Directory UMM :Journals:Journal_of_mathematics:OTHER:

p

-Adi Monodromy of the Universal

Deformation of a HW-Cyclic Barsotti-Tate Group

Yichao Tian

Received: September 5, 2009

Communicated by Peter Schneider

Abstract.

Let

k

be an algebraially losed eld of harateristi

p >

0

,and

G

beaBarsotti-Tateover

k

. Wedenoteby

S

thealgebrai

loalmoduliin harateristi

p

of

G

,by

G

theuniversaldeformation of

G

over

S

, and by

U

⊂

S

the ordinary lous of

G

. The étale partof

G

over

U

givesrisetoamonodromyrepresentation

ρ

G

ofthe fundamental group of

U

on the Tate module of

G

. Motivated by a famoustheoremofIgusa,weproveinthisartilethat

ρ

G

issurjetive if

G

isonnetedand HW-yli. This latteronditionis equivalent tosayingthatOort's

a

-numberof

G

equals

1

,anditissatisedbyall onnetedone-dimensionalBarsotti-Tategroupsover

k

.

2000 Mathematis Subjet Classiation: 13D10, 14L05, 14H30, 14B12,14D15,14L15

KeywordsandPhrases: Barsotti-Tate groups(

p

-divisiblegroups),

p

-adi monodromy representation, universal deformation, Hasse-Witt maps.

1. Introdution

1.1.

Alassialtheoremof Igusasaysthatthemonodromyrepresentation as-soiatedwithaversalfamilyofordinaryelliptiurvesinharateristi

p >

0

is surjetive [Igu,Ka2℄. This important result hasdeep onsequenes in the theoryof

p

-adimodularforms,andinpsiredvariousgeneralizations. Faltings and Chai[Ch2,FC℄extendeditto theuniversalfamilyoverthemodulispae of higherdimensional prinipally polarizedordinaryabelianvarieties in har-ateristi

p

, andEkedahl[Eke℄ generalizeditto the jaobianof theuniversal

n

-pointedurveinharateristi

p

,equippedwithasympletilevelstruture.

arithmetiappliations. Thoughithasbeenformulatedinaglobalsetting,the proofofIgusa'stheoremispurelyloal,andithasgotalsoloalgeneralizations. Gross [Gro℄ generalizedit to one-dimensional formal

O

-modules overa om-pletedisretevaluationringofharateristi

p

,where

O

istheintegrallosure of

Zp

in aniteextension of

Qp

. Wereferto Chai[Ch2℄ and Ahter-Norman [AN ℄formoreresultsonloal monodromyofBarsotti-Tategroups. Motivated bytheseresults,ithasbeenlonglyexpeted/onjeturedthatthemonodromy of a versal familyof ordinaryBarsotti-Tate groupsin harateristi

p >

0

is maximal. Theaimofthispaperistoprovethesurjetivityofthemonodromy representationassoiatedwiththeuniversaldeformationin harateristi

p

of aertainlass ofBarsotti-Tategroups.

1.2.

To desribe ourmain result, weintrodue rst thenotion of HW-yli Barsotti-Tategroups. Let

k

beanalgebraiallylosedeldofharateristi

p >

0

,and

G

beaBarsotti-Tategroupover

k

. Wedenoteby

G

∨

theSerredualof

G

, andby

Lie(

G

∨

)

itsLiealgebra. TheFrobeniushomomorphismof

G

(ordually theVershiebungof

G

∨

)induesasemi-linearendomorphism

ϕ

G

on

Lie(

G

∨

)

, alled the Hasse-Witt map of

G

(2.6.1). We say that

G

is HW-yli, if

c

=

dim(

G

∨

)

≥

1

andthere isa

v

∈

Lie(

G

∨

)

suh that

v, ϕ

G

(

v

)

,

· · ·

, ϕ

c

−1

G

(

v

)

form abasisof

Lie(

G

∨

)

over

k

(4.1). Weprovein4.7that

G

isHW-yliand non-ordinaryifand onlyifthe

a

-numberof

G

, dened previouslyby Oort,equals

1

. BasiexamplesofHW-yliBarsotti-Tategroupsaregivenasfollows. Let

r, s

berelativelyprime integerssuhthat

0

≤

s

≤

r

and

r

6

= 0

,

λ

=

s/r

,

G

λ

betheBarsotti-Tategroupover

k

whose(ontravariant)Dieudonné moduleis generatedbyanelement

e

overthenon-ommutativeDieudonnéringwiththe relation

(

F

r

−

s

−

V

s

)

·

e

= 0

(4.10). Itis easytosee that

G

λ

is HW-ylifor any

0

< λ <

1

. AnyonnetedBarsotti-Tategroupover

k

ofdimension

1

and height

h

isisomorphito

G

1

/h

[Dem ,Chap.IVŸ8℄.

Let

G

beaBarsotti-Tategroupofdimension

d

andheight

c

+

d

over

k

;assume

c

≥

1

. Wedenoteby

S

thealgebrailoalmoduliof

G

inharateristi

p

,and

by

G

betheuniversaldeformationof

G

over

S

(f. 3.8). Thesheme

S

isane of ring

R

≃

k

[[(

t

i,j

)

1≤

i

≤

c,

1≤

j

≤

d

]]

, and the Barsotti-Tategroup

G

is obtained by algebraizingthe formaluniversal deformationof

G

over

Spf(

R

)

(3.7). Let

U

betheordinarylousof

G

(i.e. theopensubshemeof

S

parametrizingthe

ordinarybersof

G

),and

η

ageometripointoverthegeneripointof

U

. For anyinteger

n

≥

1

, wedenote by

G(

n

)

the kernelof the multipliationby

p

n

on

G

, andby

T

p

(G

, η

) = lim

←−

n

G(

n

)(

η

)

the Tate module of

G

at

η

. This is afree

Zp

-module of rank

c

. Weonsider themonodromyrepresentationattahedtotheétalepartof

G

over

U

(1.2.1)

ρ

G

:

π

1

(

U

, η

)

→

Aut

Zp

(T

p

(

G

, η

))

≃

GL

c

(

Zp

)

.

Theorem

1.3

.

If

G

isonnetedand HW-yli, then the monodromy repre-sentation

ρ

G

issurjetive.

Igusa'stheoremmentionedaboveorrespondstoTheorem1.3for

G

=

G

1

/

2

(f. 5.7). Myinterest in the

p

-adimonodromy problem started withthe seond partof myPhDthesis [Ti1℄, where Iguessed 1.3 for

G

=

G

λ

with

0

< λ <

1

andproveditfor

G

1

/

3

. AfterIpostedthemanusriptonArXiv[Ti2℄,Strauh provedtheone-dimensionalaseof1.3byusingDrinfeld'slevelstrutures[Str, Theorem 2.1℄. Later on, Lau [Lau℄ proved 1.3 without the assumption that

G

is HW-yli. Byusingthe Newtonstratiation oftheuniversal

deforma-tion spaeof

G

due to Oort, Lauredued thehigherdimensional ase to the one-dimensionalasetreatedbyStrauh. Infat,StrauhandLauonsidered moregenerallythemonodromyrepresentationovereah

p

-rankstratumofthe universaldeformationspae. Inthispaper,weproviderstadierentproofof theone-dimensionalaseof1.3. Ourapproahispurelyharateristi

p

,while Strauh used Drinfeld's levelstruture in harateristi

0

. Then byfollowing Lau'sstrategy,wegiveanew(andeasier)argumenttoreduethegeneralase of 1.3 to the one-dimensional ase for HW-yli groups. The essential part of our argumentis aversalityriterion by Hasse-Witt maps of deformations of aonnetedone-dimensionalBarsotti-Tategroup(Prop. 4.11). This rite-rion anbe onsidered as ageneralizationof anothertheorem of Igusawhih laims that theHasse invariantof aversal familyof elliptiurvesin hara-teristi

p

hassimplezeros. ComparedwithStrauh'sapproah,our harater-isti

p

approahhastheadvantageofgivingalsoresultsonthemonodromyof Barsotti-Tategroupsoveradisretevaluationringofharateristi

p

.

