On the Modularity of Elliptic Curves Over Q: Wild 3-Adic Exercises

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Citation Breuil, Christophe, Brian Conrad, Fred Diamond, and Richard L. Taylor. 2001. On the modularity of elliptic curves over Q: Wild 3- adic exercises. Journal of the American Mathematical Society 14(4): 843-939.

Published Version doi:10.1090/S0894-0347-01-00370-8

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JOURNAL OF THE

AMERICAN MATHEMATICAL SOCIETY

Volume Numb er Xxxx XXXX Pages

S XX

ON THE MODULARITY OF ELLIPTIC CURVES OVER Q

WILD ADIC EXERCISES

CHRISTOPHE BREUIL BRIAN CONRAD FRED DIAMOND AND RICHARD TAYLOR

Introduction

In this pap er building on work of Wiles Wi and of Wiles and one of us RT TW we will prove the

following two theorems see x

Theorem A If E is an el liptic curve then E is modular

Theorem B If GalQQ GL F is an irreducible continuous representation with cyclotomic

determinant then is modular

We will rst remind the reader of the content of these results and then briey outline the metho d of pro of

Z If N is a p ositive integer then we let N denote the subgroup of SL consisting of matrices that

mo dulo N are of the form

The quotient of the upp er half plane by N acting by fractional linear transformations is the complex

manifold asso ciated to an ane algebraic curve Y This curve has a natural mo del Y N which N

Q C

for N is a ne mo duli scheme for elliptic curves with a p oint of exact order N We will let X N denote

e curve which contains Y N as a dense Zariski op en subset the smo oth pro jectiv

and level N is a holomorphic function f on the upp er half Recall that a cusp form of weight k

complex plane H such that

for all matrices

a b

c d

and all z H we have f az bcz d cz d f z

and jf z j Im z is b ounded on H

N of cusp forms of weight k and level N is a nite dimensional complex vector space If The space S

f S N then it has an expansion

inz

f z c f e

Received by the editors and in revised form January

Mathematics Subject Classication Primary G Secondary F

Key words and phrases Elliptic curve Galois representation mo dularity

The rst author was supp orted by the CNRS The second author was partially supp orted by a grant from the NSF The

at Rutgers third author was partially supp orted by a grant from the NSF and an AMS Centennial Fellowship He was working

University during much of the research The fourth author was partially supp orted by a grant from the NSF and by the Miller

Institute for Basic Science

American Mathematical So ciety

and we dene the Lseries of f to b e

Lf s c f n

or each prime p jN there is a linear op erator T on S N dened by F

k p

k k

f jT z p f z ip p cpz d f apz bcpz d

for any

a b

SL Z

c d

with c mo d N and d p mo d N The op erators T for p jN can b e simultaneously diagonalised on

If f is an eigenform then the the space S N and a simultaneous eigenvector is called an eigenform

corresp onding eigenvalues a f are algebraic integers and we have c f a f c f

p p p

Let b e a place of the algebraic closure of Q in C ab ove a rational prime and let Q denote the

algebraic closure of Q thought of as a Q algebra via If f S N is an eigenform then there is a unique

continuous irreducible representation

GalQQ GL Q

is unramied at p and tr Frob a f The existence of is due such that for any prime p jN l

f f p p f

to Shimura if k Sh to Deligne if k De and to Deligne and Serre if k DS Its irreducibility

is due to Rib et if k Ri and Deligne and Serre if k DS Moreover is o dd in the sense that

det of complex conjugation is Also is p otentially semistable at in the sense of Fontaine We

can cho ose a conjugate of which is valued in GL O and reducing mo dulo the maximal ideal and

semisimplifying yields a continuous representation

GalQQ GL F

which up to isomorphism do es not dep end on the choice of conjugate of

GL No w supp ose that G Q is a continuous representation which is unramied outside nitely

many primes and for which the restriction of to a decomp osition group at is p otentially semistable in

we can asso ciate b oth a pair of Ho dgeTate numb ers and a Weil the sense of Fontaine To j

Q Q Gal

Deligne representation of the Weil group of Q We dene the conductor N of to b e the pro duct over

p of the conductor of j and of the conductor of the WeilDeligne representation asso ciated

Gal Q Q

We dene the weight k of to b e plus the absolute dierence of the two Ho dgeTate to j

Q Q Gal

It is known by work of Cara numb ers of j yol and others that the following two conditions are

Q Q Gal

equivalent

for some eigenform f and some place j

for some eigenform f of level N and weight k and some place j

that When these equivalent conditions are met we call modular It is conjectured by Fontaine and Mazur

if G GL Q is a continuous irreducible representation which satises

is unramied outside nitely many primes

j is p otentially semistable with its smaller Ho dgeTate numb er

Gal Q Q

and in the case where b oth Ho dgeTate numb ers are zero is o dd

then is mo dular FM

GalQQ GL F Serre Se denes the Next consider a continuous irreducible representation

conductor N and weight k of We call modular if for some eigenform f and some place

j We call strongly modular if moreover we may take f to have weight k and level N It is known

olo ch and others that for is strongly from work of Mazur Rib et Carayol Gross Coleman and V

mo dular if and only if it is mo dular see Di Serre has conjectured that all o dd irreducible are strongly

mo dular Se

Now consider an elliptic curve E Let denote the representation of GalQQ on the resp

Q Let N E denote the conductor of E It is known that adic Tate mo dule resp the torsion of E

the following conditions are equivalent

The Lfunction LE s of E equals the Lfunction Lf s for some eigenform f

The Lfunction LE s of E equals the Lfunction Lf s for some eigenform f of weight and level

N E

For some prime the representation is mo dular

For all primes the representation is mo dular

There is a nonconstant holomorphic map X N C E C for some p ositive integer N

There is a nonconstant morphism X N E E which is dened over Q

The implications and are tautological The implication follows

from the characterisation of LE s in terms of The implication follows from a theorem of

Carayol Ca and a theorem of Faltings Fa The implication follows from a construction of

Shimura Sh and a theorem of Faltings Fa The implication seems to have b een rst noticed

by Mazur Maz When these equivalent conditions are satised we call E modular

It has b ecome a standard conjecture that all elliptic curves over Q are mo dular although at the time this

conjecture was rst suggested the equivalence of the conditions ab ove may not have b een clear Taniyama

made a suggestion along the lines as one of a series of problems collected at the TokyoNikko conference

However his formulation did not make clear whether f should b e a mo dular form or some in Septemb er

more general automorphic form He also suggested that constructions as in and might help attack

this problem at least for some elliptic curves In private conversations with a numb er of mathematicians

Shimura suggested that assertions along the lines of and might b e including Weil in the early s

true see Sh and the commentary on a in We The rst time such a suggestion app ears in print

is Weils comment in We that assertions along the lines of and follow from the main result of that

pap er a construction of Shimura and from certain reasonable supp ositions and natural assumptions

That assertion is true for CM elliptic curves follows at once from work of Hecke and Deuring Shimura

Sh went on to check assertion for these curves

approach to Theorem A is an extension of the metho ds of Wiles Wi and of Wiles and one of us Our

RT TW We divide the pro of into three cases

is irreducible we show that is mo dular j If

Gal QQ

is absolutely irreducible we show that is If j is reducible but j

E E

GalQ Q Gal QQ

mo dular

If is reducible and is absolutely reducible then we show that E is j j

E E

Gal Gal QQ QQ

isogenous to an elliptic curve with j or and so from tables of mo dular invariant

elliptic curves of low conductor is mo dular

is mo dular and then that is In each of cases and there are two steps First we prove that

mo dular In case this rst step is our Theorem B and in case it is a celebrated theorem of Langlands

and Tunnell L T In fact in b oth cases E obtains semistable reduction over a tame extension of Q and

the deduction of the mo dularity of from that of was carried out in CDT by an extension of the

metho ds of Wi and TW In the third case we have to analyse the rational p oints on some mo dular curves

help in CDT of small level This we did with Elkies

b e as in that theorem Twisting by a quadratic It thus only remained to prove Theorem B Let

character we may assume that j falls into one of the following cases see x

Gal Q Q

is unramied at

I has order

I has order and j has conductor

GalQ Q

I has order

such that and j GalQ Q F is induced from a character

Gal Q Q

p p p

where we use the Artin map normalised to take uniformisers to arithmetic Frob enius to identify

with a character of Q

We will refer to these as the f and cases resp ectively

the technique of Minkowski and Klein ie the observation that the mo duli space of elliptic curves Using

with full level structure has genus see for example Kl Hilb ert irreducibility and some lo cal computa

tions of Manoharmayum Man we nd an elliptic curve E with the following prop erties see x

is surjective onto GL F

and

j F or in the f case either

E 3

3 E

e and is p eu rami

in the f case

3 E

in the f case F j

3 E

in the f case

3 E

es ramie and is tr

in the f case

3 E

and is tres ramie

in the f case

Gal Q Q

er k

and is tres ramie Q is nonsplit over

We are using the terms tr es ramie and peu ramie in the sense of Serre Se We are also letting denote

the mo d cyclotomic character and the second fundamental character I F ie

p p p

8 8 8

mo d

We will often use the equality without further remark We emphasise that for a general elliptic

the representation do es not have the ab ove form rather we are placing a curve over Q with

E E

signicant restriction on E

In each case our strategy is to prove that is mo dular and so deduce that is mo dular Again

is mo dular and then an analogue of the arguments of we use the LanglandsTunnell theorem to see that

Wi and TW to conclude that is mo dular This was carried out in Di in the cases f and f

is not imp ortant This leaves j and in CDT in the case f In these cases the particular form of

the cases f and which are complicated by the fact that E now only obtains go o d reduction

over a wild extension of Q In these cases our argument relies essentially on the particular form we have

We do not b elieve that our metho ds for deducing the j dep ending on j obtained for

3 E E

Gal Q Q

mo dularity of from that of would work without this key simplication It seems to b e a piece of

undeserved go o d fortune that for each p ossibility for we can nd a choice for j j for which

