Canonical Subgroups of Barsotti-Tate Groups

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Canonical subgroups of Barsotti-Tate groups

By Yichao Tian

SECOND SERIES, VOL. 172, NO. 2 September, 2010

anmaah Annals of Mathematics, 172 (2010), 955–988

Canonical subgroups of Barsotti-Tate groups

By YICHAO TIAN

Abstract

Let S be the spectrum of a complete discrete valuation ring with fraction field of characteristic 0 and perfect residue field of characteristic p 3. Let G be a truncated Barsotti-Tate group of level 1 over S. If “G is not too supersingular”, a condition that will be explicitly expressed in terms of the valuation of a certain determinant, then we prove that we can canonically lift the kernel of the Frobenius endomorphism of its special fiber to a subgroup scheme of G, finite and flat over S. We call it the canonical subgroup of G.

1. Introduction

1.1. Let ᏻK be a complete discrete valuation ring with fraction field K of characteristic 0 and perfect residue field k of characteristic p > 0. We put S Spec.ᏻK / D and denote by s (resp. ) its closed (resp. generic) point. Let G be a truncated Barsotti-Tate group of level 1 over S. If Gs is ordinary, then the kernel of its Frobe- nius endomorphism is a multiplicative group scheme and can be uniquely lifted to a closed subgroup scheme of G, finite and flat over S. If we do not assume Gs ordinary but only that “G is not too supersingular”, a condition that will be explicitly expressed in terms of the valuation of a certain determinant, then we will prove that we can still canonically lift the kernel of the Frobenius endomorphism of Gs to a subgroup scheme of G, finite and flat over S. We call it the canonical subgroup of G. Equivalently, under the same condition, we will prove that the Frobenius endomorphism of Gs can be canonically lifted to an isogeny of truncated Barsotti- Tate groups over S. This problem was first raised by Lubin in 1967 and solved by himself for 1-parameter formal groups[16]. A slightly weaker question was asked by Dwork in 1969 for abelian schemes and answered also by him for elliptic curves[9]: namely, could we extend the construction of the canonical subgroup in the ordinary case to a “tubular neighborhood” (without requiring that it lifts the kernel of the Frobenius)? The dimension one case played a fundamental role in the pioneering work of Katz on p-adic modular forms[14]. For higher-dimensional abelian schemes, Dwork’s conjecture was first solved by Abbes and Mokrane[1]; our approach is a generalization of their results. Later, there have been other proofs, 955 956 YICHAO TIAN always for abelian schemes, by Andreatta and Gasbarri[3], Kisin and Lai[15] and Conrad[7].

1.2. For an S-scheme X, we denote by X1 its reduction modulo p. The valuation vp of K, normalized by vp.p/ 1, induces a truncated valuation ᏻS 0 D 1 nf g ! ޑ Œ0; 1/. Let G be a truncated Barsotti-Tate group of level 1 and height h over S, \ G_ be its Cartier dual, and d be the dimension of the Lie algebra of Gs over k. The Lie algebra Lie.G / of G is a free ᏻS -module of rank d h d, canonically 1_ 1_ 1 D isomorphic to Hom.S1/fppf .G1; އa/ ([12, 2.1]). The Frobenius homomorphism of އa over S1 induces an endomorphism F of Lie.G1_/, which is semi-linear with respect to the Frobenius homomorphism of ᏻS1 . We deﬁne the Hodge height of Lie.G1_/ to be the truncated valuation of the determinant of a matrix of F (see 3.11). This invariant measures the ordinarity of G.

1.3. Following[1], we construct the canonical subgroup of a truncated Barsotti- Tate group over S by the ramification theory of Abbes and Saito[2]. Let G be a commutative finite and flat group scheme over S. In[1], the authors defined a a canonical exhaustive decreasing filtration .G ; a ޑ 0/ by finite, flat and closed 2 subgroup schemes of G. For a real number a 0, we put Ga Gb, where C D[b>a b runs over rational numbers.

THEOREM 1.4. Assume that p 3, and let e be the absolute ramiﬁcation index of K and j e=.p 1/. Let G be a truncated Barsotti-Tate group of level 1 D over S, d be the dimension of the Lie algebra of Gs over k. Assume that the Hodge height of Lie.G1_/ is strictly smaller than 1=p. Then, j d (i) the subgroup scheme G C of G is locally free of rank p over S; j (ii) the special ﬁber of G C is the kernel of the Frobenius endomorphism of Gs.

1.5. Statement (i) was proved by Abbes and Mokrane[1] for the kernel of multiplication by p of an abelian scheme over S. We extend their result to truncated Barsotti-Tate groups by using a theorem of Raynaud to embed G into an abelian scheme over S. To prove statement (ii), which we call “the lifting property of the canonical subgroup”, we give a new description of the canonical filtration of a finite, flat and commutative group scheme over S killed by p in terms of congruence groups. Let K be an algebraic closure of the fraction field of S, ᏻK x x be the integral closure of ᏻK in K. Put S Spec.ᏻK /. For every ᏻK with x x D x 2 x 0 vp./ 1=.p 1/, Sekiguchi, Oort and Suwa[20] introduced a finite and flat Ä Ä group scheme G of order p over S (see 7.1); following Raynaud, we call it the x congruence group of level . If vp./ 0, G is isomorphic to the multiplicative pD group scheme p Spec.ᏻK ŒX= X 1/ over S and if vp./ 1=.p 1/, then D x x D G is isomorphic to the constant etale´ group scheme ކp. For general ᏻK with 2 x 0 1=.p 1/, there is a canonical S-homomorphism G p, such that Ä Ä x W ! K is an isomorphism. For a finite, flat and commutative group scheme G over ˝ x CANONICAL SUBGROUPS OF BARSOTTI-TATE GROUPS 957

S killed by p, induces a homomorphism

.G/ HomS .G; G/ G_.K/ HomS .G; p/: W x ! x D x We prove that it is injective, and its image depends only on the valuation a D v ./; we denote it by G .K/Œea, where e is the absolute ramification index of p _ x K (the multiplication by e will be justified later). Moreover, we get a decreasing Œa e exhaustive filtration .G_.K/ ; a ޑ Œ0; p 1 /. x 2 \ THEOREM 1.6. Let G be a finite, flat and commutative group scheme over S killed by p. Under the canonical perfect pairing

G.K/ G .K/ p.K/; x _ x ! x there is, for any rational number a ޑ 0, 2 ( Œ e a ep G .K/ p 1 p ; if 0 a ; Ga .K/ _ x Ä Ä p 1 C x ? ep D G_.K/; if a > p 1 : x Andreatta and Gasbarri[3] have used congruence groups to prove the existence of the canonical subgroup for abelian schemes. This theorem explains the relation between the approach via the ramiﬁcation theory of[1] and this paper, and the approach of[3].

1.7. This article is organized as follows. For the convenience of the reader, we recall in Section 2 the theory of ramification of group schemes over a complete discrete valuation ring, developed in[2] and[1]. Section 3 is a summary of the results in[1] on the canonical subgroup of an abelian scheme over S. Section 4 consists of some preliminary results on the fppf cohomology of abelian schemes. In Section 5, we define the Bloch-Kato filtration for a finite, flat and commutative group scheme over S killed by p. Using this filtration, we prove Theorem 1.4(i) in Section 6. Section 7 is dedicated to the proof of Theorem 1.6. Finally in Section 8, we complete the proof of Theorem 1.4(ii).

1.8. This article is a part of the author’s thesis at Universite´ Paris 13. The author would like to express his great gratitude to his thesis advisor Professor A. Abbes for leading him to this problem and for his helpful comments on earlier versions of this work. The author thanks Professors W. Messing and M. Raynaud for their help. He is also grateful to the referee for his careful reading and very valuable comments.

1.9. Notation. In this article, ᏻK denotes a complete discrete valuation ring with fraction field K of characteristic 0, and residue field k of characteristic p > 0. Except in Section 2, we will assume that k is perfect. Let K be an algebraic closure x of K, ᏳK Gal.K=K/ be the Galois group of K over K, ᏻK be the integral closure D x x x of ᏻK in K, mK be the maximal ideal of ᏻK , and k be the residue field of ᏻK . x x x N x 958 YICHAO TIAN

We put S Spec.ᏻK /, S Spec.ᏻK /, and denote by s and (resp. s and ) D x D x N N the closed and generic point of S (resp. of S) respectively. x We fix a uniformizer of ᏻK . We will use two valuations v and vp on ᏻK , normalized respectively by v./ 1 and vp.p/ 1; so we have v evp, where e D D D is the absolute ramification index of K. The valuations v and vp extend uniquely to K; we denote the extensions also by v and vp. For a rational number a 0, we x put ma x K vp.x/ a and Sa Spec.ᏻK =ma/. If X is a scheme over S, D f 2 xI g x D x then we will denote respectively by X, Xs and Xa the schemes obtained by base x N x change of X to S, s and Sa. x N x If G is a commutative, finite and flat group scheme over S, then we will denote by G_ its Cartier dual. For an abelian scheme A over S, A_ will denote the dual abelian scheme, and pA the kernel of multiplication by p, which is a finite and flat group scheme over S.

2. Ramification theory of finite flat group schemes over S

2.1. We begin by recalling the main construction of[2]. Let A be a finite and flat ᏻK -algebra. We fix a finite presentation of A over ᏻK

0 I ᏻK Œx1; : : : ; xn A 0; ! ! ! ! or equivalently, an S-closed immersion of i Spec.A/ ށn . For a rational number W ! S a > 0, let X a be the tubular neighborhood of i of thickening a ([2, 3],[1, 2.1]). It is an affinoid subdomain of the n-dimensional closed unit disc over K given by X a.K/ .x ; : : : ; x / ᏻn v.f .x ; : : : ; x // a; f I : 1 n K 1 n x D f 2 xj 8 2 g Let .X a / be the set of geometric connected components of X a. It is a finite 0 K x ᏳK -set that does not depend on the choice of the presentation ([2, Lemma 3.1]). We put (2.1.1) Ᏺa.A/ .X a /: 0 K D x For two rational numbers b a > 0, X b is an affinoid sub-domain of X a. So there is a natural transition map Ᏺb.A/ Ᏺa.A/. !

2.2. We denote by AFPᏻK the category of finite flat ᏻK -algebras, and by ᏳK -Ens the category of finite sets with a continuous action of ᏳK . Let

Ᏺ AFP ᏳK -Ens W ᏻıK ! A Spec.A/.K/ 7! x be the functor of geometric points. For a ޑ>0, (2.1.1) gives rise to a functor 2 a Ᏺ AFP ᏳK -Ens: W ᏻıK ! For b a 0, we have morphisms of functors a Ᏺ Ᏺa and a Ᏺb Ᏺa, W ! b W ! satisfying the relations a b a and a a b for c b a 0. D ı b c D b ı c CANONICAL SUBGROUPS OF BARSOTTI-TATE GROUPS 959

a a To stress the dependence on K, we will denote Ᏺ (resp. Ᏺ ) by ᏲK (resp. ᏲK ). These functors behave well only for ﬁnite, ﬂat and relative complete intersection algebras over ᏻK (EGA IV 19.3.6). We refer to[2, Props. 6.2 and 6.4] for their main properties.