1.4.

Let

A

=

k

[[

π

]]

be the ringof formal powerseries over

k

in thevariable

π

,

K

itsfrationeld,and

v

thevaluationon

K

normalizedby

v

(

π

) = 1

. We

x analgebrai losure

K

of

K

, and let

K

sep

be the separable losure of

K

ontainedin

K

,

I

betheGaloisgroupof

K

sep

over

K

,

I

p

⊂

I

bethewildinertia subgroup,and

I

t

=

I/I

p

thetameinertiagroup.Foreveryinteger

n

≥

1

,there is a anonial surjetive harater

θ

p

n

−1

:

I

t

→

F

×

p

n

(5.2), where

Fp

n

is the nitesubeldof

k

with

p

n

elements.

Weput

S

= Spec(

A

)

. Let

G

beaBarsotti-Tategroupover

S

,

G

∨

beitsSerre dual,

Lie(

G

∨

)

the Lie algebra of

G

∨

, and

ϕ

G

the Hasse-Witt map of

G

, i.e. thesemi-linearendomorphismof

Lie(

G

∨

)

induedbytheFrobeniusof

G

. We dene

h

(

G

)

tobethevaluationofthedeterminantofamatrixof

ϕ

G

,andall ittheHasseinvariant of

G

(5.4). Weseeeasilythat

h

(

G

) = 0

ifandonlyif

G

isordinaryover

S

,and

h

(

G

)

<

∞

ifandonlyif

G

isgeneriallyordinary. If

G

isonneted ofheight

2

anddimension

1

,then

h

(

G

) = 1

isequivalenttothat

G

isversal(5.7).

G

(1)(

K

)

isan

Fp

c

-vetorspaeofdimension

1

onwhih the induedationof

I

t

isgiven by the surjetiveharater

θ

p

c

−1

:

I

t

→

F

×

p

c

.

This proposition is an analog in harateristi

p

of Serre's result[Se3, Prop. 9℄onthetamenessofthemonodromyassoiatedwithone-dimensionalformal groupsoveratraitofmixedharateristi. Wereferto5.8fortheproofofthis propositionandmoreresultsonthe

p

-adimonodromyofHW-yli Barsotti-Tategroupsoveratraitin harateristi

p

.

1.6.

This paper is organized as follows. In Setion 2, we review some well knownfatsonordinaryBarsotti-Tategroups.Setion3ontainssome prelim-inariesonthe Dieudonné theoryand thedeformationtheoryof Barsotti-Tate groups. In Setion 4, after establishing some basi properties of HW-yli groups,wegivethe fundamental relationbetween theversality ofa Barsotti-TategroupandtheoeientsofitsHasse-Wittmatrix(Prop. 4.11). Setion 5isdevotedtothestudyofthemonodromyofaHW-yliBarsotti-Tategroup overaomplete trait of harateristi

p

. Setion 6is totallyelementary, and ontainsariterion(6.3)forthesurjetivityofahomomorphismfroma pro-nite groupto

GL

n

(

Zp

)

. Setion 7 is the heart of this work, and it ontains aproof of Theorem1.3 in the one-dimensionalase. Finallyin Setion 8,we follow Lau's strategy and omplete the proof of 1.3 by reduing the general asetotheone-dimensionalasetreatedinSetion7.

TheproofinSetion7of1.3in theone-dimensionalaseisbasedonan indu-tionontheheight

n

+ 1

≥

2

of

G

. Thease

n

= 1

isjust thelassialIgusa's theorem (5.7). For

n

≥

2

,by lemmas6.3 and 6.5,it suesto provethe fol-lowingtwostatements: (a)theimage ofredutionmodulo

p

of

ρ

G

ontainsa non-splitCartansubgroup;(b)underasuitablebasis,theimageof

ρ

G

ontains

all matrixof theform

B

b

0

1

with

B

∈

GL

n

−1

(

Zp

)

and

b

∈

M

(

n

−1)×1

(

Zp

)

. Therststatementfollowseasilyfrom1.5byonsideringaertainbasehange of

G

to aompletedisretevaluationring. Toprove(b),weonsider the for-mal ompletion

Spec(

R

′

)

of theloalization of theloal moduli

S

= Spec(

R

)

of

G

at thegeneripointof thelouswhere theuniversal deformation

G

has

p

-rank

≤

1

(7.4). Thering

R

′

is aomplete regularring ofdimension

n

−

1

, and theBarsotti-Tategroup

G

′

=

G

⊗

R

R

′

hasaonneted partof height

n

and an étale partof height

1

. Let

K

0

be the residue eld of

R

′

, and

K

0

an algebrai losure of

K

0

. In order to apply the indution hypothesis, we on-sider the set of

k

-algebrahomomorphisms

σ

:

R

′

→

f

R

′

=

K

0

[[

t

1

,

· · ·

, t

n

−1

]]

liftingthenaturalinlusion

K

0

→

K

0

. Thekeypointisthat, thenaturalmap

σ

7→

G

f

R

′

,σ

=

G

′

⊗

R

′

,σ

f

R

′

givesabijetionbetweenthesetofsuh

σ

'sandtheset ofdeformationsof

G

K

0

=

G

′

⊗

R

′

K

₀

to

R

f

′

;moreover,weanomputeexpliitly theHasse-Wittmap oftheonnetedomponent

G

◦

f

R

′

,σ

of

G

f

R

′

,σ

(Lemma7.8). Fromtheversalityriterionforone-dimensionalBarsotti-Tategroupsinterms oftheHasse-Witt mapestablishedinSetion 4(Prop. 4.11),itfollows imme-diately that there exists a

σ

suh that the Barsotti-Tate group

G

◦

f

is onnetedand one-dimensionalof height

n

, istheuniversaldeformationof its losed ber. Wex suh a

σ

. Then the set of all

σ

′

with

G

◦

f

R

′

,σ

′

≃

G

◦

f

R

′

,σ

as deformations of their ommon losed ber is atually a group isomorphi to

Ext

1

f

R

′

(

Qp

/

Zp

,

G

◦

f

R

′

,σ

)

(Prop. 3.10). Let

σ

1

be the element orresponding

toneutralelementin

Ext

1

f

R

′

(

Qp

/

Zp

,

G

◦

f

R

′

,σ

)

. Applyingtheindution hypothesis to

G

◦

f

R

′

,σ

1

, we seethat themonodromy groupof

G

f

R

′

,σ

1

, henethat of

G

,

on-tainsthesubgroup

GL

n

−1

(

Zp

) 0

0

1

underasuitablebasisoftheTatemodule

(7.5.3). In order to onludethe proof, we need another

σ

2

suh that

G

f

R

′

,σ

2

hasthesameonnetedomponentas

G

f

R

′

,σ

1

,and thattheinduedextension betweentheTatemoduleoftheétalepartof

G

f

R

′

,σ

2

and thatof

G

◦

R

′

,σ

2

is

non-trivialafterredutionmodulo

p

(see 7.5and7.5.4). Toverifytheexisteneof suha

σ

2

,wereduetheproblemtoasimilarsituationoveraompletetraitof harateristi

p

(see7.9),andweuseariterionofnon-trivialityof extensions byHasse-Wittmaps(5.12).

1.7. Acknowledgement.

This paperis an expandedversionof theseond partofmyPh.D.thesisatUniversityParis13. Iwouldliketoexpressmygreat gratitude to my thesis advisor Prof. A. Abbesfor his enouragementduring thiswork,andalsoforhisvarioushelpful ommentsonearlierversionsofthis paper. I also thank heartily E. Lau, F. Oort and M.Strauh for interesting disussionsandvaluablesuggestions.

