then D coincides with the lower ramification subgroup scheme Gb forew/(p − 1) < b ≤ e(1 − w)/(p − 1).
To prove the proposition, we may assume that the residue field k is perfect. Let
M be the object of the category Mod1,φ/S1 corresponding to G via the anti-equivalence
G(−). We identify the k-algebras ÕK and k[[u]]/(ue ) as before. Then we can read oﬀ
the Hasse invariant from the Kisin module M, as follows. Put M1 = M ⊗ ÕK . This canbe considered as a finite (locally) free φ-module over ÕK . Consider the natural exact
sequences of ÕK -modules
0 → (1 ⊗ φM1 )(φ∗ M1 ) → M1 → M1 /(1 ⊗ φM1 )(φ∗ M1 ) → 0.
Since G is a truncated Barsotti-Tate group of level one, we can show from the construction of the functor G(−) that we have natural isomorphisms of ÕK -modules
ωG×ÕK ' Ker(1 ⊗ φM1 : φ∗ M1 → M1 ),Lie(G ∨ × ÕK ) ' (1 ⊗ φM1 )(φ∗ M1 )and thus the first term of the exact sequence above is a free ÕK -module of rank h − d.
Moreover we can show that this term is stable under φM1 and the exact sequence ofÕK -modules above splits. Then the Hasse invariant Ha(G) is equal to the truncatedp-adic valuation of the determinant of the φM1 -action on the module of the first termof the exact sequence.
We choose a basis e1 . . . , eh of M such that e1 , . . . , eh−d (resp. eh−d+1 , . . . , eh ) isa lift of a basis of the ÕK -module of the first (resp. third) term of the exact sequence
above. Then we have
(
)
P1 P2
φM (e1 , . . . , eh ) = (e1 , . . . , eh )ue P 3 ue P 4
for some matrices Pi with entries in k[[u]], where P4 is a d×d-matrix. Set A ∈ Mh (k[[u]])to be the matrix on the right-hand side and put AM1 = SpanÕK ((e1 , . . . , ed )A). Then

14

Shin Hattori

the exact sequence above is equal to the exact sequence
0 → AM1 → M1 → M1 /AM1 → 0of φ-modules over ÕK , where we have the equality vp (det(φAM1 )) = w. Let vu be
the u-adic valuation on k[[u]] with vu (u) = 1. Since w < 1, we also have the equalityvu (det(P1 )) = ew and thus we can define the object L of the category Mod1,φ/S1 byL = ⊕h−di=1 S1 ei andφL (e1 , . . . , eh−d ) = (e1 , . . . , eh−d )P1 .
Now, using Theorem 3.2, we switch to H(M) for calculating the lower ramification
filtrations. Then the proofs of the following two lemmas are straightforward.
Let l be a positive integer, U be an element ofLemma 4.6 ([20], Lemma 3.3).
0
Ml (k[[u]]) and w be a rational number such that vu (det(U )) = ew0 . Let T be the scheme
over k[[u]] defined by the system of equations(X1p , . . . , Xlp ) = (X1 , . . . , Xl )U.
Suppose that we have the inequality w0 < (p − 1)/p. Then the natural map≥b
T (R) → Im(T (R/m≥eR ) → T (R/mR ))is a bijection for ew0 /(p − 1) < b ≤ e(1 − w0 ).
Lemma 4.7 ([20], Lemma 3.4).
There exists a unique injection ι : L → M of1,φthe category Mod/S1 such that the S1 -submodule L is a direct summand of M and the
reduction modulo ue(1−w) of the injection ι coincides with the inclusion
AM1 ⊗ (k[[u]]/(ue(1−w) )) → M1 ⊗ (k[[u]]/(ue(1−w) )).
Let b be a rational number with ew/(p − 1) < b ≤ e(1 − w). Then the map≥b≥bι ⊗ (R/m≥b
R ) also coincides with the inclusion AM1 ⊗ (R/mR ) → M1 ⊗ (R/mR ).
Hence we have a commutative diagram
H(M)(R)/ H(L)(R)



≥b
≥b
≥b
/
H(M1 ⊗ (R/mR ))(R/mR )H(L ⊗ (R/mR ))(R/m≥b
R )
VVVV
VVVV
VVVV
VVVV
VVV+
≥b
H(AM1 ⊗ (R/m≥bR ))(R/mR )Ramification correspondence and canonical subgroups

15

whose upper right vertical arrow is an injection by Lemma 4.6. Put N = M/L. Thisis an object of the category Mod1,φ/S1 which is free of rank d over S1 . Then the groupd
H(N)(R) is of order p and we have the equality
H(N)(R) = Ker(H(M)(R) → H(AM1 )(R/m≥b
R )).
Then, by an elementary calculation of the valuations of roots of the defining equationsof H(M), we can also show the following lemma, which settles the first assertion of
Proposition 4.5 (1) and the assertion (2).
Lemma 4.8 ([20], Lemma 3.5).
the subgroup H(M)e(1−w)/(p−1) (R).

The subgroup H(N)(R) of H(M)(R) is equal to

By the description of the module of invariant diﬀerentials of truncated BarsottiTate groups stated before, we can show by taking a resolution that we have the samedescription for any finite flat group scheme over OK killed by p. From this the assertion(1a) follows. Replacing K with K1 , the explicit description of defining equations of
G(L∨ ) × Spec(ÕK ) shows the assertion (1b), which in turn implies the assertion (1c).

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