LEMMA 2.3 ([1, Lemme 2.1.5]). Let K0=K be an extension (not necessarily finite) of complete discrete valuation fields with ramification index e . Let A K0=K be a finite, flat and relative complete intersection algebra over ᏻK . Then we have aeK =K 0 a a canonical isomorphism ᏲK .A0/ ᏲK .A/ for all a ޑ>0. 0 ' 2 2.4. Abbes and Saito show that the projective system of functors a a .Ᏺ ; Ᏺ Ᏺ /a ޑ 0 ! 2 a gives rise to an exhaustive decreasing filtration (ᏳK ; a ޑ 0) of the group ᏳK , 2 called the ramification filtration ([2, Prop. 3.3]). Concretely, if L is a finite Galois extension of K contained in K, and Gal.L=K/ is the Galois group of L=K, then x a a the quotient filtration .Gal.L=K/ /a ޑ 0 induced by .ᏳK /a ޑ 0 is determined by the following canonical isomorphisms:2 2 Ᏺa.L/ Gal.L=K/= Gal.L=K/a: ' a b a b For a real number a 0, we put ᏳKC b>aᏳK , and if a > 0 ᏳK b

LEMMA 2.6. Let K0=K be an extension (not necessarily finite) of complete discrete valuation fields with ramification index e , ᏻ be the ring of integers K0=K K0 of K0, and K0 be an algebraic closure of K0 containing K. Let G be an object of x a GrS and G G S Spec.ᏻK /. Then there is a canonical isomorphism G .K/ 0 0 x aeK =K D ' G 0 .K / for all a ޑ>0. 0 0 2 a b 2.7. For an object G of GrS and a ޑ 0, we denote G C b>aG and if 2 D[ a > 0, Ga Gb, where b runs over rational numbers. The construction of D\0morphism u G H of GrS induces canon- W ! ical homomorphisms ua Ga H a, ua Ga H a and ua Ga H a . W ! C W C ! C W ! 960 YICHAO TIAN

PROPOSITION 2.8 ([1, Lemmes 2.3.2 and 2.3.5]). (i) For any object G of GrS , 0 G C is the neutral connected component of G. (ii) Let u G H be a finite, flat and surjective morphism in GrS and a ޑ>0. W ! 2 Then the homomorphism ua.K/ Ga.K/ H a.K/ is surjective. x W x ! x 2.9. Let A and B be two abelian schemes over S, A B be an isogeny W ! (i.e., a finite, flat morphism of group schemes), and G be the kernel of . Let (resp. ) be the generic point of the special fiber As (resp. Bs), and bᏻ (resp. bᏻ) be the completion of the local ring of A at (resp. of B at ). Let M and L be the fraction fields of bᏻ and bᏻ respectively. Now, we have the Cartesian diagram

Spec M / Specbᏻ / A

Spec L / Specbᏻ / B:

We ﬁx a separable closure L of L containing K, and an embedding of M in L. x x x Since A B is a G-torsor, M=L is a Galois extension, and we have a canonical W ! isomorphism

(2.9.1) G.K/ ᏲK .G/ ᏲL.bᏻ/ Gal.M=L/: x D ! D Using the same arguments of ([1, 2.4.2]), we prove the following:

PROPOSITION 2.10. For all rational numbers a 0, the isomorphism (2.9.1) induces an isomorphism Ga.K/ Gal.M=L/a. x ' 3. Review of the abelian scheme case following[1]

From this section on, we assume that the residue ﬁeld k of ᏻK is perfect of characteristic p > 0.

3.1. Let X be a smooth and proper scheme over S, and X X S S. We x D x consider the Cartesian diagram

j i N Xs N / X o X N x N

s Spec k / S o Spec K N D N x N D x and the sheaves of p-adic vanishing cycles on Xs N q q (3.1.1) ‰X i R j .ޚ=pޚ.q//; D N N where q 0 is an integer and ޚ=pޚ.q/ is the Tate twist of ޚ=pޚ. It is clear that ‰0 ޚ=pޚ. By the base change theorem for proper morphisms, we have a X ' CANONICAL SUBGROUPS OF BARSOTTI-TATE GROUPS 961 spectral sequence

p;q p q p q (3.1.2) E2 .X/ H .Xs;‰X /. q/ H C .X; ޚ=pޚ/; D N H) N which induces an exact sequence (3.1.3) 1 1 u 0 1 2 0 H .Xs; ޚ=pޚ/ H .X; ޚ=pޚ/ H .Xs;‰X /. 1/ H .Xs; ޚ=pޚ/: ! N ! N ! N ! N

3.2. The Kummer’s exact sequence 0 p އm އm 0 on X induces ! ! ! ! N the symbol map

(3.2.1) h i j ᏻ ‰1 : X XÁ X x W N N N ! 0 1 1 We put U ‰ ‰ , and for a ޑ>0, X D X 2 a 1 a (3.2.2) U ‰X hX .1 i ᏻX /; D x C N x a a where by abuse of notation is an element in ᏻK with v. / a . We have a 1 ep x D U ‰X 0 if a p 1 ([1, Lemme 3.1.1]). D 1 Passing to the cohomology, we get a filtration on H .X; ޚ=pޚ/ defined by: N 0 1 1 (3.2.3) U H .X; ޚ=pޚ/ H .X; ޚ=pޚ/; N D N a 1 1 0 a 1 U H .X; ޚ=pޚ/ u .H .Xs; U ‰X /. 1//; for a ޑ>0, N D N 2 called the Bloch-Kato filtration.

THEOREM 3.3 ([1, Théorème 3.1.2]). Let A be an abelian scheme over S, pA its kernel of multiplication by p, and e ep . Then under the canonical perfect 0 D p 1 pairing

1 (3.3.1) pA.K/ H .A; ޚ=pޚ/ ޚ=pޚ; x N ! for all a ޑ 0, 2 ( e0 a 1 a U H .As; ޚ=pޚ/ if 0 a e0 A .K/ N Ä Ä I p C x ? 1 D H .A; ޚ=pޚ/ if a > e0. N

3.4. Let X be a scheme over S, X X S S. For all a ޑ>0, we put x D x 2 Sa Spec.ᏻK =ma/ and Xa X S Sa Section 1.9. We denote by D..X1/et/ x D x x D x x the derived category of abelian etale´ sheaves over X . A morphism of schemesK is x1 called syntomic, if it is ﬂat and of complete intersection. Let X be a syntomic and quasi-projective S-scheme, r and n be integers with Œr r 0 and n 1. In[13], Kato constructed a canonical object J in D..X1/et/, n;X x K Œr Œ0 x and if 0 r p 1 a morphism 'r J J , which can be roughly seen Ä Ä W n;X ! n;X as “1=pr times of the Frobenius map”. Wex refer tox[13] and[1, 4.1.6] for details of these constructions. Let ᏿n.r/X be the ﬁber cone of the morphism 'r 1 for x 962 YICHAO TIAN

0 r p 1; so we have a distinguished triangle in D..X1/et/ Ä Ä x K Œr 'r 1 Œ0 1 (3.4.1) ᏿ .r/ J J C : n X n;X n;X x ! x ! x !

The complexes ᏿n.r/X .0 r p 1/ are called the syntomic complexes of X. x Ä Ä x For our purpose, we recall here some of their properties for r 1. D 3.5. According to[13, I.3], for any integer n 1, there exists a surjective symbol map 1 ᏻ Ᏼ .᏿n.1/ /: Xn 1 X x C ! x For a geometric point x of Xs, we put N N Â Ã 1 1 S ᏻ Œ .i j ᏻ /x: x X;x N N XÁ N D x N p D N N By[13, I.4.2], the above symbol map at x factorizes through the canonical surjec- N tion ᏻ S1 =pnS1 . X; x x x x N ! N N THEOREM 3.6 ([13, I.4.3]). Assume that p 3. Let X be a smooth and quasi- 1 1 projective scheme over S. Then there is a canonical isomorphism Ᏼ .᏿1.1/X / ‰X , 1 1 x !Q1 which is compatible with the symbol maps Sx Ᏼ .᏿1.1/X /x and hX Sx 1 N ! x N x W N ! ‰X;x (3.2.1). N

3.7. Let X be a smooth and quasi-projective scheme over S. Let X and S x1 x1 be the absolute Frobenius morphisms of X and S , and let X .p/ be the scheme x1 x1 x1 deﬁned by the Cartesian diagram

w X .p/ / X x1 x1

S1 S x / S : x1 x1 For all integers q 0, we denote by F the composed morphism q w q q q 1 .p/ =d. /; X1=S1 X =S X1=S1 X1=S1 x x ! x1 x1 ! x x x x where the second morphism is induced by the Cartier isomorphism

1 q q C .p/ Ᏼ . /: X1=S1 X =S X1=S1 x x W x1 x1 ! x x

Let c be the class in ᏻS of a p-th root of . p/. We set x1 Â F c Ã (3.7.1) ᏼ Coker ᏻX ᏻX ; D x1 ! x1 Â F 1 Ã ᏽ Ker 1 1 =d.ᏻ / : X =S X =S X1 D x1 x1 ! x1 x1 x CANONICAL SUBGROUPS OF BARSOTTI-TATE GROUPS 963

PROPOSITION 3.8 ([1, 4.1.8]). The notation is as above; assume moreover that p 3. Let x be a geometric point in Xs. N N (i) There exist canonical isomorphisms Â Ã Œ1 '1 1 Œ0 ᏼ Coker Ᏼ0.J / Ᏼ0.J / ; 1;X 1;X ! x ! x Â Ã Œ1 '1 1 Œ0 ᏽ Ker Ᏼ1.J / Ᏼ1.J / ; 1;X 1;X ! x ! x so that the distinguished triangle (3.4.1) gives rise to an exact sequence

˛ 1 ˇ (3.8.1) 0 ᏼ Ᏼ .᏿1.1/X / ᏽ 0: ! ! x ! ! Pp 1 i (ii) Let e.T / i 0 T =iŠ ޚpŒT . Then the morphism ˛x in (3.8.1) is D 2 N induced by the map ᏻ D S1 =pS1 given by X1;x x x x N ! N N a e. a. 1/p 1/; 7! Q where a is a lift of a ᏻX ;x and K is a primitive p-th root of unity. Q 2 x1 2 x (iii) The composed mapN

1 1 symbol 1 ˇ Sx=pSx Ᏼ .᏿1.1/X /x ᏽx N N ! x N ! N is the unique morphism sending a ᏻ to a 1da ᏽ : X; x x 2 x N 2 N Remark 3.9. Statement (ii) of Proposition 3.8 implies that, via the canonical 1 1 isomorphism Ᏼ .᏿1.1/X / ‰X Theorem 3.6, ᏼ can be identiﬁed with the sub- e 1 1 x ' module U ‰X of ‰X deﬁned in (3.2.2).