3 E

GalQ Q

our metho ds work

Following Wiles to deduce the mo dularity of from that of we consider certain universal defor

mations of and identify them with certain mo dular deformations which we realise over certain Hecke

algebras The key problem is to nd the right lo cal condition to imp ose on these deformations at the prime

As in CDT we require that the deformations lie in the closure of the characteristic zero p oints which are

p otentially BarsottiTate ie come from a divisible group over the ring of integers of a nite extension of

Q and for which the asso ciated representation of the Weil group see for example App endix B of CDT

is of some sp ecied form That one can nd suitable conditions on the representation of the Weil group at

for the arguments of TW to work seems to b e a rare phenomenon in the wild case It is here we make

essential use of the fact that we need only treat our sp ecic pairs

E E

Our arguments follow closely the arguments of CDT There are two main new features Firstly in the

f case we are forced to sp ecify the restriction of our representation of the Weil group completely

rather than simply its restriction to the inertia group as we have done in the past Secondly in the key

computation of the lo cal deformation rings we now make use of a new description due to CB of nite at

group schemes over the ring of integers of any padic eld in terms of certain semilinear algebra data see

Br and the summary Br This description enables us to make these computations As the p ersistent

reader will so on discover they are lengthy and delicate particularly in the case f It seems miraculous

to us that these long computations with nite at group schemes in x x and x give answers completely in

accord with predictions made from much shorter computations with the lo cal Langlands corresp ondence and

the mo dular representation theory of GL Q in x We see no direct connection but cant help thinking

that some such connection should exist

Ac knowledgements We would like to thank Bas Edixhoven Rene Scho of Tom Weston and the referee for

corrections and improvements to preliminary versions of this pap er Barry Mazur for helpful conversations

ollack for help with computer calculations which we made in an earlier attempt to compute some and David P

of these lo cal deformation rings We are grateful to the Harvard University Clay fund for supp orting CB

during a key visit he paid to Harvard BC is grateful to the Institute for Advanced Study for its stimulating

unster for its hospitality FD is grateful to Harvard University and environment and the University of M

the Universite de Paris Sud for their hospitality RT is grateful to the University of California at Berkeley

for its hospitality and the Miller Institute for Basic Science for its supp ort

In this pap er denotes a rational prime In x x x and x it is arbitrary In the rest Notation

of x and in x we only assume it is o dd In the rest of the pap er we only consider

If F is a eld we let F denote a separable closure F the maximal sub extension of F which is ab elian

the Galois group GalF F If F is a padic eld ie a nite extension of Q and F F over F and G

p F

denote the inertia subgroup of GalF F We also let a p ossibly innite Galois extension we let I

F F

denote I I the Weil group of denote the arithmetic Frob enius element and W I G Frob

F F F F F

0 0 0 0 0

F F

We will consisting of elements which map to an integer p ower of Frob F ie the dense subgroup of G

F F

0 0

normalise the Artin map of lo cal class eld theory so that uniformisers and arithmetic Frob enius elements

corresp ond We ap ologise for this convention which now seems to us a bad choice However we feel it is

denote the ring of integers of F imp ortant to stay consistent with CDT We let O the maximal

F F

0 0

and Frob for I for I We write simply G for G the residue eld O and k ideal of O

p p Q p Q F F F F

p p 0 0 0 0

denote the unique unramied degree n extension of Q in Q If k is any p erfect We also let Q Frob

p p Q

p p

eld of characteristic p we also use Frob to denote the p p ower automorphism of k and its canonical lift

to the Witt vectors W k

We write for the adic cyclotomic character and sometimes for the reduction of mo dulo We write

for the second fundamental character I F ie

2 2 2

mo d

We also use and to denote the Teichmuller lifts of and

We let denote the trivial character of a group We will denote by V the dual of a vector space V

If g A B is a homomorphism of rings and if X is an Ascheme we sometimes write X for

Sp ec A

g h g h

X Y is a the pullback of X by Sp ec g We adopt this notation so that X X Similarly if

morphism of schemes over A we will sometimes write for the pullback of by Sp ec g

By nite at group scheme we always mean commutative nite at group scheme If F is a eld of

F we use without comment the canonical identication of nite characteristic with xed algebraic closure

F F mo dules and we will say that such ob jects corresp ond at F group schemes with nite discrete Gal

If R is an Dedekind domain with eld of fractions F of characteristic then by a model of a nite at F

group scheme G we mean a nite lo cally free R group scheme G and an isomorphism i G G F As

in Prop osition of Ra the isomorphism classes of mo dels for G form a lattice G i G i if there

exists a map of nite at group schemes G G compatible with i and i and we can talk ab out the inf and

sup of two such mo dels If R is also lo cal we call the a mo del G i lo callo cal if its sp ecial bre is lo callo cal

When the ring R is understo o d we sometimes simply refer to G i or even just G as an integral model of G

We use Serres terminology peu ramie and tres ramie see Se

Contents

Intro duction

Acknowledgements

Notation

Typ es

Typ es of lo cal deformations

Typ es for admissible representations

Reduction of typ es for admissible representations

The main theorems

Examples and applications

Examples

Applications

An extension of a result of Manoharmayum

Admittance

The case of

New deformation problems

Some generalities on group schemes

Filtrations

Generalities on deformation theory

Reduction steps for Theorem

Some Galois cohomology

Breuil Mo dules

Basic prop erties of Breuil mo dules

Examples

Relationship to syntomic sheaves

Base change

Reformulation

Descent data

More examples

Some lo cal elds

The case of F

Pro of of Theorem

Rank one calculations

Rank two calculations

Rank three calculations

Conclusion of pro of of Theorem

Pro of of Theorem

Rank one calculations

Mo dels for

Completion of pro of of Theorem

Pro of of Theorems and

Rank one calculations

Mo dels for

Further rank two calculations

Completion of the pro of of Theorem

Corrigenda for CDT

References

Types

Typ es of lo cal deformations By an type we mean an equivalence class of twodimensional repre

sentations

I GLD

over Q which have op en kernel and which can b e extended to a representation of W By an extended

type we shall simply mean an equivalence class of twodimensional representations

W GLD

over Q with op en kernel

is a nite extension of Q in Q Recall from CDT that a Supp ose that is an typ e and that K

continuous representation of G on a two dimensional K vector space M is said to b e of type if

is trivial is BarsottiTate over F for any nite extension F of Q such that j

the restriction of WD to I is in

the character det has nite order prime to

For the denition of BarsottiTate and of the representation WD asso ciated to a p otentially Barsotti

Tate representation see x and App endix B of CDT Similarly if is an extended typ e then we say

that is of extended type if

is trivial is BarsottiTate over F for any nite extension F of Q such that j

WD is equivalent to

the character det has nite order prime to

Note that no representation can have extended typ e unless det is of the form where has nite

order prime to and where is unramied and takes an arithmetic Frob enius element to see App endix

B of CDT Using Theorem of Br one can show that for o dd one obtains equivalent denitions of

typ e and extended typ e if one weakens the rst assumption to simply require that is p otentially

BarsottiTate

Now x a nite extension K of Q in Q Let O denote the integers of K and let k denote the residue

eld of O Let

G GLV

b e a continuous representation of G on a nite dimensional k vector space V and supp ose that End V

k G

k One then has a universal deformation ring R for see for instance App endix A of CDT

O V

We say that a prime ideal p of R is of type resp of extended type if there exist a nite extension

V O

K of K in Q and an Oalgebra homomorphism R K with kernel p such that the pushforward of the

V O

universal deformation of over R to K is of typ e resp of extended typ e

V O

Let b e an typ e and an irreducible extended typ e If there do not exist any prime ideals p of typ e

0 0

D D D D

resp R Otherwise dene R resp R to resp of extended typ e we dene R

V O V O V O V O

b e the quotient of R by the intersection of all p of typ e resp of extended typ e We will sometimes

V O

0 0

D D

write R resp R for R resp R We say that a deformation of is weakly of type resp

V O V O V O V O

weakly of extended type if the asso ciated lo cal Oalgebra map R R factors through the quotient

V O

0 0

D D D D

We say that resp is weakly acceptable for resp R R or resp R if either R

V O V O V O V O

D D

R resp OX R We say that resp is there is a surjective lo cal Oalgebra map OX

V O V O

D D D

acceptable for and if there is a surjective lo cal Oalgebra map OX R resp R if R

V O V O V O

resp OX R

V O

resp Q with valuation ring O and residue eld k then O R If K is a nite extension of K in

V O

0 0

D D D

is naturally isomorphic to R O R resp R Thus weak acceptability dep ends only

0 0 0 0

V V O k O V k O

k k

on resp and and not on the choice of K Moreover resp is acceptable for if and only if

k resp is acceptable for

Although it is of no imp ortance for the sequel we make the following conjecture part of which we already

conjectured as conjecture of CDT

Conjecture Suppose that is an type and an absolutely irreducible extended type A defor

to the ring of integers O of a nite extension K K in Q is weakly of type mation G GLM of

resp weakly of extended type if and only if M is of type resp of extended type

If is a tamely ramied typ e then we exp ect that it is frequently the case that is acceptable for

as in Conjectures and of CDT On the other hand if resp is residual representations

a wildly ramied typ e resp wildly ramied extended typ e then we exp ect that it is rather rare that

resp is acceptable for a residual representation In this pap er we will b e concerned with a few wild

cases for the prime which do turn out to b e acceptable

representations From now on we assume that is o dd If F is a nite Typ es for admissible

ab r

denote the subgroup of GL Z extension of Q we will identify F with W via the Artin map Let U

consisting of elements with upp er triangular mo d reduction Also let U denote the normaliser of U

in GL Q Thus U is generated by U and by

If is an typ e set U GL Z if is reducible and U U if is irreducible If is an extended

U In this subsection we will asso ciate to an typ e an irreducible typ e with j irreducible set U

Q with op en kernel and to an extended typ e with j irreducible an representation of U over

0 0

over of U irreducible representation Q with op en kernel We need to consider several cases which we

treat one at a time

First supp ose that j j where each is a character of W Let a denote the conductor of