1.8. Notations.

Let

S

be a sheme of harateristi

p >

0

. A BT-group over

S

stands for a Barsotti-Tate group over

S

. Let

G

be a ommutative nite groupsheme(resp. aBT-group)over

S

. Wedenoteby

G

∨

itsCartier dual(resp. itsSerredual), by

ω

G

thesheafofinvariantdierentialsof

G

over

S

, and by

Lie(

G

)

the sheaf of Lie algebras of

G

. If

S

= Spec(

A

)

is ane

and there is no risk of onfusions, we also use

ω

G

and

Lie(

G

)

to denote the orresponding

A

-modules of global setions. Weput

G

(

p

)

thepull-bak of

G

bythe absoluteFrobeniusof

S

,

F

G

:

G

→

G

(

p

)

the Frobenius homomorphism and

V

G

:

G

(

p

)

→

G

theVershiebunghomomorphism. If

G

isaBT-groupand

n

aninteger

≥

1

,wedenoteby

G

(

n

)

thekernelof themultipliationby

p

n

on

G

; wehave

G

∨

(

n

) = (

G

∨

)(

n

)

bydenition. Foran

O

S

-module

M

, wedenote by

M

(

p

)

=

O

S

⊗

F

S

M

thesalarextension of

M

bytheabsoluteFrobeniusof

O

S

. If

ϕ

:

M

→

N

beasemi-linearhomomorphismof

O

S

-modules,wedenote

by

ϕ

e

:

M

(

p

)

→

N

thelinearizationof

ϕ

,i.e. wehave

ϕ

e

(

λ

⊗

x

) =

λ

·

ϕ

(

x

)

,where

λ

(resp.

x

)isaloalsetionof

O

S

(resp. of

M

).

Starting from Setion 5,

k

will denote an algebraially losedeld of hara-teristi

p >

0

.

2.

Review of ordinary Barsotti-Tate groups

2.1.

Let

G

beaommutativegroupsheme,loally freeofnitetypeover

S

. Wehaveaanonialisomorphismofoherent

O

S

-modules[Ill,2.1℄

(2.1.1)

Lie(

G

∨

)

≃

H

om

S

fppf

(

G,

Ga

)

,

where

H

om

S

fppf

is the sheaf of homomorphisms in the ategory of abelian

fppf

-sheavesover

S

, and

Ga

isthe additivegroupsheme. Sine

G

(

p

)

a

≃

Ga

, theFrobeniushomomorphismof

Ga

induesanendomorphism

(2.1.2)

ϕ

G

: Lie(

G

∨

)

→

Lie(

G

∨

)

,

semi-linearwithrespettotheabsoluteFrobeniusmap

F

S

:

O

S

→

O

S

;weall ittheHasse-Witt map of

G

. BythefuntorialityofFrobenius,

ϕ

G

is alsothe anonialmap induedbythe Frobenius of

G

, ordually bytheVershiebung of

G

∨

.

2.2.

Byaommutative

p

-Lie algebra over

S

, wemean apair

(

L, ϕ

)

, where

L

is an

O

S

-module loally free of nite type, and

ϕ

:

L

→

L

is a semi-linear endomorphismwith respet to theabsoluteFrobenius

F

S

:

O

S

→

O

S

. When there is no risk of onfusions, we omit

ϕ

from the notation. We denote by

p

-

L

ie

S

theategoryofommutative

p

-Liealgebrasover

S

.

Let

(

L, ϕ

)

beanobjetof

p

-

L

ie

S

. Wedenoteby

U

(

L

) = Sym(

L

) =

⊕

n

≥0

Sym

n

(

L

)

,

the symmetri algebraof

L

over

O

S

. Let

I

p

(

L

)

be the ideal sheaf of

U

(

L

)

dened,foranopensubset

V

⊂

S

, by

Γ(

V,

I

p

(

L

)) =

{

x

⊗

p

−

ϕ

(

x

) ;

x

∈

Γ(

V,

U

(

L

))

}

,

where

x

⊗

p

=

x

⊗

x

⊗ · · · ⊗

x

∈

Γ(

V,

Sym

p

(

L

))

. Weput

U

p

(

L

) =

U

(

L

)

/

I

p

(

L

)

,

andallitthe

p

-envelopingalgebraof

(

L, ϕ

)

. Weendow

U

p

(

L

)

withthe stru-tureofaHopf-algebrawiththeomultipliationgivenby

∆(

x

) = 1

⊗

x

+

x

⊗

1

andtheoinversegivenby

i

(

x

) =

−

x

.

Let

G

be aommutativegroupsheme,loally freeof nite typeover

S

. We say that

G

is of oheight one if theVershiebung

V

G

:

G

(

p

)

→

G

is the zero homomorphism. We denote by

G

V

S

the ategory of suh objets. For an objet

G

of

G

V

S

,theFrobenius

F

G

∨

of

G

∨

iszero,sotheLiealgebra

Lie(

G

∨

)

isloallyfreeofnitetypeover

O

S

([DG ℄

VII

A

Théo. 7.4(iii)). TheHasse-Witt map of

G

(2.1.2)endows

Lie(

G

∨

)

withaommutative

p

-Liealgebrastruture over

S

.

Proposition

2.3 ([DG ℄

VII

A

,Théo. 7.2et 7.4)

.

The funtor

G

V

S

→

p

-

L

ie

S

denedby

G

7→

Lie(

G

∨

)

isananti-equivaleneofategories;aquasi-inverse is given by

(

L, ϕ

)

7→

Spec(

U

p

(

L

))

.

2.4.

Assume

S

= Spec(

A

)

ane. Let

(

L, ϕ

)

be anobjetof

p

-

L

ie

S

suh that

L

isfreeofrank

n

over

O

S

,

(

e

1

,

· · ·

, e

n

)

beabasisof

L

over

O

S

,

(

h

ij

)

1≤

i,j

≤

n

be the matrixof

ϕ

under the basis

(

e

1

,

· · ·

, e

n

)

, i.e.

ϕ

(

e

j

) =

P

n

1

≤

j

≤

n

. Thenthegroupshemeattahed to

(

L, ϕ

)

isexpliitlygivenby

Spec(

U

p

(

L

)) = Spec

A

[

X

1

,

· · ·

, X

n

]

/

(

X

j

p

−

n

X

i

=1

h

ij

X

i

)

1≤

j

≤

n

,

withtheomultipliation

∆(

X

j

) = 1

⊗

X

j

+

X

j

⊗

1

. BytheJaobianriterion of étaleness [EGA ,

IV

0

22.6.7℄, the nite groupsheme

Spec(

U

p

(

L

))

is étale over

S

if and only if the matrix

(

h

ij

)

1≤

i,j

≤

n

is invertible. This ondition is equivalenttothat thelinearizationof

ϕ

isanisomorphism.

Corollary

2.5

.

An objet

G

of

G

V

S

is étale over

S

,if andonly if the lin-earizationof itsHasse-Wittmap (2.1.2)is anisomorphism.

Proof. The problem being loal over

S

, we may assume

S

ane and

L

=

Lie(

G

∨

)

freeover

O

S

. ByTheorem2.3,

G

is isomorphito

Spec(

U

p

(

L

))

,and

weonludebythelastremark of2.4.

2.6.

Let

G

beaBT-groupover

S

ofheight

c

+

d

anddimension

d

. TheLie al-gebra

Lie(

G

∨

)

isan

O

S

-moduleloallyfreeofrank

c

,andanoniallyidentied with

Lie(

G

∨

(1))

([BBM℄3.3.2). WedenetheHasse-Wittmap of

G

(2.6.1)

ϕ

G

: Lie(

G

∨

)

→

Lie(

G

∨

)

to bethatof

G

(1)

(2.1.2).

2.7.