PROPOSITION 3.10 ([1, 4.1.9]). Assume that p 3. Let X be a smooth projective scheme over S, t .p 1/=p. D (i) The morphism F c ᏻX ᏻX factorizes through the quotient morphism W x1 ! x1 ᏻX ᏻX , and we have an exact sequence x1 ! xt F c (3.10.1) 0 ކp ᏻX ᏻX ᏼ 0: ! ! xt ! x1 ! ! 0 2 (ii) Let ıE H .X1; ᏼ/ H .X1; ކp/ be the cup-product with the class of 2 W x ! x (3.10.1) in Ext .ᏼ; ކp/, and

0;1 0 1 2 d2 H .Xs;‰X /. 1/ H .Xs; ކp/ W xN ! xN be the connecting morphism in (3.1.3). Then the composed morphism

0;1 d2 .1/ 0 0 1 0 1 2 H .X1; ᏼ/ H .Xs; Ᏼ .᏿1.1/X // H .Xs;‰X / H .Xs; ކp/.1/ x ! xN x ! xN ! xN coincides with ıE , where the middle isomorphism is given by Theorem 3.6, and is a chosen p-th root of unity. 964 YICHAO TIAN

0 (iii) Assume moreover that H .Xr ; ᏻX / ᏻSr for r 1 and t. Then there is x xr D D an exact sequence

Â F c Ã 1 1 1 (3.10.2) 0 H .Xs; ކp/ Ker H .X1; ᏻX / H .X1; ᏻX / ! N ! x xt ! x x1

0 ıE 2 H .X1; ᏼ/ H .Xs; ކp/: ! x ! N

3.11. Let M be a free ᏻS -module of rank r and ' M M be a semi-linear x1 W ! endomorphism with respect to the absolute Frobenius of ᏻS . Following[1], we x1 call M a '-ᏻS -module of rank r. Then '.M / is an ᏻS -submodule of M , and x1 x1 there exist rational numbers 0 a1 a2 ar 1, such that Ä Ä Ä Ä Ä r M='.M / i 1ᏻK =mai : ' ˚ D x Pr We deﬁne the Hodge height of M to be ai . For any rational number 0 t 1, i 1 Ä Ä we put M M ᏻ : D t ᏻS St D ˝ x1 x PROPOSITION 3.12 ([3, 9.1 and 9.7]). Assume that p 3. Let be an element 1 in ᏻK , and r 1 an integer. We assume that v vp./ < 2 and let M be a '-ᏻS - x D x1 module of rank r such that its Hodge height is strictly smaller than v. (i) The morphism ' M M factorizes through the canonical map M W ! ! M1 v and the kernel of ' M1 v M is an ކp-vector space of dimension r. W ! (ii) Let N0 be the kernel of the morphism M1 v M induced by ' , and N ! be the ᏻK -submodule of M1 v generated by N0. Then we have dimk.N=mK N/ x N x D dimކ N0 r. p D 3.13. We can now summarize the strategy of[1] as follows. Let A be a projective abelian scheme of dimension g over S. By Proposition 3.10(iii), we have Â Ã 1 0 ıE 2 (3.13.1) dimކ H .A1; ކp/ dimކ Ker H .A1; ᏼ/ H .A1; ކp/ p x C p x ! x Â F c Ã 1 1 dimކp Ker H .A1; ᏻA / H .A1; ᏻA / : D x xt ! x x1 By Remark 3.9 and Proposition 3.10(ii), the left-hand side equals e 1 dimކp U H .A; ޚ=pޚ/ N as in (3.2.3). Taking account of Theorem 3.3, we get

Â F c Ã j 1 1 (3.13.2) 2g dimކp pA C.K/ dimކp Ker H .A1; ᏻX / H .A1; ᏻX / ; x D x xt ! x x1 e 1 where j p 1 . Applying 3.12(i) to M H .A1; ᏻX / and c, we obtain im- D D x x1 D mediately the ﬁrst statement of Theorem 1.4 for projective abelian schemes. In fact, Abbes and Mokrane proved a less optimal bound on the Hodge height ([1, 5.1.1]). CANONICAL SUBGROUPS OF BARSOTTI-TATE GROUPS 965

4. Cohomological preliminaries

4.1. Let f X T be a proper, flat and finitely presented morphism of W ! schemes. We work with the fppf-topology on T , and denote by PicX=T the relative 1 Picard functor Rfppff .އm/. If T is the spectrum of a field, then PicX=T is representable by a group scheme 0 locally of finite type over k. We denote by PicX=T the neutral component and put S 1 0 PicX=T n 1 n PicX=T , where n PicX=T PicX=T is the multiplication by D 0 W ! n. Then PicX=T and PicX=T are open sub-group schemes of PicX=T . For a general base, PicX=T is representable by an algebraic space over T 0 ([4, Th. 7.3]). We denote by PicX=T (resp. PicX=T ) the subfunctor of PicX=T which consists of all elements whose restriction to all fibers Xt , t T , belong 2 to Pic0 (resp. Pic ). By (SGA 6 XIII, Thm 4.7), the canonical inclusion Xt =t Xt =t Pic Pic is relatively representable by an open immersion. X=T ! X=T 4.2. Let f A T be an abelian scheme. If T is the spectrum of a field, W ! 0 0 the Neron-S´ everi´ group PicA=T = PicA=T is torsion free; i.e., we have PicA=T D PicA=T . This coincidence remains true for a general base T by the definitions of 0 PicA=T and PicA=T . Moreover, PicA=T is formally smooth (cf.[18, Prop. 6.7]), and PicA=T is actually open and closed in PicA=T , and is representable by a proper and smooth algebraic space over T , i.e., an abelian algebraic space over T . By a theorem of Raynaud ([10, Ch. 1, Th. 1.9]), every abelian algebraic space over T is automatically an abelian scheme over T . So Pic0 Pic is an abelian A=T D A=T scheme, called the dual abelian scheme of A, and denoted by A_. Let H be a commutative group scheme locally free of finite type over T . Recall the following isomorphism due to Raynaud ([19, 6.2.1]):

1 (4.2.1) Rfppff .HA/ Ᏼom.H _; PicA=T /; ! where HA H T A, H is the Cartier dual of H, and Ᏼom is taken for the D _ fppf-topology on T .

PROPOSITION 4.3. Let A be an abelian scheme over a scheme T , and H a commutative group scheme locally free of ﬁnite type over T . Then there are canonical isomorphisms

(4.3.1) Ᏹxt1.A; H / Ᏼom.H ;A /; ! _ _ (4.3.2) Ext1.A; H / H0 .T; Ᏹxt1.A; H // Hom.H ;A /: ! fppf ! _ _ Proof. For any fppf-sheaf E on T ,

Ᏼom.H ; Ᏼom.E; އm// Ᏼom.H ޚ E; އm/ Ᏼom.E; H /: _ ' _ ˝ ' 966 YICHAO TIAN

Deriving this isomorphism of functors in E and putting E A, we obtain a spectral D sequence p;q p q p q E Ᏹxt .H ; Ᏹxt .A; އm// Ᏹxt .A; H /: 2 D _ H) C Since Ᏼom.A; އm/ 0, the exact sequence D 0 E1;0 Ᏹxt1.A; H / E0;1 E2;0 ! 2 ! ! 2 ! 2 induces an isomorphism 1 1 Ᏹxt .A; H / Ᏼom.H ; Ᏹxt .A; އm//: ' _ 1 Then (4.3.1) follows from the canonical identiﬁcation Ᏹxt .A; އm/ A ([8, 2.4]). ' _ For (4.3.2), the spectral sequence Ep;q Hp .T; Ᏹxtq.A; H // Extp q.A; H / 2 D fppf H) C induces a long exact sequence

0 H1 .T; Ᏼom.A; H // Ext1.A; H / ! fppf ! .1/ H0 .T; Ᏹxt1.A; H // H2 .T; Ᏼom.A; H //: ! fppf ! fppf Since Ᏼom.A; H / 0, the arrow .1/ is an isomorphism, and (4.3.2) follows by D 0 applying the functor Hfppf.T; / to (4.3.1). 4.4. The assumptions are those of Proposition 4.3. We define a canonical morphism (4.4.1) Ext1.A; H / H1 .A; H / ! fppf as follows. Let a be an element in Ext1.A; H / represented by the extension 0 ! H E A 0. Then the fppf-sheaf E is representable by a scheme over T , ! ! ! and is naturally an H -torsor over A. The image of a by the homomorphism (4.4.1) is defined to be the class of the torsor E. Since this construction is functorial in T , by passing to sheaves, we obtain a canonical morphism 1 1 (4.4.2) Ᏹxt .A; H / Rfppff .HA/: ! Via the isomorphisms (4.2.1) and (4.3.1), we check that (4.4.2) is induced by the canonical map A Pic0 Pic . _ D A=T ! A=T Since H is faithfully flat and finite over T , the inverse image of the fppf-sheaf H by f is representable by HA, i.e., we have f .H / HA. Therefore, we deduce D an adjunction morphism 0 (4.4.3) H Rfppff .HA/: ! PROPOSITION 4.5. Let f A T be an abelian scheme, and H be a com- W ! mutative group scheme locally free of finite type over T . Then the canonical maps (4.4.2) and (4.4.3) are isomorphisms. CANONICAL SUBGROUPS OF BARSOTTI-TATE GROUPS 967

Proof. First, we prove that (4.4.2) is an isomorphism. By (4.2.1) and (4.3.1), we have to verify that the canonical morphism Ᏼom.H ;A / Ᏼom.H ; Pic / _ _ ! _ A=T is an isomorphism. Let g H Pic be a homomorphism over T . For every W _ ! A=T t T , the induced morphism g H Pic falls actually in Pic , because t t_ At =t At =t 2 W ! H _ is a finite group scheme. Hence, by the definition of PicA=T , the homomorphism g factorizes through the canonical inclusion A Pic Pic ; so the _ D A=T ! A=T canonical morphism Hom.H ;A / Hom.H ; Pic / is an isomorphism. _ _ ! _ A=T Secondly, we prove that (4.4.3) is an isomorphism. For T -schemes U and G, we denote GU G T U . We must verify that for any T -scheme U , the adjunction D morphism 0 (4.5.1) '.U / H.U / Rfppff .HA/.U / H.AU / W ! D is an isomorphism. We note that H.U / HU .U / and H.AU / HU .AU /; D D therefore, up to taking base changes, it suffices to prove that '.T / (4.5.1) is an isomorphism. We remark that f is surjective, hence '.T / is injective. To prove the surjectivity of '.T /, we take an element h H.A/, i.e., a morphism of T - 2 schemes h A H; by the rigidity lemma for abelian schemes (cf.[18, Prop. W ! 6.1]), there exists a section s T H of the structure morphism H T such that W ! ! s f h. Hence we have '.T /.s/ h, and '.T / is an isomorphism. ı D D COROLLARY 4.6. Let T be the spectrum of a strictly henselian local ring, f A T an abelian scheme, and H a finite étale group scheme over T . Then W ! we have canonical isomorphisms 1 1 1 Het.A; H / Hfppf.A; H / Ext .A; H /: K ' ' Proof. The first isomorphism follows from the etaleness´ of H ([11, 11.7]). For the second one, the “local-global” spectral sequence induces a long exact sequence

1 0 1 0 1 0 Hfppf.T; Rfppff .HA// Hfppf.A; H / Hfppf.T; Rfppff .HA// ! ! ! 2 0 Hfppf.T; Rfppff .HA//: ! 0 By Proposition 4.5, we have Rfppff .HA/ H. Since T is strictly henselian and q q D H etale,´ we have Hfppf.T; H / Het.T; H / 0 for all integers q 1. Therefore, D K D 1 0 1 we obtain Hfppf.A; H / Hfppf.T; Rfppff .HA//, and the corollary follows from ! Proposition 4.5 and (4.3.2).