I I i Q

If a then set

det St det St

where St denotes the Steinb erg representation of PGL F Now supp ose that a Let denote the

a a

induction from U to GL Z of the character of U which sends

This is irreducible and do es not dep end on the ordering of and

For the next case let F denote the unramied quadratic extension of Q and s the nontrivial automor

phism of F over Q Supp ose that is the restriction to I of the induction from W to W of a character

F Q

s s

of W with Let a denote the conductor of so that a Cho ose a character of W

F Q

such that j has conductor a If a we set

j det

is the irreducible representation of GL F dened on page of CDT where

To dene for a we will identify GL Z with the automorphisms of the Z mo dule O If a is even

a a

of O O s then we let denote the induction from O O s to GL Z of the character

F F

and O s where for O

j det

O s to GL Z of If a is o dd then we let denote the induction from O where

is the dimensional irreducible representation of O O s such that j is the

a+1)2 (

O O s

direct sum of the characters

j det

for O and O s where runs over the nontrivial characters of O Z O

F F

Now supp ose is an extended typ e such that j is irreducible There is a ramied quadratic extension

F Q and a character of W such that the induction from W to W of is see x of G Let s

F F Q

denote the nontrivial eld automorphism of F over Q and also let denote the maximal ideal of the ring

of integers O of F Let a denote the conductor of so a is even and a We may cho ose a character

of W such that j has conductor a We will identify GL Q with the automorphisms of the Q

vector space F We will also identify U with the stabiliser of the pair of lattices O We dene

a a

to b e the induction from F s to U of the character of F s where

F F

det j

b e a with F and s where is a character of F O dened as follows Let

F F

character of Q with kernel Z Cho ose F such that for x we have

j x tr x

F Q

We imp ose the following conditions which determine

is a character of F O

j is nontrivial

and

N x N

F Q F Q

x ZZ

where is a uniformiser in O

Finally if is an irreducible typ e cho ose an extended typ e which restricts to on I and set

We remark that these denitions are indep endent of any choices see G

Recall that by the lo cal Langlands conjecture we can asso ciate to an irreducible admissible representation

of GL Q a twodimensional representation WD of W See x of CDT for the normalisation

we use

Lemma Suppose that is an type and that an extended type with j irreducible Suppose

also that is an innite dimensional irreducible admissible representation of GL Q over Q Then

are irreducible and

If WD j resp WD then

Q Hom

resp

Q Hom

If WD j resp WD then

Hom

resp

Hom

Pr oof The case that extends to a reducible representation of W follows from the standard theory of

principal series representations for GL Q The case that is reducible but do es not extend to a reducible

representation of W follows from Theorem of G The case of follows from Theorem of G

Thus supp ose that is an irreducible typ e and that is an extension of to an extended typ e If

denotes the unramied quadratic character of W then and so we deduce that

0 0 0

det

j is irreducible It follows that is irreducible The second and third part of the lemma for Thus

Q U

follow similarly

Reduction of typ es for admissible representations We b egin by reviewing some irreducible

representations of GL Z U and U Let denote the standard representation of GL F over

F If n and if m Z Z then we let Symm det We may think

of as a continuous representation of GL Z over F These representations are irreducible mutually

nonisomorphic and exhaust the irreducible continuous representations of GL Z over F

denote the character of U over If m m Z Z we let F determined by

m m

1 2

a b

m m

2 1

d a

c d

These representations are irreducible mutually nonisomorphic and exhaust the irreducible continuous

F representations of U over

denote the representation of If m m Z Z a F and m m then we let

m m ga f

1 2

U over on U and which F obtained by inducing the character of Q U which restricts to

m m

1 2

F then we let denote the character of U over F sends to a If m Z Z and a

fmga

which restricts to on U and which sends w to a These representations are irreducible mutually

F nonisomorphic and exhaust the irreducible nite dimensional continuous representations of U over

We will say that a reducible typ e resp irreducible typ e resp extended typ e with irreducible

F if restriction to I admits an irreducible representation of GL Z resp U resp U over

lattice contains an invariant O resp resp and if is a JordanHolder constituent of F

F of We will say that resp resp simply admits if is a JordanHolder constituent of

multiplicity one

For each of the F representations of GL Z U and U just dened we wish to dene notions of

admittance and simple admittance with resp ect to a continuous representation G GL F Let

b e a xed continuous representation G GL F

The representation admits if either

which in addition we require to b e p euramie in the case n Note that admits if and only

if the Serre weight see Se of is n

if admits The representation simply admits

nm nm

admits The representation if either

m m

1 2

m m

i j

m m

i j

m m g fm m g and m m or where f

i j i j

GL Z

admits Note that admits if and only if some irreducible constituent of Ind

m m m m

1 2 1 2

if either with m m simply admits The representation

m m

1 2

The representation simply admits if

is tres ramie

with m m admits The representation admits or if either and if

m m m m

fm m ga

2 1 1 2

1 2

j equals the central character of Note that in this case j det

fm m ga fm m ga

1 2 1 2

m m m m

2 1 1 2

The representation with m m simply admits equals the central if det j

fm m ga

1 2

character of and either

fm m ga

1 2

The representation admits if

fmga

admits

equals the central character of j det

fmga

and if

is tres ramie then

where is unramied and sends Frob enius to a

Note that j

fmga

if admits The representation simply admits

fmga fmga

We remark that the denition of admits the Cartier dual of might lo ok more natural to the reader

We are forced to adopt this version of the denition by some unfortunate choices of normalisations in CDT

We say that a reducible typ e resp irreducible typ e resp extended typ e with j irre

ducible admits a continuous representation G GL F if resp resp admits an irreducible

representation of GL Z resp U resp U over F which in turn admits We say that resp

if resp simply admits

resp resp admits a unique irreducible representation of GL Z resp U resp U

over F which admits

resp resp simply admits

and simply admits

Note that the concept of simply admits is strictly stronger than the concept admits

The starting p oint for this work was the following conjecture of which a few examples will b e veried in

Conjecture Let k be a nite subeld of F G GL k a continuous representation an

type and an extended type with irreducible restriction to I Suppose that det and det are tamely

ramied that the centraliser of the image of is k and that the image of is not contained in the centre of

Q GL

D D

resp admits if and only if R resp R ie if and only if there is a nite

V O V O

extension K of Q in Q and a continuous representation G GL O and which reduces to

has type resp has extended type

if and only if resp is acceptable for resp simply admits

is a relatively straightforward computation On the We remark that to check if or simply admits

other hand to show that or is acceptable for is at present a nontrivial undertaking The reader who

doubts us might like to compare x with x x x x x and x All the latter sections are devoted to

verifying some very sp ecial cases of this conjecture

The main theorems With these denitions we can state our two main theorems The pro ofs very

closely parallel the pro of of Theorem of CDT

Let be an odd prime K a nite extension of Q in Q and k the residue eld of K Let Theorem

G GL K

be an odd continuous representation ramied at only nitely many primes Assume that its reduction

G GL k

and is modular F urther suppose that is absolutely irreducible after restriction to Q

j has centraliser k

ate with type j is potential ly BarsottiT

admits

and is weakly acceptable for

Then is modular

Proof Note that the existence of shows that is acceptable for Now the pro of is verbatim the pro of of

Theorem of CDT see x x x x x and x of that pap er and the corrigendum at the end of

with the following exceptions this pap er

On page one should take U U V ker and

S S Sl

the pro of of Lemma one must use Lemma of this pap er in addition to the results recalled In

in x of CDT

On page replace Setting S T fr g to the end of the rst paragraph by the following

Again the key comp onent of this argument is very similar to the main idea of Kh

U where U U other Set S T fr g U U p if p T and U

S Sp Sp Sp

wise V and V V otherwise and L if p T V where V U p

S Sp S Sp Sp

Z satises the hyp otheses of The Hom O Then GL M H X I SL Z U

OU V

S S

S S S

orem F urthermore

H L H Y F k

M n

U GL Z

on L k mo dule where M is the mo dule for U GL Z dened by the action of GL F as a T

n S

Therefore m is in the supp ort of H Y F

S M

U GL Z

W e now drop the sp ecial assumption on j made in the last paragraph Twisting we see that if

F admitting j then is an irreducible representation of GL Z over

H Y F

U GL Z

Moreover if is irreducible and if is an irreducible representation of U over F which admits

j then we see using the denition of admits and Lemmas and of CDT that

H Y F H Y F

GL Z

m m

U GL Z

S S

Ind

It follows from the denition of admits and Lemma of CDT that m is in the supp ort of

so L is nonzero Using the fact that lemma holds with U replacing U H Y F

M O Hom

S S

and replacing and the discussion on page we conclude that N is nonempty

S S

Theorem Let be an odd prime K a nite extension of Q Q and k the residue eld of K Let in

G GL K

ontinuous representation ramied at only nitely many primes Assume that its reduction be an odd c

G GL k

is absolutely irreducible after restriction to Q urther suppose that and is modular F

j has centraliser k

j is potential ly BarsottiTate with extended type

admits

and is weakly acceptable for

Then is modular

Proof The existence of shows that is in fact acceptable for Again the pro of now follows very closely

that of Theorem of CDT In this case we have to make the following changes All references are to

CDT unless otherwise indicated

j One should also j and On page one should take U U V ker

S S S

U U

e e

GL Q and e to b e the extension of to U dene U to b e the group generated by U and w

S S S S S

on U which restricts to

by Hom e In the statement of Lemma one should replace Hom

S S U

In the pro of of Lemma one must use Lemma ab ove in addition to the results recalled in x

of CDT

Because is acceptable for we know that det of a Frob enius lift is for some ro ot of unity

sZ s

e e

where Thus for some s Hence e factors through the nite group G U V

S S S S

GL Q

In x cho ose M so that it is invariant for the action of U V Also in the denition of L