Let

k

be aeld of harateristi

p >

0

,

G

beaBT-groupover

k

. Reall that wehaveaanonialexatsequeneofBT-groupsover

k

(2.7.1)

0

→

G

◦

→

G

→

G

´

et

→

₀

with

G

◦

onnetedand

G

´

et

étale([Dem ℄ Chap.II, Ÿ7). This indues anexat sequeneofLiealgebras

(2.7.2)

0

→

Lie(

G

´

et∨

)

→

Lie(

G

∨

)

→

Lie(

G

◦∨

)

→

₀

,

ompatible withHasse-Wittmaps.

Proposition

2.8

.

Let

k

be a eld of harateristi

p >

0

,

G

be a BT-group over

k

. Then

Lie(

G

´

et∨

)

is the unique maximal

k

-subspae

V

of

Lie(

G

∨

)

with the followingproperties:

(a)

V

isstable under

ϕ

G

;

(b) therestritionof

ϕ

G

to

V

isinjetive.

Proof. It islearthat

Lie(

G

´

et∨

)

satisesproperty(a). Wenote that the Ver-shiebung of

G

´

et

(1)

vanishes; so

G

´

et

(1)

isin the ategory

G

V

Spec(

k

)

. Sine

k

is aeld,2.5 implies that therestritionof

ϕ

G

to

Lie(

G

´

et∨

)

, whih oinides with

ϕ

G

´

et

,isinjetive. Thisprovesthat

Lie(

G

´

et∨

)

veries(b). Conversely,let

V

beanarbitrary

k

-subspaeof

Lie(

G

∨

)

withproperties(a)and(b). Wehave toshowthat

V

⊂

Lie(

G

´

et∨

)

. Let

σ

betheFrobeniusendomorphismof

k

. If

M

is a

k

-vetorspae,for eah integer

n

≥

1

, weput

M

(

p

n

)

=

k

⊗

σ

n

M

, i.e. we

have

1

⊗

ax

=

σ

n

(

a

)

⊗

x

in

k

⊗

σ

n

M

for

a

∈

k, x

∈

M

. Sine

ϕ

G

|

V

:

V

→

V

is injetive by assumption, the linearization

ϕ

f

n

G

|

V

(pn)

:

V

(

p

n

)

→

V

of

ϕ

is injetive (hene bijetive) for any

n

≥

1

. We have

V

=

ϕ

f

n

G

(

V

(

p

n

)

)

. Sine

G

◦

is onneted, there isan integer

n

≥

1

suh that the

n

-th iterated Frobe-nius

F

n

G

◦

(1)

:

G

◦

(1)

→

G

◦

(1)

(

p

n

)

vanishes. Hene bydenition, thelinearized

n

-iterated Hasse-Witt map

ϕ

g

n

G

◦

: Lie(

G

◦∨

)

(

p

n

)

→

Lie(

G

◦∨

)

is zero. By the ompatibility of Hasse-Witt maps, we have

ϕ

f

n

G

(Lie(

G

∨

)

(

p

n

)

)

⊂

Lie(

G

´

et∨

)

; in partiular,wehave

V

=

ϕ

f

n

G

(

V

(

p

n

)

)

⊂

Lie(

G

´

et∨

)

. Thisompletestheproof.

Corollary

2.9

.

Let

k

beaeldofharateristi

p >

0

,

G

beaBT-groupover

k

. Then

G

isonnetedifandonly if

ϕ

G

isnilpotent.

Proof. Intheproofoftheproposition,wehaveseenthattheHasse-Wittmap oftheonnetedpartof

G

isnilpotent. Sotheonlyif partisveried. Con-versely,if

ϕ

G

isnilpotent,

Lie(

G

´

et∨

)

iszerobytheproposition. Therefore

G

is

onneted.

Definition

2.10

.

Let

S

be a sheme of harateristi

p >

0

,

G

be a BT-groupover

S

. Wesaythat

G

isordinary if thereexists an exatsequeneof BT-groupsover

S

(2.10.1)

0

→

G

mult

→

G

→

G

´

et

→

₀

,

suhthat

G

mult

ismultipliativeand

G

´

et

isétale.

We note that when it exists, the exat sequene (2.10.1) is unique up to a uniqueisomorphism,beausethereisnonon-trivialhomomorphismsbetweena multipliativeBT-groupandanétaleoneinharateristi

p >

0

. Theproperty ofbeingordinaryislearlystableunderarbitrarybasehangeandSerreduality. If

S

isthespetrumofaeldofharateristi

p >

0

,

G

isordinaryifandonly ifitsonnetedpart

G

◦

isofmultipliativetype.

Proposition

2.11

.

Let

G

beaBT-groupover

S

. Thefollowing onditionsare equivalent:

(a)

G

isordinaryover

S

.

(b) Forevery

x

∈

S

,the ber

G

x

=

G

⊗

S

κ

(

x

)

isordinaryover

κ

(

x

)

. () Thenite group sheme

Ker

V

G

isétaleover

S

.

(') The nitegroup sheme

Ker

F

G

isofmultipliativetype over

S

. (d) Thelinearizationof theHasse-Wittmap

ϕ

G

isan isomorphism.

First,weprovethefollowinglemmas.

Lemma

2.12

.

Let

T

beasheme,

H

beaommutativegroupshemeloallyfree of nite type over

T

. Then

H

isétale (resp. of multipliative type)over

T

if andonlyif,forevery

x

∈

T

,theber

H

⊗

T

κ

(

x

)

isétale(resp. ofmultipliative type) over

κ

(

x

)

.

Lemma

2.13

.

Let

G

be aBT-group over

S

. Then

Ker

V

G

isan objet of the ategory

G

V

S

,i.e. itisloallyfreeofnitetypeover

S

,anditsVershiebungis zero. Moreover,wehaveaanonialisomorphism

(Ker

V

G

)

∨

≃

Ker

F

G

∨

,whih indues an isomorphism of Lie algebras

Lie (Ker

V

G

)

∨

≃

Lie(Ker

F

G

∨

) =

Lie(

G

∨

)

, and the Hasse-Witt map (2.1.2) of

Ker

V

G

is identied with

ϕ

G

(2.6.1).

Proof. Thegroupsheme

Ker

V

G

isloallyfreeofnitetypeover

S

([Ill℄1.3(b)), andwehaveaommutativediagram

(Ker

V

G

)

(

p

)

V

Ker

VG

/

/

 _

Ker

V

 _

G

(

G

(

p

)

)

(

p

)

V

G

(p)

/

/

G

(

p

)

BythefuntorialityofVershiebung,wehave

V

G

(p)

= (

V

G

)

(

p

)

and

Ker

V

G

(p)

=

(Ker

V

G

)

(

p

)

. Hene the omposition of theleft vertialarrowwith

V

G

(p)

van-ishes,andtheVershiebungof

Ker

V

G

iszero.

ByCartierduality,wehave

(Ker

V

G

)

∨

= Coker(

F

G

∨

(1)

)

. Moreover,theexat sequene

· · · →

G

∨

(1)

F

G

∨

(1)

−−−−→

G

∨

(1)

(

p

)

V

G

∨

(1)

−−−−→

G

∨

(1)

→ · · ·

,

induesaanonialisomorphism

(2.13.1)

Coker(

F

G

∨

(1)

)

∼

−

→

Im(

V

G

∨

(1)

) = Ker

F

G

∨

(1)

= Ker

F

G

∨

.

Hene,wededuethat

(2.13.2)

(Ker

V

G

)

∨

≃

Coker(

F

G

∨

(1)

)

−

∼

→

Ker

F

G

∨

֒

→

G

∨

(1)

.

Sine the naturalinjetion

Ker

F

G

∨

→

G

∨

(1)

indues an isomorphismof Lie algebras,weget

(2.13.3)

Lie (Ker

V

G

)

∨

≃

Lie(Ker

F

G

∨

) = Lie(

G

∨

(1)) = Lie(

G

∨

)

.