4.7. Let T be a scheme, and G be a commutative group scheme locally free of ﬁnite type over T . We denote by އa the additive group scheme, and by Lie.G_/ the Lie algebra of G_. By Grothendieck’s duality formula ([17, II 14]), we have a canonical isomorphism

(4.7.1) Lie.G / ᏴomT .G; އa/; _ ' 968 YICHAO TIAN where we have regarded G and އa as abelian fppf-sheaves on T . If T is of characteristic p and G is a truncated Barsotti-Tate group over T , then Lie.G_/ is a locally free, of ﬁnite type ᏻT -module ([12, 2.2.1(c)]). Similarly, for an abelian scheme f A T , we have a canonical isomorphism W ! ([5, 2.5.8])

1 1 (4.7.2) Lie.A_/ ᏱxtT .A; އa/ R f .އa/: ' ' In the sequel, we will frequently use the identiﬁcations (4.7.1) and (4.7.2) without any indications. The following lemma is indicated by W. Messing.

LEMMA 4.8. Let L be an algebraically closed field of characteristic p > 0, R be an L-algebra integral over L, and M be a module of finite presentation over R, equipped with an endomorphism ' semi-linear with respect to the Frobenius of R. Then the map ' 1 M M is surjective. W ! Proof. First, we reduce the lemma to the case R L. Consider R as a filtrant D inductive limit of finite L-algebras .Ri /i I . Since M is of finite presentation, 2 there exists an i I , and an Ri -module Mi of finite presentation endowed with 2 a Frobenius semi-linear endomorphism 'i , such that M Mi R R and ' D ˝ i D 'i , where is the Frobenius on R. For j i, we put Mj Mi R Rj ˝ D ˝ i and 'j 'i R j , where j is the Frobenius of Rj . In order to prove ' 1 is D ˝ i surjective on M , it is sufficient to prove the surjectivity of 'j 1 on each Mj for j i. Therefore, we may assume that R is a finite-dimensional L-algebra, and M is thus a finite-dimensional vector space over L. S n T n We put M1 n 1 Ker.' / and M2 n 1 Im.' /. Then we have a D D decomposition M M1 M2 as '-modules, such that ' is nilpotent on M1 D ˚ and bijective on M2 (Bourbaki, Algebre` VIII 2 nı 2 Lemme 2). Therefore, it is sufficient to prove the surjectivity of ' 1 in the following two cases: (i) ' is nilpotent. In this case, the endomorphism 1 ' admits an inverse P n 1 n 1 ' . Hence it is surjective. C (ii) ' is invertible. We choose a basis of M over L, and let U .ai;j /1 i;j n D Ä Ä be the matrix of ' in this basis. The problem reduces to prove that the equation Pn p system j 1 ai;j xj xi bi .1 i n/ in X .x1; : : : ; xn/ has solutions D Dn Ä Ä D for all b .b1; : : : ; bn/ L . Since U is invertible, let V .ci;j /1 i;j n be D 2 Pn p D Ä Ä its inverse. Then the equation system j 1 ai;j xj xi bi is equivalent to p Pn D P D x ci;j xj b for 1 i n with b ci;j bj . But these n equations i j 1 D i0 Ä Ä 0 D j define a finiteD etale´ cover of Spec L of degree pn. Hence they admit solutions in L, since L is separably closed. This completes the proof.

COROLLARY 4.9. Let H be a Barsotti-Tate group or an abelian scheme over 2 S1 (see §1.9). Then Ext .H; ކp/ 0 for the fppf topology on S1. x D x CANONICAL SUBGROUPS OF BARSOTTI-TATE GROUPS 969

Proof. Let K0 be the fraction field of the ring of Witt vectors with coefficients in k; thus K is a finite extension of degree e of K . Let ᏻur be the ring of 0 K0 integers of the maximal unramified extension of K0 in K. Then ᏻ ᏻ =pᏻ x S1 K K ur ur x2 D x x is integral over the algebraically closed field k ᏻ =pᏻ . As Ext .H; އa/ 0 N K0 K0 D F 1 D ([5, Prop. 3.3.2]), the Artin-Schreier exact sequence 0 ކp އa އa 0 ! ! ! ! induces an exact sequence

1 ' 1 1 2 Ext .H; އa/ Ext .H; އa/ Ext .H; ކp/ 0: ! ! ! Since Ext1.H; އ / is a free ᏻ -module ([5, 3.3.2.1]), the corollary follows imme- a S1 diately from Lemma 4.8. x

5. The Bloch-Kato filtration for finite flat group schemes killed by p

5.1. Recall the following theorem of Raynaud ([5, 3.1.1]): Let T be a scheme, G be a commutative group scheme locally free of finite type over T . Then locally for the Zariski topology, there exists a projective abelian scheme A over T , such that G can be identified to a closed subgroup of A. In particular, if G is a commutative, finite and flat group scheme over S D Spec.ᏻK /, we have an exact sequence of abelian fppf-sheaves over S, (5.1.1) 0 G A B 0; ! ! ! ! where A and B are projective abelian schemes over S. In the sequel, such an exact sequence is called a resolution of G by abelian schemes.

5.2. Let f X Y be a morphism of proper and smooth S-schemes. For W ! q q any integer q 0, we have a base change morphism f .‰ / ‰ of p-adic s Y ! X vanishing cycles (3.1.1) (SGA 7 XIII 1.3.7.1). For q 1N, this morphism respects D a 1 a 1 the Bloch-Kato ﬁltrations (3.2.2); that is, it sends fs.U ‰Y / to U ‰X for all N a ޑ 0. 2 Passing to cohomology, we get a functorial map p q p q H .Ys;‰Y /. q/ H .Xs;‰X /. q/ N ! N for each pair of integers p; q 0. These morphisms piece together to give a mor- .p;q/ .p;q/ phism of spectral sequences (3.1.2) E2 .Y / E2 .X/, which converges to p q p q ! the map H C .Y; ޚ=pޚ/ H C .X; ޚ=pޚ/ induced by f. Therefore, we N ! N have the following commutative diagram: N (5.2.1)

d 1;0 1 1 0 1 2 2 0 / H .Ys; ޚ=pޚ/ / H .Y; ޚ=pޚ/ / H .Ys;‰Y /. 1/ / H .Ys; ޚ=pޚ/ N N N N

˛1 ˛2 ˛3 ˛4 d 1;0 1 1 0 1 2 2 0 / H .Xs; ޚ=pޚ/ / H .X; ޚ=pޚ/ / H .Xs;‰X /. 1/ / H .Xs; ޚ=pޚ/: N N N N 970 YICHAO TIAN

1 It is clear that the Bloch-Kato ﬁltration on H .X; ޚ=pޚ/ (3.2.3) is functorial in X. More precisely, the following diagram is commutative:N

a 1 1 (5.2.2) U H .Y; ޚ=pޚ/ / H .Y; ޚ=pޚ/ N N

˛2 a 1 1 U H .X; ޚ=pޚ/ / H .X; ޚ=pޚ/: N N 5.3. Let G be a commutative, ﬁnite and ﬂat group scheme over S killed by p, and 0 G A B 0 a resolution of G by abelian schemes (5.1.1). We apply ! ! ! ! the construction (5.2.1) to the morphism A B. Using Corollary 4.6, we obtain ! immediately that Â Ã 1 1 Ker ˛2 Ker Ext .B; ކp/ Ext .A; ކp/ D N ! N

Hom.G; ކp/ G_.K/. 1/; D N D x Â Ã 1 1 Ker ˛1 Ker Ext .Bs; ކp/ Ext .As; ކp/ D N ! N

Hom.Gs; ކp/ .Get/_.K/. 1/; D N D K x 0 where G G=G is the etale´ part of G (cf. 2.8. Setting N Ker ˛3, we can et D C D completeK (5.2.1) as follows:

(5.3.1)

u 0 / .Get/_.K/. 1/ / G_.K/. 1/ / N ______/ 0 K x _ x _ _

1 2 3 1;0 d .B/ 1 1 0 1 2 2 0 / H .Bs; ޚ=pޚ/ / H .B; ޚ=pޚ/ / H .Bs;‰B /. 1/ / H .Bs; ޚ=pޚ/ N N N N

˛1 ˛2 ˛3 ˛4 d 1;0.A/ 1 1 0 1 2 2 0 / H .As; ޚ=pޚ/ / H .A; ޚ=pޚ/ / H .As;‰A/. 1/ / H .As; ޚ=pޚ/: N N N N

We will show later that the morphism u is surjective.