S S

replace G by G

S S

In the pro of of Lemma replace U by U and by e

S S S S

Note that w acts naturally on Y and F In Lemma we should replace the group H Y F

S S S S

w w

by H Y F and the group H Y F by H Y F

S S S S S S

x with the pro of of the required extension of Prop osition given b elow Replace

On page the isomorphism

H Y F Hom H Y F O

S S O S S

T w linear In the next line one should not only lo calise at m but restrict to the kernel on line is

of w Because w on H Y F we see that the natural map

S S m

H Y F H Y F w

S S S S m

is an isomorphism and so the map

L Hom L O

S O S

is also an isomorphism

should b e replaced by their maximal F and H Y On page the groups H Y F

p p

S S S S

m m

S S

subgroups on which w

e e

On page one should also dene V resp V to b e the group generated by V resp V and

w GL Q Similarly dene e to b e e

e e

In Lemma replace V by V V by V and by e In the pro of of Lemma also replace U

f rr g

e e

resp U by U resp U and resp by e resp e

S frr g frr g S frr g frr g S frr g frr g S frr g

On line of page M should b e chosen as a mo del of e This is p ossible b ecause ker e has nite

s w

for some s One should also set L index in V b ecause in turn H Y F

i V M

On line we must replace V by V

i i

In the pro of of Lemma one must replace V by V and by e

In line of the pro of of Lemma to see that L is a direct summand of H Y as an O F

V M S

b ecause F is a direct summand of H Y F mo dule one needs to note that H Y

m M V M V

F w on H Y

M m V

D D

by R On line of page replace R

V O V O

of CDT Let M Proof of extension of Proposition

First supp ose that admits with m As in the pro of of m and that admits

fm fm m ga m ga

Theorem esp ecially x of CDT as mo died ab ove we have

H Y F F

fr g

m m

r g f

On the other hand

w a

H Y F F H Y F F

fr g fr g

m m m m

fr g

F H Y F

fr g

fm m ga

fr g

Thus using the denition of admits and Lemma of CDT we see that

H Y F

fr g fr g

fr g

so N N

fr g

Next supp ose that admits which in turn admits Assume that j is irreducible or p eu

fmga

rami e By twisting we may reduce to the case m As in the pro of of Theorem esp ecially x of

CDT as mo died ab ove we have

H Y F

U GL Z

fr g

Thus

w ea

H Y F

U GL Z

fr g

where ea is the Teichmuller lift of a Using the emb edding

w H Y F Q H Y F Q ea

U GL Z fr g

fr g

we deduce that

w ea

H Y F

fr g

and so

F H Y F

fr g

fga

fr g

Thus using the denition of admits and Lemma of CDT we see that

H Y F

fr g fr g

fr g

and so N N

fr g

and that j is reducible and tres ramie Finally supp ose that admits which in turn admits

fmga

By twisting we may reduce to the case m Note that Frob a As in the pro of of Theorem

esp ecially x of CDT as mo died ab ove we have

H Y F

fr g

so is Supp ose that is a cuspidal automorphic representation which contributes to H Y F

fr g

a cuspidal automorphic representation of GL A such that is the lowest discrete series with trivial

innitesimal character is a lift of of typ e fr g and hence of typ e and dim As

acts on by the Teichmuller lift of a As has a det has order prime to we see that w

U xed vector but no GL Z xed vector we see that U w on On the other hand

U U

the eigenvalue of U on reduces to a Thus w acts on by the Teichmuller lift of a so w acts

by the Teichmuller lift of a We deduce that on H Y F

fr g

w a

F H Y F

fr g

Using the denition of admits and Lemma of CDT we see that

H Y F

fr g fr g

fr g

so N N

fr g

Examples and applications

Examples Now we will sp ecialise to the case Fix an element GL Z with but

The following denitions which concern isomorphism classes of dimensional representations into

Q do not dep end on this choice We will consider the following typ es These are in fact up to GL

twist a complete list of the wildly ramied typ es which can arise from elliptic curves over Q or in the

for studying case of conductor the extended typ es We will not need this fact Rather the justication

these particular typ es can b e found in x More detailed information ab out the xed elds of these typ es

can b e found in x

corresp onds to the order homomorphism

Z Z

determined by

corresp onds to the order homomorphism

Z Z

determined by

is the unique typ e such that j corresp onds to the order homomorphism

Z Z

determined by

is the unique typ e such that j corresp onds to the order homomorphism

Z Z

determined by

corresp ond For i ZZ we will also consider the unique extended typ es whose restrictions to G

to the homomorphisms

Q Q

determined by

Subsequent sections of this pap er will b e devoted to checking the following sp ecial cases of Conjecture

Suppose that G GL F and Lemma

is tres ramie Both and simply admit

Theorem Suppose that G GL F and

is tres ramie Both and are weakly acceptable for

Lemma Suppose that G GL F

is tres ramie Both and simply admit

G GL F and Theorem Suppose that

is tres ramie Both and are weakly acceptable for

Let i ZZ Suppose that G GL F and Lemma

is tres ramie The extended type simply admits

Let i ZZ Suppose that G GL F and Theorem

is tres ramie Then is weakly acceptable for

We remark that in Theorems and we could replace weakly acceptable by acceptable

This can b e shown by using elliptic curves to construct explicit liftings of the desired typ e For Theorems

and the results of Man suce for this For Theorem a slightly more rened analysis along

the lines of x is required

We also remark that it was Lemmas and Conjecture which originally suggested

to us that we try to prove Theorems and

Conditional on the results stated in x which we will prove b elow we prove the Applications

following results

G GL F with cyclotomic Theorem Any continuous absolutely irreducible representation

determinant is modular

GL F with but The following classication will b e Proof Cho ose an element

indep endent of the choice of Then up to equivalence and twisting by a quadratic character one of the

following p ossibilities can b e attained

is tamely ramied at

is given by the character j

Q F

determined by

where i ZZ

j is given by the character

F Q

determined by

is given by the character j

Q F

determined by

is given by the character j

F Q

determined by

j is given by the character

F Q

determined by

where i ZZ

To see that one of these cases can b e attained use the following facts all of which are easy to verify

A subgroup of GL F with a nontrivial normal subgroup of p ower order is up to conjugation

contained in the normaliser of F

in GL F is generated by and an element The intersection of SL F with the normaliser of

such that and

F det and then If

such that the representation Q In each case we may cho ose an elliptic curve E of G on E

is isomorphic to and such that the representation j of G on E Q has the following form

where we use the same numb ering as ab ove

We place no restriction on

The restriction of to I has the form

and is tres ramie Use Theorem of Man

The restriction of to I has the form

and is tres ramie Use Theorem of Man

to I has the form The restriction of

and is tres ramie Use x of Man

The restriction of to I has the form

and is tres ramie Use x of Man

has the form

Use Corollary is tres ramie and remains indecomp osable when restricted to the splitting eld of

b elow

In each case cho ose such an E and x an isomorphism F E Q such that the Weil pairing on

following the conventions in x of SBT The E corresp onds to the standard alternating pairing on F

in SBT We can nd a adic pair E denes a Q rational p oint on the smo oth curve denoted X

op en set U X Q containing E such that if E denes a p oint in U then E E as

F G mo dules

edahls version of the Hilb ert Irreducibility Theorem see Theorem of E and the argument Using Ek

with E Q of x of SBT we may nd an elliptic curve E and an F G mo dule isomorphism of

such that

and the Weil pairing on E agree under the standard alternating pairing on F

the representation of G on E Q is surjective onto Aut E Q

and E denes a p oint of U

category

stable under isomorphisms and which is closed under nite pro ducts Let S b e a full sub category of S

Ssub ob jects and Squotients and which contains V We will consider the following setvalued functors

on the category of complete no etherian lo cal Oalgebras R with nite residue eld k

D R is the set of conjugacy classes of continuous representations G GL R such that

V O

mo d m is conjugate to

D R is the set of conjugacy classes of continuous representations G GL R such that

V O

mo d m is conjugate to and det

R is the set of conjugacy classes of continuous representations G GL R such that D

V O

mo d m is conjugate to and such that for each op en ideal a R the action makes R a into

an ob ject of S

D R is the set of conjugacy classes of continuous representations G GL R such that

V O

such that det and such that for each op en ideal a R the action mo d m is conjugate to

makes R a into an ob ject of S

and Each of these deformation problems is representable by ob jects which we will denote R R R

V O

R resp ectively

V O

Recall that the following sets are in natural k linear bijection with each other

m m

K R

O V

V O

to k The set of deformations of

V V Ext

k G

H G ad

These bijections give rise to an isomorphism

m H G ad m

V O

as well as bijections b etween

m m

V O

the set of deformations of to k which make k into an ob ject of S

Ext V V ie Ext in the category of discrete k G mo dules which are also ob jects of S

k G S

the subgroup H G ad H G ad corresp onding to Ext V V

k G S

H G ad H G ad so that we get an isomorphism We will set H G ad

S S

G ad H m m

V O

Reduction steps for Theorem We now b egin the pro of of Theorem Making an unram

ied twist we may supp ose that has the form

We may also supp ose that O Z

Let F F denote a totally ramied cubic Galois extension of Q Let F denote the unique cubic

such that F Q is Galois but not ab elian and let F denote a cubic subeld of extension of Q

F so F F is unramied

Let S denote the full sub category S consisting of Z G mo dules X for which there exists a nite

at O group scheme G fg g with descent data for F Q such that X G fg g Q as a Z G

mo dule By Lemma we see that S is closed under nite pro ducts sub ob jects and quotients Using

Tates theorem on the uniqueness of extensions of divisible groups from F to O Theorem of T

Thus Theorem follows from the following factors through R R we see that the map R

V Z

V Z V Z

result which we will prove in x

Theorem dim H G ad

Reduction steps for Theorem We now b egin the pro of of Theorem Making an unram

has the form ied twist we may supp ose that

We may also supp ose that O Z

p p

with norm subgroup in Q top olog denote the degree ab elian extension of Q Let F

Note that F Q is Galois We have an isomorphism ically generated by and

GalF Q C C C

I Let b e the unique element of order In later applications this will b e the square of an

F Q

Q We also let F denote the xed eld of a Frob enius lift of order so element of order in GalF

F Q is totally ramied

Q generated by We will let I denote the twosided ideal of W F F V Gal F

F V

and

consisting of ob jects X for which we can nd an ob ject Let S denote the full sub category of S

such that X G fg g G fg g of FD Q as a Z G mo dule By Lemma we see

F Q I

that S is closed under nite pro ducts sub ob jects and quotients

R Q such that the corre Now cho ose a nite extension K Q and continuous map of rings f