ItremainstoprovetheompatibilityoftheHasse-Wittmapswith(2.13.3). We note thatthedual ofthemorphism(2.13.2)is theanonialmap

F

:

G

(1)

→

Ker

V

G

= Im(

F

G

(1)

)

indued by

F

G

(1)

. Hene by (2.1.1), the isomorphism

(2.13.3)isidentiedwiththefuntorialmap

H

om

S

fppf

(Ker

V

G

,

Ga

)

→

H

om

S

fppf

(

G

(1)

,

Ga

)

indued by

F

, and its ompatibility with theHasse-Witt maps followseasily

fromthedenition (2.1.2).

Proofof 2.11. (a)

⇒

(b). Indeed,theordinarityof

G

isstablebybasehange. (b)

⇒

(). ByLemma 2.12,it suestoverifythat for everypoint

x

∈

S

, the ber

(Ker

V

G

)

⊗

S

κ

(

x

)

≃

Ker

V

G

x

isétaleover

κ

(

x

)

. Sine

G

x

isassumedtobe ordinary,itsonnetedpart

(

G

x

)

◦

(

G

x

)

◦

isanisomorphism,and

Ker

V

G

x

isanoniallyisomorphito

Ker

V

G

´

et

x

⊂

(

G

´

et

x

)

(

p

)

≃

(

G

(

p

)

x

)

´

et

,soourassertionfollows.

(

c

)

⇔

(

d

)

. ItfollowsimmediatelyfromLemma 2.13andCorollary2.5.

()

⇔

('). By2.12, wemayassume that

S

is thespetrumof a eld. So the ategoryof ommutativenite groupshemes over

S

isabelian. Wewill just prove()

⇒

(');theonverseanbeprovedbyduality. Wehaveafundamental short exatsequeneofnitegroupshemes

(2.13.4)

0

→

Ker

F

G

→

G

(1)

F

−

→

Ker

V

G

→

0

,

where

F

isinduedby

F

G

(1)

,That induesaommutativediagram

0

/

/

Ker

F

G

(

p

)

V

′

/

/

G

(1)

(

p

)

F

(p)

/

/

V

G(1)

Ker

V

G

(

p

)

/

/

V

′′

0

0

/

/

Ker

F

G

/

/

G

(1)

F

/

/

Ker

V

G

/

/

0

where vertial arrows are the Vershiebung homomorphisms. We have seen that

V

′′

= 0

(2.13). Therefore, by the snake lemma, we have a long exat sequene

(2.13.5)

0

→

Ker

V

′

→

Ker

V

G

(1)

α

−

→

Ker

V

G

(

p

)

→

→

Coker

V

′

→

Coker

V

G

(1)

β

−

→

Ker

V

G

→

0

,

wherethemap

α

istheFrobeniusof

Ker

V

G

and

β

istheomposedisomorphism

Coker(

V

G

(1)

)

≃

G

(1)

/

Ker

F

G

(1)

∼

−

→

Im(

F

G

(1)

)

≃

Ker

V

G

.

Then ondition() isequivalentto that

α

is anisomorphism; itimplies that

Ker

V

′

= Coker

V

′

= 0

, i.e. the Vershiebung of

Ker

F

G

is an isomorphism, andhene(').

()

⇒

(a). Foreveryinteger

n >

0

, wedenote by

F

n

G

the omposed homomor-phism

G

F

G

−−→

G

(

p

)

−−−−→ · · ·

F

G

(p)

−−−−−−→

F

G

(pn

−

1 )

G

(

p

n

)

,

andby

V

n

G

theomposedhomomorphism

G

(

p

n

)

V

G

(pn

−

1)

−−−−−−→

G

(

p

n

−

1

)

V

G

(pn

−

2 )

−−−−−−→ · · ·

V

G

−−→

G

;

F

n

G

and

V

n

G

are isogeniesof BT-groups. Fromtherelation

V

n

G

◦

F

G

n

=

p

n

, we dedueanexatsequene

(2.13.6)

0

→

Ker

F

n

G

→

G

(

n

)

F

n

where

F

n

isinduedby

F

n

G

. For

1

≤

j < n

,wehaveaommutativediagram

(2.13.7)

G

(

p

n

)

V

n

−

j

G

(pj)

/

/

V

n

G

E

E

E

"

"

E

E

E

E

E

G

(

p

j

)

V

G

j

|

|

yy

yy

yy

yy

G.

One noties that

Ker

V

n

−

j

G

(pj)

= (Ker

V

n

−

j

G

)

(

p

j

)

by the funtoriality of Ver-shiebung. Sineallmapsin(2.13.7)areisogenies,wehaveanexatsequene

(2.13.8)

0

→

(Ker

V

n

−

j

G

)

(

p

j

)

i

′

n

−

j,n

−−−−→

Ker

V

G

n

p

n,j

−−→

Ker

V

G

j

→

0

.

Therefore, ondition () implies by indution that

Ker

V

n

G

is an étale group sheme over

S

. Hene the

j

-th iteration of the Frobenius

Ker

V

n

−

j

G

→

(Ker

V

G

n

−

j

)

(

p

j

)

is an isomorphism, and

Ker

V

n

−

j

G

is identied with a losed subgroupshemeof

Ker

V

n

G

bytheomposedmap

i

n

−

j,n

: Ker

V

G

n

−

j

∼

−

→

(Ker

V

G

n

−

j

)

(

p

j

)

i

′

n

−

j,n

−−−−→

Ker

V

G

n

.

Welaimthatthekernelofthemultipliationby

p

n

−

j

on

Ker

V

n

G

is

Ker

V

n

−

j

G

.

Indeed,fromtherelation

p

n

−

j

·

Id

G

(pn)

=

F

n

−

j

G

(pj)

◦

V

n

−

j

G

(pj)

,wededuea ommu-tativediagram (withoutdottedarrows)

(2.13.9)

Ker

V

n

G

/

/

p

n

−

j

p

n,j

$

$

I

I

I

I

I

G

(

p

n

)

p

n

−

j

V

n

−

j

G

(pj

)

#

#

G

G

G

G

G

G

G

G

G

Ker

V

G

j

_

_

_

_

_

_

_

_

_

/

/

i

j,n

z

z

u

u

u

u

u

G

(

p

j

)

F

n

−

j

G

(pj)

{

{

ww

ww

ww

ww

w

Ker

V

n

G

/

/

G

(

p

n

)

.

Itfollowsfrom(2.13.8)thatthesubgroup

Ker

V

n

G

of

G

(

p

n

)

issentby

V

n

−

j

G

(pj)

onto

Ker

V

G

j

. Thereforediagram(2.13.9)remainsommutativewhenompleted by

thedottedarrows,heneourlaim. Itfollowsfromthelaimthat

(Ker

V

n

G

)

n

≥1

onstitutesanétaleBT-groupover

S

,denoted by

G

´

et

. Byduality,wehavean exatsequene

(2.13.10)

0

→

Ker

F

j

G

→

Ker

F

G

n

→

(Ker

F

n

−

j

G

)

(

p

j

)

→

0

.

Condition(')impliesbyindutionthat

Ker

F

n

G

isofmultipliativetype. Hene the

j

-th iteration of Vershiebung

(Ker

F

n

−

j

G

)

(

p

j

)

→

Ker

F

G

n

−

j

is an

isomor-phism. Wededuefrom(2.13.10)that

(Ker

F

n

G

)

n

≥1

formamultipliative BT-groupover

S

thatwedenoteby

G

mult

. Thentheexatsequenes(2.13.6)give

Corollary

2.14

.

Let

G

be aBT-groupover

S

,and

S

ord

bethe lousin

S

of the points

x

∈

S

suhthat

G

x

=

G

⊗

S

κ

(

x

)

isordinary over

κ

(

x

)

. Then

S

ord

isopenin

S

,andtheanonial inlusion

S

ord

→

S

is ane.

Theopensubsheme

S

ord

of

S

isalledtheordinarylous of

G

.