Definition 5.4. The assumptions are those of Section 5.3. We call the Bloch- a Kato filtration on G_.K/, and denote by .U G_.K/; a ޑ 0/, the decreasing x 0 x 2 and exhaustive filtration defined by U G .K/ G .K/, and for a ޑ>0; _ x D _ x 2 a 1 a 1 (5.4.1) U G_.K/ 2 .U H .B; ޚ=pޚ//.1/: x D N ep PROPOSITION 5.5. Let e , G be a commutative finite and flat group 0 D p 1 scheme over S killed by p, and 0 G A B 0 be a resolution of G by ! ! ! ! abelian schemes (5.1.1). CANONICAL SUBGROUPS OF BARSOTTI-TATE GROUPS 971

(i) For all a ޑ 0, 2 Â Ã a a 1 ˛2.1/ a 1 (5.5.1) U G_.K/ Ker U H .B; ޚ=pޚ/.1/ U H .A; ޚ=pޚ/.1/ x ' N ! N Â Ã 1 0 a 1 u N.1/ H .Bs; U ‰ .B// ; ' \ N 0 1 where N.1/ is identiﬁed to a subgroup of H .Bs;‰ .B// by 3.1/ in (5.3.1). N (ii) The morphism u G_.K/. 1/ N in (5.3.1) is surjective. In particular, W x ! 1;0 N is contained in the kernel of the morphism d2 .B/ in (5.3.1). (iii) Under the canonical perfect pairing

(5.5.2) G.K/ G .K/ p.K/; x _ x ! x we have, for all a ޑ 0; 2 ( e0 a a U G_.K/ if 0 a e0; G C.K/? x Ä Ä x D G .K/ if a > e . _ x 0 a In particular, the filtration .U G_.K/; a ޑ 0/ does not depend on the resolution x 2 of G by abelian schemes. Proof. Statement (i) is obvious from Definition 5.4 and diagrams (5.2.2) and (5.3.1). For (ii) and (iii), thanks to Lemma 2.6, we need only to prove the proposition after a base change ᏻK ᏻK , where K =K is a finite extension. Therefore, up to ! 0 0 such a base change, we may add the following assumptions. (1) We may assume that k is algebraically closed, K contains a primitive p-th root of unity, and G.K/ G.K/. x D (2) For X A or B, we consider the Cartesian diagram D i j Xs / X o X

s s / S o DN 1 1 and the etale´ sheaf ‰X;K i R j .ޚ=pޚ/ over Xs. By an argument as in the D 0 1 0 1 proof of ([1, 3.1.1]), we may assume that H .Xs;‰ / H .Xs;‰ /. X D X;K Since Ua‰1 0 for a e ([1, Lemme 3.1.1]), we have Ue0 G .K/ X D 0 _ x D Ker.u/.1/ .G / .K/: Statement (iii) for a 0 follows immediately from Propo- D et _ x D sition 2.8(i). TheK pairing (5.5.2) induces a perfect pairing 0 (5.5.3) G .K/ Im.u/.1/ p.K/: C x ! x 0 In particular, dimކ Im.u/.1/ dimކ G .K/ . p D p C x Let (resp. ) be the generic point of Bs Bs (resp. of As As), and be D N D N x a geometric point over . Then , by the morphism , induces a geometric x ! 972 YICHAO TIAN point over , denoted by . Let ᏻ (resp. ᏻ) be the local ring of B at (resp. x of A at ), ᏻ (resp. ᏻ) be the henselization of ᏻ at (resp. of ᏻ at ). We x x x h x h denote by bᏻ and bᏻ the completions of ᏻ and ᏻ, and by .bᏻ/ (resp. by .bᏻ/) x x the henselization of bᏻ (resp. of bᏻ) at (resp. at ). Let L0 (resp. M0) be the ur x ur x fraction field of ᏻ (resp. of ᏻ), L0 (resp. M0 ) be the fraction field of ᏻ (resp. x of ᏻ). We denote by L (resp. by M ) the fraction field of bᏻ (resp. of bᏻ), and by ur x ur h h L (resp. by M ) the fraction field of .bᏻ/ (resp. of .bᏻ/). We notice that h x x is a uniformizing element in bᏻ and in .bᏻ/, x ur ur h ᏻ / L0 / L o .bᏻ/ Ox O O O x

ᏻ / L0 / L o bᏻ: Since G Ker.A B/, we have an identification Gal.M=L/ G.K/ D ! ' D G.K/. We fix a separable closure L of L, and an embedding of M in L, which x x x induces a surjection ' Gal.L=L/ Gal.M=L/. By Proposition 2.10, we have a W ax a! '.Gal.L=L/ / Gal.M=L/ G .K/, for all a ޑ>0. In particular, we have x D D x 2 (5.5.4) '.Gal.L=Lur // Gal.M=M Lur / Gal.M ur =Lur / G0 .K/: x D \ D D C x ur Since L =L is unramified, we have, for all a ޑ>0, 2 (5.5.5) Gal.M ur =Lur /a Gal.M=L/a Ga: D D Let B be the composition of the canonical morphisms 0 1 0 1 1 H .Bs;‰B / H .Bs;‰B;K / .‰B;K / D ! x 1 ur 1 ur H .Spec L ; p/ H .Spec L ; p/ D 0 ! and define A similarly. By functoriality and (5.5.4), we have a commutative diagram (5.5.6) 0 1 0 1 0 / N.1/ / H .Bs;‰ / / H .As;‰ / _ _ B _ A B A

inf res / H1.G0 .K/; / / H1.Spec Lur ; / / H1.Spec M ur ; /; 0 C x p p p where the lower horizontal row is the “inﬂation-restriction” exact sequence in Ga- lois cohomology. By ([6, Prop. 6.1]), B and A are injective. Hence Im.u/.1/ 1 0 N.1/ is identiﬁed with a subgroup of H .G C.K/; p/. By assumption (1), we 1 0 0 x have H .G .K/; p/ Hom.G .K/; p.K//, which has the same dimension C x D C x x over ކ as G0 .K/. Hence by the remark below (5.5.3), we get p C x 1 0 0 (5.5.7) Im.u/.1/ N.1/ H .G .K/; p/ Hom.G .K/; p.K//: D ' C x D C x x This proves statement (ii) of the proposition. CANONICAL SUBGROUPS OF BARSOTTI-TATE GROUPS 973

Statement (iii) for a 0 has been proved above. Since the filtration Ga is D decreasing and U0G .K/ G .K/, we may assume, for the proof of (iii), that _ x D _ x 0 < a e . It suffices to prove that, under the pairing (5.5.3), we have Ä 0 a 0 e a 1 G .K/ Im.u/.1/ H .Bs; U 0 ‰ /: C x ? D \ B From (5.5.6) and (5.5.7), we check easily that (5.5.3) is identified with the canonical pairing 0 1 0 G .K/ H .G .K/; p/ p.K/: C x C x ! x Hence we are reduced to prove that, under this pairing, we have

a 1 0 0 e a 1 (5.5.8) G .K/ H .G .K/; p/ H .Bs; U 0 ‰ /; C x ? D C x \ B 1 ur where the “ ” is taken in H .Spec L ; p/ (5.5.6). \ ur 1 ur Let h .L / H .Spec L ; p/ be the symbol morphism. We deﬁne a W ! 1 ur decreasing ﬁltration on H .Spec L ; p/ in a similar way as (3.2.2) by

0 1 1 b 1 b h U H H ; and U H h.1 .bᏻ// for all integers b > 0. D D C x b 1 Œb 1 We extend this deﬁnition to all b ޑ 0 by setting U H U H , where Œb 2 D denotes the integer part of b. By ([1, Lemme 3.1.3]), for all b ޑ>0, we have Â Ã 2 0 b 1 1 b 1 ur H .Bs; U ‰ / U H .Spec L ; p/ : Therefore, the right-hand side of B D B (5.5.8) is

1 0 .e a/ 1 ur (5.5.9) H .G .K/; p/ U 0 H .Spec L ; p/: C x \ We identify H1.G0 .K/; / with the character group C x p ur ur Hom.Gal.M =L /; p.K// x I then by ([1, Prop. 2.2.1]), the subgroup (5.5.9) consists of the characters such that .Gal.M ur =Lur /.e0 a/ / 0: In view of (5.5.5), we obtain immediately (5.5.8), C D which completes the proof.

Remark 5.6. The proof of 5.5(iii) follows the same strategy as the proof of ([1, Th. 3.1.2]). The referee points out that we can also reduce Proposition 5.5(iii) to Theorem 3.3. Indeed, the commutative diagram

0 / G / A / B / 0 p p p 0 / G / A / B / 0 induces, by the snake lemma, an exact sequence of ﬁnite group schemes 0 G ! ! pA pB G 0: We consider the following perfect pairing: ! ! ! 974 YICHAO TIAN

1 . ; /A 1 pA.K/ pA.K/ H .A; ކp/ / ކp H .A; ކp/ x x N ON ˛2

. ; /B a 1 1 pB C.K/ / pB.K/ pB.K/ H .B; ކp/ / ކp H .B; ކp/ x x x N O 2 . ; / a G ? G .K/ / G.K/G.K/ G .K/. 1/ / ކp G .K/: C x x x _ x _ x By functoriality, the homomorphism is adjoint to ˛2 and is adjoint to 2; i.e., we have ..x/; f /B .x; ˛2.f //A and . .y/; g/B .y; 2.g//G, for all D 1 D x pA.K/, y pB.K/, f H .B; ކp/ and g G_.K/. By Proposition 2.8(ii), 2 x 2 x 2 N a 2 xa induces a surjective homomorphism pB C.K/ G C.K/ for all a ޑ 0. x ! x 2 Now statement 5.5(iii) follows from (5.4.1) and Theorem 3.3 applied to B.

6. Proof of Theorem 1.4(i)

6.1. For r ޑ>0, we denote by އa;r the additive group scheme over Sr (see 2 x 1.9). Putting t 1 1=p, we identify އa;t with an abelian fppf-sheaf over S1 D x by the canonical immersion i St S1. Let F އa;1 އa;1 be the Frobenius W x ! x W ! homomorphism, and c a p-th root of . p/. It is easy to check that the morphism F c އa;1 އa;1, whose cokernel is denoted by ސ, factorizes through the W ! canonical reduction morphism އa;1 އa;t , and we have an exact sequence of ! abelian fppf-sheaves on S . x1 F c (6.1.1) 0 ކp އa;t އa;1 ސ 0: ! ! ! ! ! This exact sequence gives (3.10.1) after restriction to the small etale´ topos .X / x1 et for a smooth S-scheme X. K

PROPOSITION 6.2. Assume p 3. Let G be a truncated Barsotti-Tate group of level 1 over S, and t 1 1=p. Then there exists the equality D Â Ã e F c dimކ U G .K/ dimކ Ker Lie.G / Lie.G / ; p _ x D p xt_ ! x1_ where UeG .K/ is the Bloch-Kato filtration Definition 5.4, and the morphism on _ x the right-hand side is obtained by applying the functor HomS .G1; _/ to the map x1 x F c އa;t އa;1. W ! 6.3. Before proving this proposition, we first deduce Theorem 1.4(i). Let G be a truncated Barsotti-Tate group of level 1 and height h over S satisfying the assumptions of Theorem 1.4, d be the dimension of Lie.Gs_/ over k, and d h d. It follows from Propositions 5.5(iii) and 6.2 that D e Â F c Ã dimކ G p 1 C.K/ h dimކ Ker Lie.G / Lie.G / : p x D p xt_ ! x1_ CANONICAL SUBGROUPS OF BARSOTTI-TATE GROUPS 975

Since Lie.G1_/ is a free ᏻS -module of rank d , we obtain immediately Theorem x x1 1.4(i) by applying Proposition 3.12 to c and M Lie.G /. D D x1_ The rest of this section is dedicated to the proof of Proposition 6.2.