V Z

sp onding representation G GL O is of typ e Let G b e the corresp onding divisible group

has a unique extension to a over Q By Tates theorem Theorem of T the base change of G to F

divisible group G over O By the uniqueness of this extension it is also equipp ed with descent data

Q and with an action of O compatible with the canonical structure on the generic fg g relative to F

bre

p p p

corresp ond to We will use the notation of App endix B of Q Let e GalQ

CDT in particular WD and D G except that we will write F and F in place of and Then

WDe but not WD e is a scalar

and det WD e

Thus WD e Hence on D G Q we have

and F e WD e

We conclude that on D G we have

and so F V

m m

O factors In particular I annihilates D G and for all m the map f mo d R

K V Z

S S

and Theorem follows from the factors through R R Hence the map R through R

V Z

V Z V Z V Z

following result which we will prove in x

Theorem dim H G ad

Reduction steps for Theorem We now b egin the pro of of Theorem We may supp ose

that O Z

p p

with norms the subgroup of Q denote the degree ab elian extension of Q Let F

p p

and where is the unique lift of i to Z with top ologically generated by

Note that F Q is Galois We identify

GalF Q h i h i h i

p p

where corresp onds to and has order corresp onds to and has order and

corresp onds to and has order We also let F denote the xed eld of f g so F Q is totally

i i

ramied

We will let I denote the twosided ideal of W F F V GalF Q generated by

F V

and F

We remark that the ideal I is unchanged if we change our choice of

In x we will prove the following result and explain the unusual lo oking notation

Theorem There are objects G fg g G fg g G fg g and G fg g in the

with the fol lowing properties category FD

F Q I

For r s and we have Q as G modules G fg g

For r s and there is a short exact sequence in FD

F Q I

G fg g G fg g G fg g

rs rs rs

such that G fg g and G fg g have order and for al l a b f g possibly equal the

rs rs

natural map

Ext G fg g G G fg g G fg g fg g Ext

a b b a

rs rsQ rsQ rs

FD F G

F Q I

is injective

with an action of k such If k F is a nite eld extension and if G fg g is an object of FD

F Q I

Q is isomorphic to k then for some r s or the that G fg g

is weakly ltered by fG fg g G fg g g object G fg g of FD

rs rs F Q I

For r s and we have F on D G while F on D G

Note that for all a b p ossibly equal we must have

if a b

G fg g HomG fg g G fg g HomG fg g

b a b a

rsQ rs rs rsQ

F if a b

consisting For r s and we let S denote the full sub category of S

irs

for some ob ject H fg g of FD of ob jects X which are isomorphic to H fg g

F Q I fGfg g g

By Lemma Corollary and Theorem we see that S is closed under nite pro ducts

irs

sub ob jects and Squotients In x we will also prove the following two results S

Theorem For r s and we have

G ad dim H

irs

Theorem For r s and and for any N there exists a continuous repre

sentation

GL F T T G

N Q

such that

det

G fg g Q for some object G fg g of FD where I F de

N Q N N i

F Q I FfGfg g g

Q generated by I and F notes the twosided ideal of W F F V Gal F

k T T and mo d T

We are not asserting that and G are indep endent of the choice of r s though in fact we b elieve

N N

that is indep endent of this choice

From these results we can easily draw the following consequence

Corollary For r s and we have

irs

F T R

V Z

irs

F T and that if R is an Artinian Proof By Theorems and we see that R

V Z

irs

quotient of R corresp onding to a necessarily unique see Lemma ob ject G fg g of

V Z

FD then F on D G

F Q I fGfg g g

irs

Now supp ose R is any Artinian quotient of R which corresp onds to an ob ject G fg g of the

V Z

and consider the exact sequences category FD Let G G fg g

F Q I fGfg g g

G G G

and

G G GG

By Lemma we have exact sequences

G fg g K fg g

and

K fg g G fg g H fg g

such that the comp osite in FD

Q I fGfg g g F

G fg g K fg g G fg g

is multiplication by In particular we have exact sequences

D K D G

and

D H D G D K

such that the comp osite

D G D K D G

is multiplication by As F V on D H we see that F and V factor through maps D K D G

ie we can write F F and V V for some endomorphisms F and V of D G Thus F V equals

zero on D GD G and so D K We conclude that K so that G and R

Thus

S S

irs irs

F T R R

V Z V Z

We now mo dify the argument in x Cho ose a nite extension K Q and continuous map of rings

f R Let Q such that the corresp onding representation G GL O is of extended typ e

V Z K

G b e the corresp onding divisible group over Q By Tates theorem Theorem of T G has a unique

extension to a divisible group G over O By the uniqueness of this extension G comes equipp ed with

descent data fg g relative to F Q and with an action of O compatible with the canonical structure on

the generic bre

p p p

Q corresp ond to We will use the notation of App endix B of Let e GalQ

CDT in particular WD and D G except that we will write F and F in place of and Then

WDe

and WD e WD WD

Thus on D G Q we have

F e WD e

and F

We conclude that on D G we have

and F

Hence also

F V

and F

In particular I annihilates D G

O corresp onds to such that G fg g Thus G fg g is an ob ject of FD

K K K Q K

F Q I

By Theorem we see that G fg g is weakly ltered by fG fg g G fg g g for some

rs rs

r s or We will prove r s By Theorem and Lemma

is ltered by G fg g is strongly ltered by fG fg g G fg g g As G fg g

K K Q

rs rs

using Theorem we see that G fg g is weakly ltered by G fg g For all m we have

G fg g G fg g G

fg g is also weakly and hence strongly ltered by G fg g for the so for all m the ob ject G

irs

m m

By factors through R O same r s Thus for all m the map f mo d p R

K V Z

V Z

and Theorem R Corollary we see that r s so the map R factors through R

V Z

V Z V Z

follows from Theorem

Some Galois cohomology In this section we will b egin the pro ofs of Theorems and

We will let S denote one of the categories S S or S We will let in the cases S and

irs

otherwise In all cases

is tres ramie

The maps and induce a commutative diagram with exact rows and columns

Ext

F G

Ext Ext

F G F G

Ext Ext Ext

F G F G F G

denote the comp osite map We will let

Ext Ext

F G F G

and resp the induced mapping

ker Ext

F G

resp

ker Ext

F G

resp the induced mapping We will also let

ker Ext Ext

F G F I

resp

Ext ker Ext

F I F G

If we reinterpret our Extgroups as cohomology groups and use the isomorphism our diagram

b ecomes

H G F

H G ad H G

H G H G F H G

takes the form Fix a basis of F so that

Then any extension of by in characteristic may b e represented by a matrix

where the co cycle

Z G ad

H G ad Moreover represents the class of this extension in Ext

F G

H G

if then H G F and H G F

if and only if H G F and H G ad

In particular we have on H G ad ker

We have an exact sequence

where the rst map sends

y x x

y y

and the second map sends

a b

c a

Thus we get an exact sequence

H G H G H G ad

ker with H G We also have an exact sequence and so we may identify H G ad

which gives rise to an exact sequence

F H G H G H G F H G

with H G ad ker then the latter map H G H G F If we identify H G

is identied with

The sequence Lemma

F H G H G H I F

is exact

Proof The key p oint is that is tres ramie compare with Prop osition of Di It suces to show

that the comp osite

F H G F H G H G

is injective Supp ose that x H G F maps to zero in H G then by Tate duality x is annihilated

by the image of the map H G F H G coming from the short exact sequence

Cartier dual to As is tres ramie we see that the image of

H G F H G Q Q

is not contained in Z Z Thus

F H G F HomQ F x HomQ Z F H G

of Se must b e zero see Prop osition of x of chapter XIV

Corollary The maps

H G ad ker H I F

and

H G ad ker H I F

have the same kernel and this has dimension over F

Theorems and now follow from the following results which we will prove later One

advantage of these new formulations is that with one exception they refer only to Ext and make

concepts which we found tricky to translate into the language of no mention of the determinant or ad

integral mo dels

Theorem Ext H G is the zero map

Ext H I F is the zero map

H H I F is the zero map G ad

Theorem Ext H G is the zero map

Ext H I F is the zero map

Theorem Suppose that i ZZ and r s or

Ext H G is the zero map

irs

Ext H I F or Ext H I F is the zero map Either

S S

irs irs

The deduction of Theorems and from these results is immediate

Breuil Modules

In this section we recall some results from Br see also the summary Br and give some slight

extensions of them Three of the authors ap ologise to the fourth for the title of this section but they nd

that the term Breuil mo dule is much more convenient than ltered mo dule

Throughout this section will b e an o dd rational prime and R will b e a complete discrete valuation ring

with fraction eld F of characteristic zero and p erfect residue eld k of characteristic

prop erties of Breuil mo dules We will x a choice of uniformiser of R and let Basic

u G u E u

b e the Eisenstein p olynomial which is the minimal p olynomial of over the fraction eld of W k so

G u W k u is a p olynomial with unit constant term G W k and degree at most e The

th e

p ower map on k uu is denoted and we dene

c G u k uu

It is very imp ortant to keep in mind that these denitions as well as many of the denitions b elow dep end

on the choice of the uniformiser

The category of torsion Breuil modules or torsion Breuil mo dules over R or simply Breuil mo dules

or Breuil mo dules over R is dened to b e the category of triples M M where

M is a nite free k uu mo dule

e e

M is a k uu submo dule of M containing u M

M M is semilinear and has image whose k uu span is all of M

A morphism M M N N is a morphism f M N of k uu mo dules such that f M N

and f f on M We dene the rank of M M to b e the rank of M over k uu Breuil

mo dules form an additive category not ab elian in general in the obvious manner and this category do es

or mo d The induced semilinear map of not dep end on the choice of It is denoted mo d

R F

k vector spaces

M u M M u M

M u M M u Mu M u M In particular a map of is bijective b ecause it is onto and

Breuil mo dules

M M M M

is an isomorphism if and only if the map M M on underlying k uu mo dules is an isomorphism