3.

Preliminaries onDieudonné Theory andDeformation Theory

3.1.

Wewill use freely theonventions of 1.8. Let

S

bea sheme of hara-teristi

p >

0

,

G

beaBarsotti-Tategroupover

S

, and

M(

G

) =

D

(

G

)

(

S,S

)

be theoherent

O

S

-moduleobtainedbyevaluatingthe(ontravariant)Dieudonné rystalof

G

atthetrivialdividedpowerimmersion

S ֒

→

S

[BBM,3.3.6℄. Reall that

M(

G

)

isan

O

S

-moduleloally freeof nitetypesatisfyingthefollowing properties:

(i)Let

F

M

:

M(

G

)

(

p

)

→

M(

G

)

and

V

M

:

M(

G

)

→

M(

G

)

(

p

)

bethe

O

S

-linear maps indued respetivelyby the Frobenius and theVershiebung of

G

. We havethefollowingexatsequene:

· · · →

M(

G

)

(

p

)

F

M

−−→

M(

G

)

V

M

−−→

M(

G

)

(

p

)

→ · · ·

.

(ii) There is a onnetion

∇

:

M(

G

)

→

M(

G

)

⊗

O

S

Ω

1

S/

Fp

for whih

F

M

and

V

M

arehorizontalmorphisms.

(iii)Wehavetwoanonialltrationson

M(

G

)

by

O

S

-modulesloally freeof nitetype:

(3.1.1)

0

→

ω

G

→

M(

G

)

→

Lie(

G

∨

)

→

₀

,

alled theHodge ltration on

M

(

G

)

[BBM,3.3.5℄,andtheonjugate ltration on

M

(

G

)

(3.1.2)

0

→

Lie(

G

∨

)

(

p

)

φ

G

−−→

M(

G

)

→

ω

G

(

p

)

→

0

,

and2.3.4℄) (3.1.3)

0

0

0

ω

(

G

p

)

ω

G

ψ

G

/

/

ω

(

p

)

G

/

/

M

(

G

)

(

p

)

F

M

/

/

M

(

G

)

V

M

/

/

6 6

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

M

(

G

)

(

p

)

/

/

,

Lie(

G

∨

)

(

p

)

(

φ

G

6

l

l

l

l

l

l

l

l

l

l

l

l

l

l

ϕ

f

G

/

/

Lie(

G

∨

)

Lie(

G

∨

)

(

p

)

0

0

0

where the olumns are the Hodge ltrations and the anti-diagonal is the onjugate ltration. By funtoriality, we see easily that

ϕ

f

G

above is noth-ing but the linearization of the Hasse-Witt map

ϕ

G

(2.6.1), and the mor-phism

ψ

∗

G

: Lie(

G

)

(

p

)

→

Lie(

G

)

, whih is obtained by applying the funtor

H

om

O

S

(

_

,

O

S

)

to

ψ

G

,isidentiedwiththelinearization

ϕ

g

G

∨

of

ϕ

G

∨

.

The formation of these strutures on

M(

G

)

ommutes with arbitrary base hanges of

S

. In the sequel, we will use

(M(

G

)

, F

M

,

∇

)

to emphasize these strutureson

M(

G

)

.

3.2.

Inthereminderofthis setion,

k

willdenoteanalgebraiallylosedeld of harateristi

p >

0

. Let

S

beashemeformally smoothover

k

suh that

Ω

1

S/

Fp

= Ω

1

S/k

is an

O

S

-module loally free of nite type, e.g.

S

= Spec(

A

)

with

A

aformally smooth

k

-algebrawith anite

p

-basisover

k

. Let

G

bea BT-groupover

S

. Weput

KS

tobetheomposed morphism

(3.2.1)

KS :

ω

G

→

M

(

G

)

∇

−→

M

(

G

)

⊗

O

S

Ω

1

S/k

pr

−→

Lie(

G

∨

)

⊗

O

S

Ω

1

S/k

whih is

O

S

-linear. We put

T

S/k

=

H

om

O

S

(Ω

1

S/k

,

O

S

)

, and dene the Kodaira-Spener map of

G

(3.2.2)

Kod :

T

S/k

→

H

om

O

S

(

ω

G

,

Lie(

G

∨

))

tobethemorphisminduedby

KS

. Wesaythat

G

isversal if

Kod

issurjetive.

3.3.

Let

r

be an integer

≥

1

,

R

=

k

[[

t

1

,

· · ·

, t

r

]]

,

m

be the maximal ideal of

R

. We put

S

= Spf(

R

)

,

S

= Spec(

R

)

, and for eah integer

n

≥

0

,

S

n

= Spec(

R/

m

n

+1

)

. Bya BT-group

G

overtheformalsheme

S

,wemean

a sequene of BT-groups

(

G

n

)

n

≥0

over

(

S

n

)

n

≥0

equipped with isomorphisms

Aordingto[deJ,2.4.4℄,thefuntor

G

7→

(

G

×

S

S

n

)

n

≥0

induesanequivalene ofategoriesbetweentheategoryofBT-groupsover

S

andtheategoryof BT-groups over

S

. For a BT-group

G

over

S

, the orresponding BT-group

G

over

S

isalledthealgebraization of

G

. Wesaythat

G

isversal over

S

,ifits algebraization

G

isversalover

S

. Sine

S

is loal,byNakayama'sLemma,

G

or

G

isversalifand onlyiftheredutionof

Kod

modulothemaximalideal (3.3.1)

Kod

0

:

T

S/k

⊗

O

S

k

−→

Hom

k

(

ω

G

0

,

Lie(

G

∨

0

))

issurjetive.

3.4.

Wereallbriey thedeformationtheoryof aBT-group. Let

A

L

k

bethe ategory of loal artinian

k

-algebraswith residueeld

k

. We notie that all morphisms of

A

L

k

are loal. A morphism

A

′

→

A

in

A

L

k

is alled a small extension,ifit issurjetiveanditskernel

I

satises

I

·

m

A

′

= 0

, where

m

A

′

is themaximalidealof

A

′

.

Let

G

0

be a BT-group over

k

, and

A

an objet of

A

L

k

. A deformation of

G

0

over

A

is a pair

(

G, φ

)

, where

G

is a BT-group over

Spec(

A

)

and

φ

is

an isomorphism

φ

:

G

⊗

A

k

∼

−

→

G

0

. Whenthere is norisk of onfusions, we

will denote adeformation

(

G, φ

)

simply by

G

. Twodeformations

(

G, φ

)

and

(

G

′

, φ

′

)

over

A

are isomorphi if there exists an isomorphism of BT-groups

ψ

:

G

−

∼

→

G

′

over

A

suhthat

φ

=

φ

′

◦

(

ψ

⊗

A

k

)

. Let'sdenoteby

D

thefuntor whih assoiateswith eah objet

A

of

A

L

k

the set of isomorphsm lassesof deformations of

G

0

over

A

. If

f

:

A

→

B

is a morphism of

A

L

k

, then the map

D

(

f

) :

D

(

A

)

→ D

(

B

)

is given by extension of salars. We all

D

the deformation funtor of

G

0

over

A

L

k

.

Proposition

3.5([Ill ℄,4.8)

.

Let

G

0

beaBT-groupover

k

ofdimension

d

and height

c

+

d

,

D

bethe deformation funtorof

G

0

over

A

L

k

.

(i) Let

A

′

→

A

be a small extension in

A

L

k

with ideal

I

,

x

= (

G, φ

)

be an element in

D

(

A

)

,

D

x

(

A

′

)

be the subset of

D

(

A

′

)

with image

x

in

D

(

A

)

. Then the set

D

x

(

A

′

)

isanonempty homogenous spaeunder the group

Hom

k

(

ω

G

0

,

Lie(

G

∨

0

))

⊗

k

I

.