1 LEMMA 6.4. Let G be a Barsotti-Tate group of level 1 over S, t 1 p . 1 1 D Then the morphism Ext .G1; ކp/ Ext .G1; އa;t / induced by the morphism W x ! x ކp އa;t in (6.1.1) is injective. ! Proof. By ([12, Th. 4.4(e)]), there exists a Barsotti-Tate group H over S such p that we have an exact sequence 0 G H H 0; which induces a long ! ! ! ! exact sequence

1 p 1 1 2 Ext .H1; ކp/ Ext .H1; ކp/ Ext .G1; ކp/ Ext .H1; ކp/: x ! x ! x ! x 1 2 It is clear that the multiplication by p on Ext .H1; ކp/ is 0, and Ext .H1; ކp/ 0 x x D by Corollary 4.9; hence the middle morphism in the exact sequence above is an 2 isomorphism. Similarly, using Ext .H1; އa;t / 0 ([5, 3.3.2]), we prove that the 1 1 x D natural map Ext .H1; އa;t / Ext .G1; އa;t / is an isomorphism. So we get a x ! x commutative diagram

Ext1.H ; ކ / / Ext1.G ; ކ / x1 p x1 p

H Ext1.H ; އ / / Ext1.G ; އ /; x1 a;t x1 a;t where the horizontal maps are isomorphisms. Now it sufﬁces to prove that H is injective. Let ދ be the fppf-sheaf on S determined by the following exact sequences: x1

0 ކp އa;t ދ 0 0 ދ އa;1 ސ 0: ! ! ! ! I ! ! ! ! Applying the functors Extq.H ; /, we get x1 1 H 1 Hom.H1; ދ/ Ext .H1; ކp/ Ext .H1; އa;t / x ! x ! x I 0 Hom.H1; ދ/ Hom.H1; އa;1/ Hom.H1; ސ/: ! x ! x ! x Since Hom.H ; އ / 0 ([5, 3.3.2]), the injectivity of follows immediately. x1 a;1 H D

6.5. Assume that p 3. Let G be a commutative finite and flat group scheme killed by p over S, and 0 G A B 0 be a resolution of G by abelian ! ! ! ! F c schemes (5.1.1). We denote ᏼ.B/ Coker.ᏻB ᏻB / (3.7.1), and similarly D x1 ! x1 for ᏼ.A/. According to Remark 3.9, we have an identification

0 e 1 0 0 (6.5.1) H .Bs; U ‰B / H .Bs; ᏼ.B// H .B1; ᏼ.B// N ' N D x 976 YICHAO TIAN

0 1 as submodules of H .Bs;‰B /; in the last equality, we have identiﬁed the topos N .Bs/et with .B1/et. We denote N K x K Â Ã 1 F c 1 Ker.B; F c/ Ker H .B1; އa;t / H .B1; އa;1/ D x ! x Â F c Ã Ker Lie.B / Lie.B / ; D xt_ ! x1_ Â Ã 0 ıE 2 Ker.B; ıE / Ker H .B1; ᏼ.B// H .Bs; ކp/ ; D x ! N where ıE is the morphism deﬁned in Proposition 3.10(2); we also have similar notations for A. Since the exact sequence (3.10.2) is functorial in X, we have a commutative diagram

1 (6.5.2) 0 / H .Bs; ކp/ / Ker.B; F c/ / Ker.B; ıE / / 0 N

ˇ1 ˇ2 ˇ3 1 0 / H .As; ކp/ / Ker.A; F c/ / Ker.A; ıE / / 0: N LEMMA 6.6. The assumptions are those of Section 6.5. (i) In diagram (6.5.2), Â Ã 1 1 (6.6.1) Ker ˇ1 Ker Ext .Bs; ކp/ Ext .As; ކp/ .Get/_.K/. 1/ D N ! N D K x I Â F c Ã (6.6.2) Ker ˇ2 Ker Lie.G / Lie.G / D xt_ ! x1_ I 0 0 1 (6.6.3) Ker ˇ3 H .Bs; ᏼ.B// N.1/ H .Bs;‰B /; D N \ N where N.1/ is as deﬁned in (5.3.1). (ii) We have the equality

e (6.6.4) dimކ U G .K/ dimކ Ker ˇ1 dimކ Ker ˇ3: p _ x D p C p In particular, Â Ã e F c (6.6.5) dimކ U G .K/ dimކ Ker Lie.G / Lie.G / : p _ x p xt_ ! x1_

Moreover, the equality holds in (6.6.5) if and only if the morphism Coker ˇ1 ! Coker ˇ2 induced by diagram (6.5.2) is injective.

1 Proof. (i) By Corollary 4.6, we have a canonical isomorphism Ext .Xs; ކp/ 1 N ' H .Xs; ކp/ for X A or B. Hence formula (6.6.1) follows easily by applying the N q D functors Ext . ; ކp/ to the exact sequence 0 Gs As Bs 0. Applying the i ! N i! N ! N ! morphism of functors F c Ext . ; އa;t / Ext . ; އa;1/ to the exact sequence W ! CANONICAL SUBGROUPS OF BARSOTTI-TATE GROUPS 977

0 G A B 0, we obtain the following commutative diagram: ! ! ! ! / Lie.G / / Lie.B / / Lie.A / 0 xt_ xt_ xt_ F c F c F c / Lie.G / / Lie.B / / Lie.A /; 0 x1_ x1_ x1_ where we have used (4.7.1) and (4.7.2). Formula (6.6.2) follows immediately from this diagram. For (6.6.3), using (6.5.1), we have a commutative diagram

.1/ 0 1 Ker.B; ıE / / H .Bs;‰B / N

ˇ3 .2/ 0 1 Ker.A; ıE / / H .As;‰A/; N where the maps .1/ and .2/ are injective. Hence we obtain Â Ã 0 1 0 1 Ker ˇ3 Ker.B; ıE / Ker H .Bs;‰ .B// H .As;‰ .A// D \ N ! N Ker.B; ıE / N.1/: D \ 1;0 The morphisms ıE and d2 are compatible in the sense of Proposition 3.10, and 0 Proposition 5.5 implies that Ker.B; ıE / N.1/ H .Bs; ᏼ.B// N.1/, which \ D N \ proves (6.6.3). (ii) By the isomorphism (5.5.1) and the surjectivity of the morphism u in (5.3.1), we have Â Ã e e 1 (6.6.6) dimކp U G_.K/ dimކp G_.K/. 1/ U H .B; ޚ=pޚ/ x D x \ N Â Ã Â Ã 0 e 1 dimކp .Get/_.K/. 1/ dimކp N H .Bs; U ‰ .B//. 1/ : D K x C \ N Then the equality (6.6.4) follows from (i) of this lemma and (6.5.1). The remaining part of (ii) follows immediately from diagram (6.5.2).

6.7. Proof of Proposition 6.2. We choose a resolution 0 G A B 0 ! ! ! ! of G by abelian schemes (5.1.1). By Lemma 6.6, we have to prove that if G is a truncated Barsotti-Tate group of level 1 over S, the morphism 12 Coker ˇ1 W ! Coker ˇ2 induced by diagram (6.5.2) is injective. By Corollary 4.6, we have H1 .X ; ކ / H1 .X ; ކ / Ext1.X ; ކ / for et s p et x1 p x1 p K N D K D X A or B. Thus the morphism ˇ1 is canonically identiﬁed to the morphism D1 1 Ext .B1; ކp/ Ext .A1; ކp/ induced by the map A B. Applying the functors i x ! x ! Ext . ; ކp/ to 0 G1 A1 B1 0, we obtain a long exact sequence ! x ! x ! x ! 1 1 1 2 Ext .B1; ކp/ Ext .A1; ކp/ Ext .G1; ކp/ Ext .B1; ކp/: x ! x ! x ! x 978 YICHAO TIAN

2 1 Since Ext .B1; ކp/ 0 by Corollary 4.9, we have Coker ˇ1 Ext .G1; ކp/. The x D D x commutative diagram

F c 0 / Ker.B; F c/ / Lie.B / / Lie.B / t_ 1_

ˇ2 F c 0 / Ker.A; F c/ / Lie.A / / Lie.A / t_ 1_ 1 induces a canonical morphism Coker ˇ2 Coker Ext .G1; އa;t /. Let 1 1 W ! D x Ext .G1; ކp/ Ext .G1; އa;t / be the morphism induced by the map ކp އa;t W x ! x ! in (6.1.1). Then we have the following commutative diagram:

1 12 Ext .G1; ކp/ Coker ˇ1 / Coker ˇ2 x D

) Ext1.G ; އ /: x1 a;t

Now Lemma 6.4 implies that is injective, hence so is 12. This completes the proof.

7. The canonical ﬁltration in terms of congruence groups

./ 7.1. Recall the following deﬁnitions in[20]. For any ᏻK , let Ᏻ be 1 2 x the group scheme Spec.ᏻK ŒT; 1 T / with the comultiplication given by T x 7! T 1 1 T T T; the co-unitC by T 0 and the coinverse by T T . ˝ C ˝ C ˝ D 7! 1 T If v./ e=.p 1/, we put C Ä .1 T /p 1 P.T / C ᏻK ŒT D p 2 x

./ .p/ and let Ᏻ Ᏻ be the morphism of ᏻK -group schemes defined on the W ! x level of Hopf algebras by T P.T /. We denote by G the kernel of , so we 7! have G Spec ᏻK ŒT =P.T / : We call it, following Raynaud, the congruence D x group of level . It is a finite, flat group scheme over S Spec.ᏻK / of rank p. x D x

7.2. For all ᏻK with v./ e=.p 1/, let 2 x Ä p G p Spec.ᏻK ŒX=.X 1// W ! D x be the homomorphism given on the level of Hopf algebras by X 1 T . Then 7! C K is an isomorphism, and if v./ 0, is an isomorphism. For all ; ᏻK ˝ x D 2 x with v. / v./ e=.p 1/, let G G be the map deﬁned by the Ä Ä ; W ! homomorphism of Hopf algebras T .= /T . We have . 7! D ı ; CANONICAL SUBGROUPS OF BARSOTTI-TATE GROUPS 979

7.3. Let ᏻK with v./ e=.p 1/, A be an abelian scheme over S. 2 x Ä Define (7.3.1) .A/ Ext1 .A; G / Ext1 .A; / S S p W x ! x to be the homomorphism induced by the canonical morphism G p, where, W ! by abuse of notation, A denotes also the inverse image of A over S, and Ext1 means x S the extension in the category of abelian fppf-sheaves over S. Similarly, letxG be a x commutative finite and flat group scheme killed by p over S; we define

(7.3.2) .G/ HomS .G; G/ HomS .G; p/ G_.K/ W x ! x D x to be the homomorphism induced by . If G pA, where A is an S-abelian Dp scheme, the natural exact sequence 0 pA A A 0 induces a commutative ! ! ! ! diagram

Hom . A; G / Ext1 .A; G / (7.3.3) S p / S x x

.pA/ .A/ Ext1 .A; /; pA_.K/ / S p x x where horizontal maps are isomorphisms (4.3.2). Hence, .A/ is canonically identiﬁed to .pA/.