Lemma Suppose that

M M M

is a complex of Breuil modules The fol lowing are equivalent

The underlying sequence of k uu modules is exact

The underlying sequence of k uu modules is exact as is the sequence

M M M

The complex of vector spaces

M u M u M u

is exact

Proof The second statement clearly implies the rst The rst implies the third as Breuil mo dules are free

over k uu It remains to show that the third condition implies the second Using Nakayamas lemma and

the freeness of Breuil mo dules we see that

M M M

we see that the natural map is an exact sequence of k uu mo dules Using the bijectivity of

f M M

is surjective mo dulo u and therefore is surjective It remains to check that the inclusion of k uu mo dules

M kerf is an equality Since f is compatible with f M M via the inclusions M M M M

and also via the maps and it is obvious that kerf kerf M and that kerf M Since

Then kerf contains M which in turn contains u M we see that M kerf is a Breuil mo dule

M kerf dened via the identity map on M is a map of Breuil mo dules which is an M M

isomorphism on underlying k uu mo dules so it must b e an isomorphism of Breuil mo dules This forces

kerf M

When the equivalent conditions of this lemma are met we call the sequence of Breuil mo dules

M M M

exact

For any Breuil mo dule M M we dene the Frob enius endomorphism M M by

u m m

where c is dened as in Note that this dep ends on our choice of uniformiser

We let N W k u W k u denote the unique continuous W k linear derivation satisfying N u u

This op erator extends to any Breuil mo dule More precisely we have the following lemma ie N u

Lemma Let M be an object of mo d There is a unique additive operator N M M the

mono dromy op erator satisfying the three conditions

N sx N sx sN x s k uu x M

N N on M

N M uM

Mor eover any morphism of Breuil modules M M automatical ly commutes with N

M M Br Proof Lets start with unicity Recall we have an isomorphism k uu e

k u u

Supp ose there are two op erators N and N satisfying and ab ove so N N is

k uu linear and satises and M uM Thus

M M uM u M

so M k uu e M u M Iterating M M u M so M u M

k u u

and so on As u we get For the existence let N N on

k uu e M M

k u u

and note N satises N sx N sx sN x Call a derivation of M any additive op erator satisfying this

relation and dene successive derivations of M by the formula

N s x N s x sN x

j j

for j Note that N is well dened by the following observations

N u s u N s and N ux ux uN x imply that N u s x N s ux

j j j j

e e

If x then x u M see of Lemma of Br and so N x u M and N x

j j

As N M uM we have N N M u M so N N for j This N satises

j j j j j

and

The reason for intro ducing Breuil mo dules and putting the factor c in the denition of is the

following theorem

Theorem Given the choice of uniformiser for R there is a contravariant functor M from

nite at R group schemes which are kil led by to mo d and a quasiinverse functor G

If G is a nite at R group scheme kil led by then G has rank if and only if M G has rank r

If G is a nite at R group scheme kil led by then there is a canonical k linear isomorphism

D G k M GuM G

k Frob

this identication F Frob corresponds to and V Frob corresponds to the composite Under

V M u M M u M M u M

G G G

is a diagram of nite at group schemes over R which are kil led by and if

M G M G M G

is the corresponding diagram of Breuil modules then the diagram of nite at group schemes is a short

exact sequence if and only if the diagram of Breuil modules is a short exact sequence

Proof See x Prop osition Theorem Theorem and the pro of of Theorem of

Br In of Br it is shown that M Gu M G can b e k linearly identied with the crystalline

Dieudonne mo dule of G k In of BBM the crystalline Dieudonne mo dule of G k is identied with

D G k The equivalence of the two notions of exactness can b e deduced from the compatibility of

k Frob

M with Dieudonne theory from Lemma and from the fact that a complex of nite at group schemes

over R is exact if and only if its sp ecial bre is exact see for example Prop osition of deJ

Examples For r e an integer and for a k dene a Breuil mo dule Mr a by

Mr a k uu e

e r

Mr a k uu u e

u e ae

It is easy to check that is well dened and uniquely determined by the given conditions We will refer

to e as the standard generator of Mr a and write Gr a for G Mr a The following lemma is easy to

check

Lemma Any Breuil module of rank over k uu is isomorphic to some Mr a

There is a nonzero morphism Mr a Mr a if and only if r r r r mo d and

r r

aa k Al l such morphisms are then of the form e bu e where b aa

The modules Mr a and Mr a are isomorphic if and only if r r and aa k or

equivalently if and only if there are nonzero morphisms Mr a Mr a and Mr a Mr a

If we order the Mr a by setting Mr a Mr a if there is a nonzero morphism Mr a

as G runs over models of a xed nite at Mr a then the set of isomorphism classes of M Gs

F group scheme G of order is wel l ordered

Mr a we have N e so N On

Gr a is etale resp multiplicative if and only if r e resp r

and Ge G ZZ G

The Cartier dual of Gr G is Ge r

Proof The rst three parts are easy computations For the fourth part note that two nite at group

schemes G and G of order over R have isomorphic generic bres if and only if there is a nonzero morphism

G G or G G The fth part is another easy computation and the sixth part follows on computing the

Dieudonne mo dule using Theorem

By of Br we see that the ane R algebra of the group scheme attached to Mr a is

R X X X

where ea is a lift of a to W k This has constant generic b er if and only if eaG F is an

th th

p ower This o ccurs if and only if r e mo d and aG k is a p ower Thus

Me G corresp onds to the etale group scheme ZZ over R

Next we show that the group scheme G corresp onding to the Breuil mo dule M is isomorphic to

By using the relation b etween Breuil mo dules and Dieudonne mo dules see Theorem we see that the

Dieudonne mo dule of the closed b er of G is isomorphic to the Dieudonne mo dule of the closed b er of

This forces G since we may consider Cartier duals and observe that a nite at R group scheme G is

etale if and only if its sp ecial bre is etale and then x of b o ok IV of EGA may b e used

This establishes the seventh part The nal part follows from parts four and seven

maxr se e

Now supp ose that r s e are integers and cho ose a b k f u k uu We can dene

an extension class

Ms b Ms b r a f Mr a

in mo d by

e e

Ms b r a f k uu e k uu e

s r

Ms b r a f hu e u e f ei

s r

u e be and u e f e ae

the standard generator of Ms b maps to e

e maps to and e maps to the standard generator in Mr a

The following lemma is also easy to check

Lemma Any extension of Mr a by Ms b in mo d is isomorphic to Ms b r a f for

maxr se e

some f u k uu

Two such extensions Ms b r a f and Ms b r a f are isomorphic as extension classes if and only

s r

f f u h bau h

for some h k uu in which case one such isomorphism xes e and sends e to e bah e

maxr se e e

We remark that f u k uu is required so that Ms b r a f u Ms b r a f We will

write Gs b r a f for G Ms b r a f

We will also need some slight extensions of these results to allow for co ecients To this end let k F b e

a nite extension linearly disjoint from k and write k k for the eld k k For r e an integer and

for a k k dene a Breuil mo dule Mk r a with an action of k by

Mk r a k k uu e

e r

Mk r a k k uu u e

u e ae

We will let denote the automorphism of k k u which is the identity on k and which raises elements of

k u to the p ower The following lemma is easy to check

Lemma Any Breuil module with an action of k which is free of rank k k over k uu is

isomorphic to some Mk r a

There is a nonzero morphism Mk r a Mk r a if and only if r r r r mo d and

r r

aa bb for some b k k Al l such morphisms are then of the form e b u e

where b k k and b b aa

The modules Mk r a and Mk r a are isomorphic if and only if r r and aa k k

On Mk r a we have N e and so N

G Mk r a is etale resp multiplicative if and only if r e respr

maxr se e

w cho ose r s e integers a b k k and f u k k uu We dene an extension No

class

Mk s b Mk s b r a f Mk r a

with an action of k by in mo d

e e

Mk s b r a f k k uu e k k uu e

s r

Mk s b r a f hu e u e f ei

s r

u e be and u e f e ae

to e the standard generator of Mk s b maps

e maps to and e to the standard generators in Mk r a

the following lemma is easy to check Then

Lemma Any extension of Mk r a by Mk s b in mo d with a compatible action of k

maxr se e

is isomorphic to Mk s b r a f for some f u k k uu

Two such extensions Mk s b r a f and Mk s b r a f are isomorphic as extensions if and only

r s

bau h f f u h

e and sends e to e bahe for some h k k uu in which case one such isomorphism xes

We will write Gk r a and Gk r a s b f for G Mk r a and G Mk r a s b f resp ectiv ely

to syntomic sheaves Let us rst recall some of the notations of Br and Br Relationship

Let Spf R b e the small adic formal syntomic site over R S the adic completion of W k u

syn i N

S Sp ecS and for any X S S E Spf R

n n n syn

cr is

O X H X E O

n n cr is

E X

n n

cr is n n

E u It turns out O where X X R is viewed over E via the thickening Sp ec R

n n R n

is the sheaf of S mo dules on Spf R asso ciated to the presheaf cf the pro of of Lemma in Br

n syn

D P DP

n n

X W W k u X O W k uu W X O

n n X n n X

k W W k

1 1

n n

n th

the subscript means we twist by the n p ower of the Frob enius when sending W k to W k u Here

n n

and the exp onent DP means we take the divided p ower envelop e with resp ect to the kernel of the canonical

map

X O W k u W X O

n X n n X

W k

n 1

n n1

w w w w s su w

where w is a lo cal lifting of w these divided p owers b eing required to b e compatible with the usual divided

i i

x ie we take the divided p ower envelop e relative to the usual divided p ower structure p owers x

on the maximal ideal of W k Note that the latter map induces a canonical surjection of sheaves of