(ii)Thefuntor

D

ispro-representable byaformallysmoothformalsheme

S

over

k

ofrelativedimension

cd

,i.e.

S

= Spf(

R

)

with

R

≃

k

[[(

t

ij

)

1≤

i

≤

c,

1≤

j

≤

d

]]

, andthere existsa uniquedeformation

(

G

, ψ

)

of

G

0

over

S

suhthat,for any objet

A

of

A

L

k

and any deformation

(

G, φ

)

of

G

0

over

A

, there is a unique homomorphism of loal

k

-algebras

ϕ

:

R

→

A

with

(

G, φ

) =

D

(

ϕ

)(

G

, ψ

)

. (iii)Let

T

S

/k

(0) =

T

S

/k

⊗

O

S

k

bethetangentspaeof

S

atitsuniquelosed point,

Kod

0

:

T

S

/k

(0)

−→

Hom

k

(

ω

G

0

,

Lie(

G

∨

0

))

betheKodaira-Spenermapof

G

evaluatedatthelosedpointof

S

.Then

Kod

0

isbijetive,anditanbedesribedasfollows. Foranelement

f

∈

T

S

/k

(0)

,i.e. ahomomorphism ofloal

k

-algebras

f

:

R

→

k

[

ǫ

]

/ǫ

2

,

Kod

0

(

f

)

isthedierene of deformations

[

G

⊗

R

(

k

[

ǫ

]

/ǫ

2

)]

−

[

G

₀

⊗

k

(

k

[

ǫ

]

/ǫ

2

)]

,

whih isawell-denedelement in

Hom

k

(

ω

G

0

,

Lie(

G

∨

Remark

3.6

.

Let

(

e

j

)

1≤

j

≤

d

beabasisof

ω

G

0

,

(

f

i

)

1≤

i

≤

c

beabasisof

Lie(

G

∨

0

)

. Inviewof3.5(iii),weanhooseasystemofparameters

(

t

ij

)

1≤

i

≤

c,

1≤

j

≤

d

of

S

suhthat

Kod

0

(

∂

∂t

ij

) =

e

∗

j

⊗

f

i

,

where

(

e

∗

j

)

1≤

j

≤

d

is thedual basisof

(

e

j

)

1≤

j

≤

d

. Moreover,if

m

is themaximal idealof

R

,theparameters

t

ij

aredetermineduniquelymodulo

m

2

.

Corollary

3.7(

Algebraization of the universal deformation

)

.

The assumptions being those of (3.5), we put moreover

S

= Spec(

R

)

and

G

the algebraization of the universal formal deformation

G

. Then the BT-group

G

is versal over

S

, and satises the following universal property: Let

A

be a noetherian omplete loal

k

-algebrawith residue eld

k

,

G

beaBT-group over

A

endowed with an isomorphism

G

⊗

A

k

≃

G

0

. Then there exists a unique

ontinuoushomomorphismofloal

k

-algebras

ϕ

:

R

→

A

suhthat

G

≃

G

⊗

R

A

. Proof. Bythelastremarkof3.3,

G

islearlyversal. Itremainstoprovethatit satisestheuniversalpropertyin theorollary. Let

G

beadeformationof

G

0

overanoetherianomplete loal

k

-algebra

A

with residueeld

k

. Wedenote by

m

A

themaximalidealof

A

,andput

A

n

=

A/

m

n

+1

A

foreahinteger

n

≥

0

. Thenby3.5(b), thereexistsauniqueloalhomomorphism

ϕ

n

:

R

→

A

n

suh that

G

⊗

A

n

≃

G

⊗

R

A

n

. The

ϕ

n

'sform aprojetivesystem

(

ϕ

n

)

n

≥0

,whose projetivelimit

ϕ

:

R

→

A

answersthequestion.

Definition

3.8

.

Thenotationsarethoseof(3.7). Weall

S

theloalmoduliin harateristi

p

of

G

0

,and

G

theuniversaldeformationof

G

0

inharateristi

p

.

Ifthereisnoonfusions,wewillomitin harateristi

p

forshort.

3.9.

Let

G

be a BT-group over

k

,

G

◦

be its onneted part, and

G

´

et

be its étale part. Let

r

be theheightof

G

´

et

. Then wehave

G

´

et

≃

(

Qp

/

Zp

)

r

, sine

k

isalgebraiallylosed. Let

D

G

(resp.

D

G

◦

)bethedeformationfuntorof

G

(resp.

G

◦

) over

A

L

k

. If

A

is anobjetin

A

L

k

and

G

is adeformation of

G

(resp.

G

◦

)over

A

, wedenoteby

[

G

]

itsisomorphismlassin

D

G

(

A

)

(resp. in

D

G

◦

(

A

)

).

Proposition

3.10

.

The assumptions areas above, let

Θ :

D

G

→ D

G

◦

be the

morphismoffuntorsthatmapsadeformationof

G

toitsonnetedomponent. (i)The morphism

Θ

isformally smooth ofrelative dimension

r

.

(ii)Let

A

beanobjetof

A

L

k

,and

G

◦

beadeformationof

G

◦

over

A

. Thenthe subset

Θ

−1

A

([

G

◦

])

of

D

G

(

A

)

is anonially identied with

Ext

1

A

(

Qp

/

Zp

,

G

◦

)

r

, where

Ext

1

A

means the group of extensions in the ategory of abelian

fppf

-sheaves on

Spec(

A

)

.

Proof. (i)Sine

D

G

and

D

G

◦

arebothpro-representablebyanoetherianloal omplete

k

-algebraandformallysmoothover

k

(3.5),byaformalompletion versionof[EGA ,

IV 17

.

11

.

1(

d

)

℄, weonlyneedtohekthatthetangentmap

is surjetive with kernel of dimension

r

over

k

. By 3.5(iii),

D

G

(

k

[

ǫ

]

/ǫ

2

)

(resp.

D

G

◦

(

k

[

ǫ

]

/ǫ

2

)

) is isomorphi to

Hom

k

(

ω

G

,

Lie(

G

∨

))

(resp.

Hom

k

(

ω

G

◦

,

Lie(

G

◦∨

))

) by the Kodaira-Spener morphism. In view of the

anonialisomorphism

ω

G

≃

ω

G

◦

,

Θ

k

[

ǫ

]

/ǫ

2

orrespondstothemap

Θ

′

k

[

ǫ

]

/ǫ

2

: Hom

k

(

ω

G

,

Lie(

G

∨

))

→

Hom

k

(

ω

G

,

Lie(

G

◦∨

))

indued by the anonial surjetion

Lie(

G

∨

)

→

Lie(

G

◦∨

)

. It is lear that

Θ

′

k

[

ǫ

]

/ǫ

2

is surjetive of kernel

Hom

k

(

ω

G

,

Lie(

G

´

et∨

))

, whih has dimension

r

over

k

.

(ii) Sine

G

´

et

is isomorphito

(

Qp

/

Zp

)

r

, everyelementin

Ext

1

A

(

Qp

/

Zp

,

G

◦

)

r

deneslearlyanelementof

D

G

(

A

)

withimage

[

G

◦

]

in

D

G

◦

(

A

)

. Conversely,for any

G

∈ D

G

(

A

)

withonnetedomponentisomorphito

G

◦

,theisomorphism

G

´

et

≃

(

Qp

/

Zp

)

r

liftsuniquelytoanisomorphism

G

´

et

≃

(

Qp

/

Zp

)

r

beause

A

is henselian. Theanonialexatsequene

0

→

G

◦

→

G

→

G

´

et

→

₀

showsthat

G

omesfromanelementof

Ext

1

A

(

Qp

/

Zp

,

G

◦

)

r

.

4.

HW-yli Barsotti-Tate Groups

Definition

4.1

.