LEMMA 7.4. Let ; ᏻK with v. / v./ e=.p 1/, G be a commutative 2 x Ä Ä ﬁnite and ﬂat group scheme killed by p over S.

(i) .G/ is injective.

(ii) The image of .G/ is contained in that of .G/.

(iii) The image of .G/ depends only on v./, and it is invariant under the action of the Galois group Gal.K=K/. x Proof. There is a commutative diagram

.G/ (7.4.1) HomS .G_;G_/ / HomS .ޚ=pޚ;G_/ x x .1/ .2/ Hom.G_;G_/ / Hom.ޚ=pޚ;G_/; N N where the horizontal maps are induced by ޚ=pޚ G , and the vertical _ W ! _ maps are induced by the base change S. Since is an isomorphism over N ! x the generic point Section 7.2, the map .2/ is an isomorphism. Hence statement (i) follows from the fact that .1/ is injective by the ﬂatness of G and G. Statement (ii) follows easily from the existence of the morphism G ; W ! G with . The ﬁrst part of (iii) follows immediately from (ii). Any D ı ; 980 YICHAO TIAN

Gal.K=K/ sends the image of .G/ isomorphically to the image of .G/, 2 x ./ which coincides with the former by the ﬁrst assertion of (iii).

7.5. Filtration by congruence groups. Let a be a rational number with 0 Ä a e=.p 1/, and G be a commutative finite and flat group scheme over S killed Ä Œa by p. We choose ᏻK with v./ a, and denote by G_.K/ the image of 2 x D x .G/. By Lemma 7.4, G .K/Œa depends only on a, and not on the choice of . _ x Then G .K/Œa; a ޑ Œ0; e=.p 1// is an exhaustive decreasing filtration of _ x 2 \ G .K/ by Gal.K=K/-groups. _ x x

7.6. Let ᏻK with 0 v./ e=.p 1/, f A S be an abelian scheme, 2 x Ä Ä W ! and f A S its base change by S S. In ([3, 6]), Andreatta and Gasbarri N W x ! x x !1 1 consider the homomorphism .A/ H .A; G / H .A; p/ induced by , 0 W fppf x ! fppf x where by abuse of notation, G denotes also the fppf-sheaf G restricted to A. We x have a commutative diagram

'.G / Ext1 .A; G / H1 .A; G / (7.6.1) S / fppf x x

.A/ 0 .A/

'.p/ Ext1 .A; / H1 .A; /; S p / fppf p x x where the horizontal arrows are the homomorphisms (4.4.1).

LEMMA 7.7. (i) The homomorphisms '.G/ and '.p/ in (7.6.1) are isomorphisms. In particular, the homomorphism 0 .A/ is canonically isomorphic to .A/ (7.3.1). (ii) The canonical morphism H1 .A; / H1.A ; / is an isomorphism. Let fppf x p p 1 Œv./ ! N H .A; p/ be the image of 0 .A/ composed with this isomorphism. Then via N 1 1 Œv./ the canonical isomorphism H .A; p/ pA_.K/, the subgroup H .A; p/ N ' x N is identiﬁed to A .K/Œv./. p _ x Proof. (i) For H G or p, the “local-global” spectral sequence induces D an exact sequence (7.7.1) 1 0 1 .H / 0 1 0 Hfppf.S;Rfppff .HA// Hfppf.A; HA/ Hfppf.S;Rfppff .HA//: ! x N x ! x x ! x N x By Proposition 4.5 and (4.3.2), there are isomorphisms

H0 .S;R1 f .H // H0 .S; Ᏹxt1 .A; H // Ext1 .A; H /: fppf fppf A fppf S S x N x ' x x ' x 1 1 Therefore, we obtain a homomorphism .H / Hfppf.A; HA/ Ext .A; H /. We W x x ! S check that the composed map .H / '.H / is the identity morphism onxExt1 .A; H /; S ı x in particular, .H / is surjective. By Proposition 4.5, we have also R0 f .H / fppf N A 1 x D HS ; on the other hand, it follows from ([3, Lemma 6.2]) that Hfppf.S;HS / 0. x x x D CANONICAL SUBGROUPS OF BARSOTTI-TATE GROUPS 981

Hence .H / is injective by the exact sequence (7.7.1), and '.H / and .H / are both isomorphisms. (ii) There is a commutative diagram

'.p/ Ext1 .A; / H1 .A; / S p / fppf p x x .1/ .2/ 1 1 Ext.A; p/ / H .A; p/; N N N where the vertical maps are base changes to the generic ﬁbers, and the horizontal morphisms are (4.4.1), which are isomorphisms in our case by Corollary 4.6 and statement (i). The morphism .1/ is easily seen to be an isomorphism by (4.3.2), hence so is the morphism .2/. The second part of statement (ii) is a consequence of (i).

The following proposition, together with Proposition 5.5, implies Theorem 1.6.

PROPOSITION 7.8. Let G be a commutative finite and flat group scheme over S killed by p. Then, for all rational numbers 0 a e=.p 1/, we have Ä Ä G .K/Œa UpaG .K/, where U G .K/ is the Bloch-Kato filtration Defini- _ x D _ x _ x tion 5.4.

Proof. Let 0 G A B 0 be a resolution of G by abelian schemes ! ! ! ! (5.1.1). We have, for all ᏻK with 0 v./ e=.p 1/, a commutative diagram 2 x Ä Ä Hom .G; G / Ext1 .B; G / Ext1 .A; G / 0 / S / S / S x _ x _ x _

.G/ .B/ .A/ 0 / G .K/ / B .K/ / A .K/: _ x p _ x p _ x Hence, for all rational numbers a satisfying 0 a e=.p 1/, we have by 7.7(ii) Ä Ä Œa Œa 1 Œa (7.8.1) G_.K/ G_.K/ pB_.K/ G_.K/ H .B; p/ : x D x \ x D x \ N 1 Œa According to ([3, Th. 6.8]), the filtration .H .B; p/ ; 0 a e=.p 1// coin- pa 1 N e Ä Ä cides with the filtration .U H .B; p/; 0 a p 1 / (3.2.3). Hence by (7.8.1) N Ä Ä and (5.4.1), the two filtrations G .K/Œa and UpaG .K/ on G .K/ coincide. _ x _ x _ x This completes the proof.

8. The lifting property of the canonical subgroup

In this section, by abuse of notation, އa will denote the additive group both over S and over S. For a rational number r > 0, we denote by އ , ./ and G x a;r Ᏻr ;r the base changes to S of the respective group schemes. xr 982 YICHAO TIAN

8.1. Following[21], for a; c ᏻK with ac p, we denote by Ga;c the group p 2 x D scheme Spec ᏻK Œy=.y ay/ over ᏻK with comultiplication x x p 1 i p i cwp 1 X y y y y 1 1 y 7! ˝ C ˝ C 1 p wi ˝ wp i i 1 D and the counit given by y 0, where wi .1 i p 1/ are universal constants D Ä Ä in ᏻK with v.wi / 0 (see[21, p. 9]). Tate and Oort proved that .a; c/ Ga;c x D 7! gives a bijection between equivalence classes of factorizations of p ac in ᏻK and D x isomorphism classes of ᏻK -group schemes of order p, where two factorizations x p a1c1 and p a2c2 are called equivalent if there exists u ᏻK such that D p 1 D 1 p 2 x a2 u a1 and c2 u c1. D D Let ᏻK with 0 vp./ 1=.p 1/. There is a factorization p a./c./ 2 x Ä Ä D p 1 such that G G . More explicitly, we may take c./ ..1 p// and a./;c./ wp 1 p ' D a./ , and we notice that vp.a.// 1 .p 1/vp./ is well deﬁned D c./ D independently of the factorization p a./c./. D LEMMA 8.2 ([3, Lemmas 8.2 and 8.10]). Let 0 < r 1 be a rational number p 1 Ä and ᏻK with vp./ 1 1=p and vp. / r. 2 x Ä (i) Let r be the morphism of groups schemes ./ Ᏻr Spec ᏻS ŒT އa;r Spec ᏻS ŒX D xt ! D xr Pp 1 i 1 T i deﬁned on the level of Hopf algebras by X . / . Then is an 7! i 1 i r isomorphism. Moreover, the following diagram is commutativeD

./ ;r .p/ Ᏻr / Ᏻr

p r r F a./ އa;r / އa;r ; where F is the Frobenius homomorphism and a./ ᏻK is as introduced in Sec- 2 x tion 8.1. i ./ r (ii) Let ı be the composed morphism G Ᏻr އa;r : Then ı ;r ;r ! ! ;r generates Lie.G;r_ / HomS .G;r ; އa;r / as an ᏻS -module. ' xr xr 1 1 LEMMA 8.3 ([3, Lemma 8.3]). Let ᏻK with p.p 1/ vp./ p 1 , and 2 x Ä Ä r .p 1/vp./. Then the following diagram is commutative D ./ ;1 .p/ Ᏻ1 / Ᏻ1

p 1;r r r F a./ ./ އ އ Ᏻr / a;r / a;1; where 1;r is the reduction map. CANONICAL SUBGROUPS OF BARSOTTI-TATE GROUPS 983

1 8.4. Let ᏻK with vp./ p , t 1 1=p, and G be a commutative finite 2 x D D and flat group scheme killed by p over S. We define ˆG to be (8.4.1) t ı ˆG HomS .G;G/ HomS .Gt ;G;t / HomS .Gt ; އa;t / Lie.Gt_/; W x x ! xt x ! xt x D x where t is the canonical reduction map, and ı is the morphism induced by the element ı;t HomS .G;t ; އa;t / Lemma 8.2(ii). 2 xt 1 PROPOSITION 8.5. Let ᏻK with vp./ p , t 1 1=p, and G be a trun- 2 x D D cated Barsotti-Tate group of level 1 over S, satisfying the hypothesis of Theorem 1.4. Then we have an exact sequence