S mo dules on Spf R

n syn

cr is

O O

cr is

the kernel of this surjection For any n let S S b e We denote by J where O X X O

n n n n X

ie ie

u u

cr is

The sheaf O is equipp ed with the the unique lifting of Frob enius such that u u and

i i

crystalline Frob enius which is also induced by the map su w w su w w on

the ab ove presheaf Here on W k u is Frob enius on W k and takes u to u Since divides

n n

cr is cr is

cr is

x x we get J j O for all n so we can dene an S linear by the usual atness

n n

trick see x of Br Let N S S b e the unique W k linear derivation such that N u u and

n n n

e e e e

u Finally dene N u eu u ie

i i i

cr is cr is

N O O

n n

to b e the unique W k linear morphism of sheaves which on the presheaf is given by N s w

n i

P P

cr is

N s w s w Note that N N so N N on J

G as a formal scheme over R Let G b e a nite at group scheme over R which is killed by Viewing

it is an ob ject in Spf R Viewing it as a sheaf of groups on Spf R its asso ciated Breuil mo dule is

syn syn

dened as

cr is e

G O k uu M G Hom

S ves of groups shea

e cr is

k uu in M G M G image of Hom G J

sheaves of groups

is induced by

cr is cr is

where the S mo dule structures are induced by the compatible S actions on O and J see x and

e e e

x of Br Here S k uu is the surjection that sends u to u u to u for i l and

i i

u to for i l

We record for future reference the following straightforward observation

If we denote by resp pr i f g the coproduct resp the two projections Lemma

G G G

Sp ecR

oups F on Spf R we have then for any sheaf of commutative gr

syn

Hom G F fx F G j pr pr x g

sheaves of groups

cr is cr is

The op erator N on O induces an op erator N on Hom G O hence on M G

sheaves of groups

Lemma The above operator N on M G coincides with the operator N dened in L emma

Proof By unicity in Lemma we only have to prove that N satises N M G uM G since the

other conditions are automatically satised Its enough to prove that N x N x uM G for

DP e cr is

Thus one also has b ecause it is so on k u X O any x M G But u N on O

e cr is

u N on Hom G O hence on M G This implies N M G u M G uM G

groups

since M G is free over k uu

Base change In this section we will examine the relationship of the functor M with two instances

of base change First we consider unramied base change

b e a p erfect eld of characteristic which is an extension of k and R R W k Cho Let k ose

W k

as uniformiser in R If X Spf R let X Spf R X and dene

syn

Spf R

cr is cr is cr is cr is

X O O X J X and J X

n n n n

cr is

in the pro of of of Br we have that O is the sheaf on Spf R asso ciated to the presheaf As

syn

X W k u W X O

n n

W k

W k u W k X O

n n k X

W k

Dene S as S but with k instead of k There is a canonical isomorphism of sheaves

cr is cr is cr is

O W k O S O

n S

W k

n n n n n

coming from the obvious isomorphism

n n

e e

n n

W k uu W k X O W k uu W X O

n n k X n n X

W k W k

1 1

n n

cr is cr is

and one easily sees it induces an isomorphism J W k J Moreover we have the following

W k

n n

obvious lemma

Lemma The diagram of sheaves on Spf R

syn

cr is cr is

J W k J

W k

n n

1 1

cr is cr is

O W k O

W k

n n

is commutative

the identication from x Lemma and Lemma for n together with obvious Using

functorialities we obtain after tensoring by k uu the following corollary

Let G be a nite at group scheme over R which is kil led by Let k k be an extension Corollary

of elds with k perfect and let a uniformiser for R R W k Then there is a canonical

W k

isomorphism in the category mo d

M G k M G k G G M M

k k

compatible with composites of such residue eld extensions

R R For any We will now turn to the case of base change by a continuous automorphism g

P P P

i i i g

s g w u and s w u where g and act on W k through s w u W k u let

i i i

their action on k Cho ose H u W k u such that g H Notice that H u W k u

g g g

P P

i i i

Dene gb W k u W k u by gb w u g w u H u

i i g

Lemma There is a unique element tu W k u such that if is dened by w u

g g g i

w u tu one has bg gb

i g g

Proof One has to solve in W k u

H u tuH u H u

g g g

where the two sides clearly b elong to W k u As H u W k u there is a unique K

g g

uW k u such that K uH K u u so we have

g g g

H K u tu H K u

g g g g

For any ob ject M of mo d dene M M by the following formula

x x tuN x

g g

where N is as in Lemma

For any X Spf R let X Spf R X and dene

syn

g Spf R

cr isg cr is g cr isg cr is g

O X O X and J X J X

n n n n

cr isg

Then O is the sheaf on Spf R asso ciated to the presheaf

syn

X W k u W X O

X n n

W k

W k u W R X O

n n g R X

W k

ei+ j ei+j

u u

eij

for j e Let gb S S b e the unique ring isomorphism such that gbw g w H u

n n i i g

i i

i There is a canonical isomorphism of sheaves

cr isg cr is

O S O

g n S

n n n

coming from the obvious gbsemilinear isomorphism

n n

e e

n n

W k uu W R X O W k uu W X O

n n g R X n n X

W k W k

1 1

n n

s w w bg s w w

n n

cr isg

cr is

J and one easily sees it induces an isomorphism J S

S g n

n n

Dene S S as in Lemma and dene

g n n

cr is cr is

O O

n n

P P

to b e the unique morphism of sheaves which is induced by s w s w on

g i i g

cr is cr is

the presheaf see x and note that this is well dened Since J O we can dene

n n

cr is

Lemma The diagram of sheaves on Spf R

syn

cr isg

cr is

J J S

S g n

g 1

cr isg

cr is

O S O

S g n

n n

cr is

eover we have on J is commutative Mor

N log tu

where N is dened as in x

n cr is

and and lo oking on the ab ove presheaves it is completely Proof By working mo dulo ie with J

straightforward

Let G b e a nite at group scheme over R which is killed by Note that thanks to Lemma and the

cr is cr is

is O formula for in Lemma the op erator M G M G induced by the map J

g g

n n

precisely the op erator denoted earlier in this section see Using this together with Lemma

Lemma for n and obvious functorialities we obtain after tensoring by k uu the following

corollary

Corollary Let g R R be a continuous automorphism

Let G be a nite at group scheme over R which is kil led by Then there is a canonical isomorphism

in the category mo d

e e g g

k uu k uu M G M G M G M G

e e

g k uu g k uu

If f G G is a morphism of nite at R group schemes kil led by and M f is the corresponding

then M f also commutes with the and there is a commutative diagram morphism in mo d

in mo d

f M

e e

M G k uu M G k uu

e e

k uu g k uu g

o o

M f

g g

M G M G

W k u such If g g are two continuous automorphisms of R and if we choose the unique H

g g

1 2

g g bg gb on W k u then on that d

e e e

k uu k uu k uu e M G e M G e

g g k uu g k uu g k uu d

2 1 2 1

one has

g g g g

2 1 1 2

Corollary Let G be a nite at group scheme over R which is kil led by To give a morphism of

schemes g G G such that the diagram of schemes

G G

Sp ecg

Sp ecR Sp ec R

is commutative and the induced morphism G Sp ec R G is an morphism of group schemes over

g Sp ecR

R is equivalent to giving an additive map bg M G M G such that both of the fol lowing hold

F or al l s k uu and x M G gbsx gbsgbx

bg M G M G and bg bg bg tugb N with t as in Lemma and N as in

g g

Lemma

Proof Note that the last condition is equivalent to bg gb The rst two conditions are equivalent

which is equivalent to the last two by Corollary to giving a morphism gb M G M G in mo d

Finally we make some computations that concern the dep endence of the ab ove compatibilities on the

e e e

choice of H u Let f u b e an element of k uu u k uu and dene for any M in mo d

the additive map M M via

f u N

where N is as in Lemma Using k uu e M M we extend to all of M by the

k u u

formula

i i i

u x u f u x

f f

for x M If x M one checks that

i i i

u x u x u x

f f f

so is well dened Moreover it is clear that M M Let

f f

cr is cr is

O O

b e the unique isomorphism of sheaves coming from the semilinear isomorphism of presheaves

e e

k uu X O k uu X O

k X k X

1 1

su w w su f u w w

n n

see

Let G b e a nite at group scheme over R killed by and recall that

e cr is

k uu M G Hom G O

sheaves of groups

cr is

Lemma The operator on M G is induced by the operator on O

f f

cr is

Proof One can check that the op erator on O satises log f N where N

f f

is dened as in x and log f is the usual expansion of log in S which makes sense b ecause of the

e e e

assumption that u jf and b ecause of the divided p owers u After tensoring with k uu we get

f u N on M G which clearly implies the two s are the same

f f

Let g and cho ose H u f u for some f E uW k u see the start of x for the denition

k uu of E u Recall from Corollary that we have a canonical isomorphism M G e

g k uu b

M G

Lemma The map is the composite M G M G M G where the rst map is the one

in Corol lary and the second comes from the obvious isomorphism G G In other words once

H u f u has been chosen M G M G is the map corresponding to the identity G G

g f G

under the equivalence of Corol lary

The pro of is straightforward by lo oking at the usual presheaves and using Lemma We remark that

is not necessarily the identity even though is However with f is the identity

f G f

ulation In this section we will reformulate Corollary Reform

Lemma There is a unique element t u W k u such that if is dened by w u

g g g i

w u t u one has bg gb

i g g

Proof One has to solve in W k u

u H u u t u H u t u

g g g g

As H u W k u there is a unique L uW k u such that L uH u u Applying L to

g g g g g

u K uH K u cf the pro of of Lemma we get L u K u We must solve

g g g g g

K u H u

g g

t u

Lemma There is a unique u uW k u such that if N N then N gb gb N

g g g g

there is a unique u uW k u such that if N N then bg N N gb Moreover Similarly

g g g g

N N and N N

g g g g g g g g

Proof Since N is a derivation so is N for any W k u One has to solve uN uH u uH u

g g g

H N u

and uH u which amounts to

H u

N H u

H u

N H K u

g g

H K u

g g

where K is as in the pro of of Lemma The commutation relations with the Frob enius follow from