Let

S

beashemeofharateristi

p >

0

,

G

beaBT-group over

S

suhthat

c

= dim(

G

∨

)

isonstant. Wesaythat

G

isHW-yli,if

c

≥

1

andthereexists anelement

v

∈

Γ(

S,

Lie(

G

∨

))

suh that

v, ϕ

G

(

v

)

,

· · ·

, ϕ

c

G

−1

(

v

)

generate

Lie(

G

∨

)

asan

O

S

-module,where

ϕ

G

istheHasse-Wittmap(2.6.1)of

G

.

Remark

4.2

.

It islear that aBT-group

G

over

S

isHW-yli, ifand only if

Lie(

G

∨

)

is freeover

O

S

and there exists abasis of

Lie(

G

∨

)

over

O

S

under whih

ϕ

G

isexpressedbyamatrixoftheform

(4.2.1)















0 0

· · ·

0

−

a

1

1 0

· · ·

0

−

a

2

0 1

· · ·

0

−

a

3

.

. .

. .

.

. . .

0 0

· · ·

1

−

a

c















,

where

a

i

∈

Γ(

S,

O

S

)

for

1

≤

i

≤

c

.

Lemma

4.3

.

Let

R

bealoal ringofharateristi

p >

0

,

k

beitsresidueeld. (i)A BT-group

G

over

R

isHW-yliif andonlyif sois

G

⊗

k

.

(ii) Let

0

→

G

′

→

G

→

G

′′

→

₀

bean exat sequeneof BT-groups over

R

. If

G

isHW-yli, then sois

G

′

. In partiular, if

R

ishenselian, the onneted partof aHW-yliBT-groupover

R

isHW-yli.

Proof. (i) The property of being HW-yli is learly stable under arbitrary base hanges, so the only if part is lear. Assume that

G

0

=

G

⊗

k

is HW-yli. Let

v

be an element of

Lie(

G

∨

(

v, ϕ

G

0

(

v

)

,

· · ·

, ϕ

c

−1

G

0

(

v

))

isabasisof

Lie(

G

∨

0

)

. Let

v

beanyliftof

v

in

Lie(

G

∨

)

. ThenbyNakayama'slemma,

(

v, ϕ

G

(

v

)

,

· · ·

, ϕ

c

−1

G

(

v

))

isabasisof

Lie(

G

∨

)

. (ii)Bystatement(i),wemayassume

R

=

k

. TheexatsequeneofBT-groups induesanexatsequeneof Liealgebras

(4.3.1)

0

→

Lie(

G

′′∨

)

→

Lie(

G

∨

)

→

Lie(

G

′∨

)

→

₀

,

and theHasse-Wittmap

ϕ

G

′

is induedby

ϕ

G

byfuntoriality. Assume that

G

isHW-yliand

G

∨

hasdimension

c

. Let

u

beanelementof

Lie(

G

∨

)

suh that

u, ϕ

G

(

u

)

,

· · ·

, ϕ

c

G

−1

(

u

)

form abasis of

Lie(

G

∨

)

over

k

. Wedenote by

u

′

the image of

u

in

Lie(

G

′∨

)

. Let

r

≤

c

bethemaximalintegersuh thatthevetors

u

′

, ϕ

G

′

(

u

′

)

,

· · ·

, ϕ

r

G

−1

′

(

u

′

)

arelinearlyindependentover

k

. Itiseasyto seethattheyformabasisofthe

k

-vetorspae

Lie(

G

′∨

)

. Hene

G

′

isHW-yli.

Lemma

4.4

.

Let

S

= Spec(

R

)

be an ane sheme of harateristi

p >

0

,

G

be aHW-yliBT-group over

R

with

c

= dim(

G

∨

)

onstant,and















0 0

· · ·

0

−

a

1

1 0

· · ·

0

−

a

2

0 1

· · ·

0

−

a

3

. . .

. .

.

. . .

0 0

· · ·

1

−

a

c















∈

M

c

×

c

(

R

)

,

be amatrixof

ϕ

G

. Put

a

c

+1

= 1

,and

P

(

X

) =

P

c

i

=0

a

i

+1

X

p

i

∈

R

[

X

]

.

(i)Let

V

G

:

G

(

p

)

→

G

bethe Vershiebung homomorphismof

G

. Then

Ker

V

G

is isomorphi to the group sheme

Spec(

R

[

X

]

/P

(

X

))

with omultipliation given by

X

7→

1

⊗

X

+

X

⊗

1

.

(ii) Let

x

∈

S

,and

G

x

bethe breof

G

at

x

. Put

(4.4.1)

i

0

(

x

) = min

0≤

i

≤

c

{

i

;

a

i

+1

(

x

)

6

= 0

}

,

where

a

i

(

x

)

denotestheimageof

a

i

intheresidueeldof

x

. Thentheétalepart of

G

x

has height

c

−

i

0

(

x

)

, andthe onnetedpartof

G

x

has height

d

+

i

0

(

x

)

. In partiular,

G

x

isonnetedifandonly if

a

i

(

x

) = 0

for

1

≤

i

≤

c

.

Proof. (i)By2.3and2.13,

Ker

V

G

isisomorphitothegroupsheme

Spec

R

[

X

1

, . . . , X

c

]

/

(

X

1

p

−

X

2

,

· · ·

, X

c

p

−1

−

X

c

, X

c

p

+

a

1

X

1

+

· · ·

+

a

c

X

c

)

with omultipliation

∆(

X

i

) = 1

⊗

X

i

+

X

i

⊗

1

for

1

≤

i

≤

c

. By sending

(

X

1

, X

2

,

· · ·

, X

c

)

7→

(

X, X

p

,

· · ·

, X

p

c

−

1

)

, weseethat the abovegroupsheme

(ii)Bybasehange,wemayassumethat

S

=

x

= Spec(

k

)

andhene

G

=

G

x

. Let

G

(1)

bethekernelofthemultipliationby

p

on

G

. Thenwehaveanexat sequene

0

→

Ker

F

G

→

G

(1)

→

Ker

V

G

→

0

.

Sine

Ker

F

G

is an innitesimal group sheme over

k

, we have

G

(1)(

k

) =

(Ker

V

G

)(

k

)

,where

k

isanalgebrailosureof

k

. Bythedenitionof

i

0

(

x

)

,we

have

P

(

X

) =

Q

(

X

p

i

0(

x)

)

,where

Q

(

X

)

isanadditivesepearablepolynomialin

k

[

X

]

with

deg(

Q

) =

p

c

−

i

0(

x

)

. Henetherootsof

P

(

X

)

in

k

form an

Fp

-vetor spae of dimension

c

−

i

0

(

x

)

. By (i),

(Ker

V

G

)(

k

)

an be identied with the additivegrouponsisting oftherootsof

P

(

X

)

in

k

. Therefore,theétalepart of

G

hasheight

c

−

i

0

(

x

)

,andtheonnetedpartof

G

hasheight

d

+

i

0

(

x

)

.

4.5.

Let

k

beaperfeteldofharateristi

p >

0

,and

α

p

= Spec(

k

[

X

]

/X

p

)

be thenitegroupshemeover

k

withomultipliationmap

∆(

X

) = 1

⊗

X

+

X

⊗

1

. Let

G

beaBT-groupover

k

. FollowingOort,weall

a

(

G

) = dim

k

Hom

k

fppf

(

α

p

, G

)

the

a

-number of

G

, where

Hom

k

fppf

means the homomorphisms in the ate-gory of abelian

fppf

-sheavesover

k

. Sine the Frobenius of

α

p

vanishes, any morphismof

α

p

in

G

fatorizethrough

Ker(

F

G

)

. Thereforewehave

Hom

k

fppf

(

α

p

, G

) = Hom

k

−

gr

(

α

p

,

Ker(

F

G

))

= Hom

k

−

gr

(Ker(

F

G

)

∨

, α

p

)

= Hom

p

-

L

ie

k

(Lie(

α

p

)

,

Lie(Ker(

F

G

)))

,

where

Hom

k

−

gr

denotes thehomomorp