ˆG F a./ (8.5.1) 0 HomS .G;G/ HomS .Gt ; އa;t / HomS .G1; އa;1/; ! x x ! xt x ! x1 x where a./ is as defined in Section 8.1. Proof. From Lemma 8.3, we deduce a commutative diagram (8.5.2) ./ ;1 .p / Hom .G1;G / S1 ;1 / HomS .G1; Ᏻ1 / / HomS .G1; Ᏻ1 / x x x1 x x1 x p 1;t 1;t0 1 ./ t F a./ Hom .Gt ;G / Hom .Gt ; އa;t / Hom .G1; އa;1/; St ;t / HomS .Gt ; Ᏻt / / St / S1 x x xt x x x x x where the upper row is exact and 1;t and 1;t0 are reduction maps. Therefore, the composition of ˆG with the morphism F a./ in (8.5.1) factorizes through the upper row of (8.5.2), and thus equals 0. Let L be the kernel of the map F a./ in (8.5.1). Then ˆG induces a map ˆ0 HomS .G;G/ L. We have to prove that W x x ! ˆ0 is an isomorphism. Let d be the rank of Lie.G1_/ HomS .G1; އa;1/ over ᏻS , and recall that x D x1 x x1 vp.a.// 1=p. Since G satisfies the assumptions of Theorem 1.4, applying D Proposition 3.12 to Lie.G / and the operator F a./, we see that the group x1_ L is an ކp-vector space of dimension d . On the other hand, HomS .G;G/ is e x x identified with G p 1 .K/ by Theorem 1.6. Thus it is also an ކ -vector space C x ? p of dimension d by Theorem 1.4(i). Therefore, to finish the proof, it suffices to prove that ˆ0 is surjective. By Lemma 8.2(i), we have the following commutative diagram

F a./ 0 / L / HomS .Gt ; އa;t / / HomS .G1; އa;1/ xt x x1 x

˛ 1;t F a./ 0 / HomS .Gt ;G;t / / HomS .Gt ; އa;t / / HomS .Gt ; އa;t /; xt x xt x xt x where 1;t is the reduction map. The composed morphism ˛ ˆ is the canonical ı 0 reduction map, whose injectivity will imply the injectivity of ˆ0. Thus the following lemma will conclude the proof of the proposition. 984 YICHAO TIAN

LEMMA 8.6. Assume that p 3. Let t 1 1=p, ᏻK with vp./ 1=p, D 2 x D and G be a commutative, ﬁnite, ﬂat group scheme killed by p over S. Then the reduction map

t HomS .G;G/ HomS .Gt ;G;t / W x x ! xt x is injective.

Proof. We put G S S Spec.A/, where A is a Hopf algebra over ᏻK with the x D x comultiplication . An element f HomS .G;G/ is determined by an element 2 x x x A satisfying 2 .x/ x 1 1 x x x; D ˝ C ˝ C ˝ .1 x/p 1 (8.6.1) P .x/ C 0: D p D Suppose that t .f / 0, which means x mt A. We want to prove that in fact x 0. D 2 D Let us write x ay where a p 1 2 is an integer, and y A. Substituting D 2 x in (8.6.1), we obtain p 1 ! X 1 p .ay/p i.a 1/yi 0: C p i C D i 1 D 1 p Since vp. p i / 0 for 1 i p 1 and A is flat over ᏻK , we see easily a 1 D Ä Ä that y C y1 for some y1 A. Continuing this process, we find that x D 2 2 a ޑ>0 maA 0 (1.9). \ 2 D LEMMA 8.7. Let G be a Barsotti-Tate group of level 1 and height h over S, and H be a flat closed subgroup scheme of G. We denote by d the dimension of Lie.Gs/ over k, and d h d. Then the following conditions are equivalent: D (i) The special fiber Hs of H coincides with the kernel of the Frobenius of Gs. (i ) The special fiber H of H .G=H / coincides with the kernel of the 0 s? ? D _ Frobenius of Gs_. d (ii) H has rank p over S and dim Lie.Hs/ d. k (ii ) H has rank pd over S and dim Lie.H / d . 0 ? k s? Proof. We have two exact sequences 0 H G G=H 0; 0 H G H 0: ! ! ! ! ! ? ! _ ! _ !

Denote by FGs (resp. by VGs ) the Frobenius (resp. the Verschiebung) of Gs. As- sume that (i) is satisfied; then we have H Coker.V / by duality. Since G s_ D Gs_ is a Barsotti-Tate group of level 1, H coincides with Im.V / Ker.F /. s? Gs_ D Gs_ Conversely, if H Ker.F /, we have also Hs Ker.FG /. This proves the s? D Gs_ D s equivalence of (i) and (i0). If (i) or (i0) is satisfied, then (ii) and (ii0) are also satisfied (SGA 31 VIIA dimk Lie.Hs / 7.4). Assume (ii) satisfied. Since Ker.FHs / has rank p (loc. cit.) and is CANONICAL SUBGROUPS OF BARSOTTI-TATE GROUPS 985

contained in both Ker.FGs / and Hs, condition (ii) implies that these three groups have the same rank; hence they coincide. This proves that (ii) implies (i). The equivalence of (i0) and (ii0) is proved in the same way.

8.8. Proof of Theorem 1.4(ii). By Lemma 8.7, the following lemma will complete the proof of 1.4(ii).

LEMMA 8.9. Let G be a Barsotti-Tate group of level 1 and height h over S, satisfying the hypothesis of Theorem 1.4, d be the dimension of Lie.Gs_/ over e k, and d h d. Let H be the ﬂat closed subgroup scheme G p 1 C, and D H .G=H / . Then H has rank pd over S and dim Lie.H / d . ? D _ ? k s? d d Proof. Since H has rank p over S by Theorem 1.4(i), H ? has rank p over S and dimކp .G=H /.K/ d . Let ᏻK with vp./ 1=p, and t 1 1=p. x D 2 x D D The canonical projection G G=H induces an injective homomorphism !

(8.9.1) HomS .G=H;G/ HomS .G;G/: x x x ! x x e Œ p By Theorem 1.6, we see that H ?.K/ HomS .G=H;G/ is orthogonal to e x D x x x p 1 .G=H / C.K/ under the perfect pairing .G=H /.K/ H ?.K/ p.K/. As e x x x ! xe H G p 1 C, Proposition 2.8(ii) implies that the group scheme .G=H / p 1 C is D trivial. Hence,

dimކp HomS .G=H;G/ dimކp H ?.K/ d dimކp HomS .G;G/; x x x D x D D x x and the canonical map (8.9.1) is an isomorphism. By the functoriality of ˆG (8.4.1), we have a commutative diagram

(8.9.2) HomS .G=H;G/ HomS .G;G/ x x x x x

ˆG=H ˆG Lie.H / / Lie.G /; xt? xt_ where the lower row is an injective homomorphism of ᏻS -modules. Put N0 xt D Hom .G;G /, M Lie.G / and M Lie.G / ᏻ Lie.G /. By S 1_ t 1_ ᏻS St t_ x x D x D x ˝ x1 x D x Proposition 8.5(ii), N0 is identiﬁed with the kernel of F a./ Mt M . Let W ! N be the ᏻK -submodule of Mt generated by N0. Applying Proposition 3.12(ii) to the morphismx F a./, we get

dimk.N=mK N/ dimކp N0 d : N x D D By (8.9.2), N is contained in M Lie.H / M . By applying Lemma 8.10 (ii) 0 D xt? below to N M , we obtain 0

(8.9.3) d dimk.N=mK N/ dimk.M 0=mK M 0/: D N x Ä N x 986 YICHAO TIAN

Let !H be the module of invariant differentials of Ht? over ᏻS . Then xt? x xt !H !H ᏻS k and s? t? xt N N D x ˝ M Lie.H / Hom .! ; ᏻ /: 0 t? ᏻS H St D x D xt xt? x

Applying Lemma 8.10 (i) to !H , we obtain xt?

(8.9.4) dim .M 0=m M 0/ dim ! : k K k Hs? N x D N N

From the relations Lie.H ?/ k k Lie.H ?/ Hom .! ; k/, we deduce s N s k Hs? N ˝ D N D N N (8.9.5) dim ! dimk Lie.H ?/: k Hs? s N N D The desired inequality dim Lie.H / d then follows from (8.9.3), (8.9.4) and k s? (8.9.5).

LEMMA 8.10. Let t be a positive rational number, M be an ᏻ -module of St ﬁnite presentation. x (i) Put M Hom .M; ᏻ /. Then we have ᏻS St D xt x

dimk.M =mK M / dimk.M=mK M /: N x D N x

(ii) If N is a finitely presented ᏻS -submodule of M , then dimk.N=mK N/ xt N x Ä dimk.M=mK M/. N x Proof. Since M is of finite presentation, up to replacing K by a finite extension, we may assume that there exist a positive integer n and a finitely generated ᏻ =nᏻ -module M , where is a uniformizer of ᏻ , such that ᏻ K K 0 K St n x D ᏻK = ᏻK and M M0 ᏻK ᏻK . Note that there exist integers 0 < a1 ar n x x D ˝ x Ä Ä such that we have an exact sequence of ᏻK -modules

r ' r (8.10.1) 0 ᏻ ᏻ M0 0; ! K ! K ! ! a where ' is given by .xi /1 i r . i xi /1 i r . In order to prove (i), it sufﬁces Ä Ä 7! Ä Ä to verify that dim .M =M / dim .M0=M0/, where k 0 0 D k n M Hom .M0; ᏻK = ᏻK /: 0 D ᏻK Let n r 'n n r .ᏻK = ᏻK / .ᏻK = ᏻK / M0 0 ! ! ! be the reduction of (8.10.1) modulo n. Applying the functor n HomᏻK . ; ᏻK = ᏻK / to the above exact sequence, we get

' n r n n r 0 M0 .ᏻK = ᏻK / .ᏻK = ᏻK / ! ! ! CANONICAL SUBGROUPS OF BARSOTTI-TATE GROUPS 987 with ' 'n. Hence M is isomorphic to the submodule n D 0 r n ai n i 1. ᏻK = ᏻK / ˚ D n r of .ᏻK = ᏻK / , and

dim .M =M / r dim .M0=M0/: k 0 0 D D k For statement (ii), by the same reasoning, we may assume that there exists a

ﬁnite ᏻK -submodule N0 of M0 such that N N0 ᏻK ᏻK .We need to prove D ˝ x that dim .N0=N0/ dim .M0=M0/. Let M0 be the kernel of M0 of the k Ä k multiplication by . We have an exact sequence of Artinian modules 0 M0 M0 M0 M0=M0 0: ! ! ! ! ! By the additivity of length of Artinian modules, we obtain dim . M0/ k D dim .M0=M0/. Similarly, we have dim . N0/ dim .N0=N0/. The asser- k k D k tion follows from the fact that N0 is a submodule of M0. Remark 8.11. If we could prove the exact sequence (8.5.1) without knowing a priori the rank of HomS .G;G/ for vp./ 1=p, then we would get another proof x x D of the existence of the canonical subgroup of G. Since then, by Proposition 3.12 and (8.5.1), Hom .G;G / has ކ -rank d under the assumptions of Theorem 1.4. S x p Then we identifyx it to be a subgroup of G .K/ by .G/ (7.3.2), and deﬁne H _ x to be the subgroup scheme of G determined by H.K/? HomS .G; G/: The x D x arguments in this section imply that H is the canonical subgroup of G. For abelian schemes, this approach is due to Andreatta-Gasbarri[3].

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(Received June 16, 2006) (Revised October 2, 2007)

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