N N gb bg N bg bg N gb bg bg N N gb and the fact gb is bijective on

g g g g

W k u

We also denote by N N and N N the corresp onding derivations on k uu For any ob ject

g g g g

M of mo d dene M M by the formula

x x t uN x

g g

where N is as in Lemma and we recall that we dened in One checks that u

g g

e e

see Note that we also have gb bg N gb bg N bg gb u u c

g g g g g

gb N N gb in k uu

then there is a unique operator N M M satisfying Lemma Let M be an object of mo d

the three conditions

N sx N sx sN x s k uu x M

g g g

N x N x x M where y u y if y M

g g g g g g

N M uM

The same statement holds for N and

g g g

Proof The pro of is the same as for Lemma using the fact we still have isomorphisms

k uu e M M

k u u

resp with replacing

g g

Lemma For M an object of mo d N N and N N where N N are as in Lemma

g g g g g g

as in Lemma and N as in Lemma

g g

Proof By unicity of N one has to check N satises the three conditions of Lemma The rst and

g g

e e e e

last are obvious Note that N u x N u x so u x u x which implies on M

g g

is as in Lemma One computes

uN u N t u N

g g g g

uN u N

g g g

But the equality N u N u in W k u from Lemma yields

g g g g

u N t u u W k u

g g g

We thus get N N hence condition For N the pro of is completely similar

g g g g g

and gb M M be an additive map such that for al l Lemma Let M be an object of mo d

s k uu and x M gbsx gbsgbx and gbM M If bg gb then bg N N bg

g g

Similarly if bg bg then N gb gb N

g g

Proof We prove the rst case the other one b eing the same As in the pro of of Lemma we dene

N N with N N for i large enough using k uu e M M It is enough to

g g g g i g

k u u

show bg N N bg for all i Supp ose bg N N gb and let s k uu and x M then

i g i i g i

N gbs x N gbs gbx

g i g i g

N gbs gbx bg s N gbx

g g g g i

gbN s x gbs bg N x

g i

gbN s x gbs N x

gb N s x

so gb N N gb by linearity One easily checks by a similar computation that N bg gb N hence

i g i g

the result follows by induction

Lemma Let M be an object of mo d and bg M M an additive map such that for al l s

k uu and x M bg sx bg sgbx and gbM M Then the fol lowing two conditions are equivalent

bg gb gb tugb N

and bg gb t uN bg

the other Proof One has to show gb bg is equivalent to bg gb We prove

g g

case b eing the same On M we have gb gb b ecause as in the pro of of Lemma

g g

By Lemmas and we have gb N bg N gb Thus we get from using N N

gb bg bg tugb u N gb

g g

Playing the same game over W k u with the relation gb gb gb tugb N which is easily

checked to hold in W k u we again end up with gb bg bg tugb u N bg in W k u

g g

But we also have in W k u the equality

gb gb t uN gb

Thus gb tugb u t u in k uu so relation holds

g g g

We can now derive the variant of Corollary which we will use

Corollary Let G be a nite at R group scheme kil led by Let g R R be a continuous au

e e

tomorphism choose H u W k u such that g H and dene gb k uu k uu by

g g

i i i

gbw u g w u H u To give a morphism of schemes g G G such that the diagram of schemes

i i g

G G

Sp ecg

Sp ecR Sp ec R

is commutative and the induced morphism G Sp ec R G is an morphism of group schemes over

g Sp ecR

R is equivalent to giving an additive map bg M G M G such that both of the fol lowing hold

For al l s k uu and x M G gbsx gbsgbx

bg M G M G and bg t uN bg with t as in Lemma and N as in Lemma

g g

Moreover g is an isomorphism if and only if gb is Assume these are isomorphisms Choose H such that

d d

g u gb u on W k u ie H u gb uu Then the map g that corresponds to g is

as above then g g H equal to bg Also if g g are two automorphisms of R and if we choose H

g g

2 1

u such that gb gb u uH corresponds to gb bg provided we choose H

g g g g

2 1 2 1

Proof The equivalence is clear thanks to Corollary and Lemma The fact that g g corre

sp onds to gb bg is automatic using Corollary and the functor G M G Applying this to g g

and g g we see that g g corresp onds to g bg But by Lemma corresp onds to

G G

with f dened by g gbu u f in W k u We see that f and that is the identity on

f f

M G Thus g gb on M G

Descent data Assume now that R is endowed with a continuous left faithful action of a nite group

Then b ecomes the Galois group of the fraction eld F of R over some subeld For each g cho ose

H u W k u so that g H with the one condition that H u Recall from Lemma

g g

that this uniquely determines elements t u W k u such that

u H u u t u H u t u

g g g g

u E uW k u such that Moreover for any pair g g there is obviously a unique f

g g

1 2

u gb gb u \ g g u f

g g

1 2

the unique k linear map M M such that for then we will denote by If M is an ob ject of mo d

g g

1 2

x M we have

x u N x where N is as in Lemma f

g g g g

1 2 1 2

i i i

x u x u f u and

g g g g g g

1 2 1 2 1 2

by See x where we denoted

f g g

g g 1 2

1 2

Supp ose that G is a nite at R group scheme Recall that by descent data on G for we mean isomor

phisms of nite at group schemes

g G G

for g such that

g h h g

for all g h Equiv alently we may think of g as a map of schemes G G over g Sp ec R Sp ec R

which induces an isomorphism of group schemes G G In this picture the compatibility condition simply

b ecomes

g h hg

Suppose that G is a nite at R group scheme kil led by Fix H u as above for al l g Theorem

To give descent data on G relative to is equivalent to giving additive bijections bg M G M G

for al l g so that bg takes M G into M G and

i i

bg w u m g w uH u gbm for m M G w k

bg t uN bg on M G

and gb bg gdg

g g

1 2

The above equivalence is functorial in G and is compatible with classical Dieudonn e theory in the

fol lowing sense if the action fgbg on M G corresponds to descent data fg g on G then the g

g induced on the contravariant Dieudonne module D G and the g semilinear map semilinear map D

bg mo d u induced on M GuM G are compatible via the isomorphism of Theorem

Proof Part is a consequence of Corollary Lemma and the choice H The functoriality in

g g

G k the functoriality follows from Corollary and the last statement there comes from G k

of the isomorphism in Theorem and the reduction mo dulo u of Corollary

Supp ose that R Then we may take H u for all g With this choice we see that t

g g

for all g g In this case to give bijections gb M G M G as in the lemma and f

g g g g

1 2 1 2

is equivalent to giving an R semilinear action on M G which commutes with u and and preserves

M G Thus M G M G is a Breuil mo dule over R from which we can recover M G by

tensoring with W k over W k In other words etale descent for group schemes translates in the obvious

manner for Breuil mo dules if we cho ose to b e invariant

are not very To build an action of on G using Theorem the conditions bg gb gdg

g g

1 2

convenient to check in practice since there are to o many of them It is useful to have the following variant

Z where is the free group on d generators and a group surjection Cho ose d

d d d

The group still acts on R via its quotient and for each i f dg cho ose elements H u

This determines isomorphisms b on W k u and k uu and by W k u such that H

i i

comp osition isomorphisms b for all Note that if ker then H u u f u for some

f E uW k u For such denote by the unique k vector space endomorphism of any ob ject M of

such that for x M we have mo d

N on the image of f u

i i i

and u x u f u x

where N is as in Lemma See x where we denoted by Let R b e a subset of ker such

that ker is the smallest normal subgroup of containing R

the above notation to give descent data on G for is equivalent to giving additive Corollary With

bijections b M G M G for j f dg so that b takes M G into M G and

j j

i i

u b m for m M G w k b w u m w uH

j j j

uN b on M G b t

j j

n n

m 1

i for j m and if we dene R where i f dg n Z and i if

j j j j

i i

m 1

n n

1 m

b b b then b

i i

1 m

Proof Straightforward from Corollary and Lemma

We dene a category DD of Breuil modules with descent data for in the obvious way This

F F

exact if the under category is additive but not necessarily ab elian We call a complex in DD

F F

lying complex in mo d to is exact In the natural way we extend M to a functor from FD

F F

More examples In this section we will determine the p ossible descent data on a rank one Breuil

mo dule Let b e as in section x

Lemma Suppose that G is a nite at R group scheme of order and that its generic bre admits

descent data over F Then there is unique descent data on G over F extending any choice of descent

data on G F over F If M G Mr a and if satises mo d then

be H u e

r e r th

root of H u in k uu with constant term where H u denotes the unique

We remark that since Aut Mr a ZZ by consideration of the geometric generic bre the choice

of isomorphism M G Mr a do es not matter

Proof We rst claim two such nite at group schemes G and G have isomorphic generic bres if and only

if there is a nonzero morphism G G or G G By Lemma we see that if G is a nite at F

group scheme then the lattice of mo dels for G over R is well ordered Supp ose all the integral mo dels are

G for G G For any isomorphism G G must then induce isomorphisms G

i n i

all i n The rst part of the lemma follows

Let M Mr a so M is a free k uu mo dule of rank with the usual basis element e The submo dule

r r

M is spanned by u e and u e ae From Theorem we have N which implies that

b b

For H mo d Clearly

i i i

b cu e cu H u b e

is a bijection if and only if b e b on the e for some unit k uu Evaluating b

element u e M we get

H u

in k uu Thus

for some unit F

Since Breuil mo dule descent data always induces a k linear action of the inertia group on the k vector

space M u M and in this case dim M u M the action of the element of p ower order on M u M

must b e trivial Thus

Some local fields

In order to apply the metho ds of x we need some more explicit information ab out the elds F intro duced

in x In this section we will collect this essentially elementary information In each case we will give an

explicit description of the Galois group GalF Q This is needed to carry out the delicate Breuil mo dule

calculations in subsequent sections We will also sp ecify a uniformiser of F and partially calculate the

following p olynomials and p ower series dep ending on our choice of

Gu W k u a p olynomial of degree at most eF Q such that has minimal p olynomial

eF Q

Gu over Q u

eF Q

c Gu mo d u

u of degree at most eF Q such For GalF Q the unique p olynomial H u W k

that H

as in x In some cases p ower series t and f

The case of F Recall that corresp onds to the order homomorphism

Z GL Q

is determined by

where det and but Recall also that F F is any totally ramied cubic Galois extension

of Q We may take F F Q where is a ro ot of X X One may check that the other

ro ots of X X are and so GalF Q is generated by one element which

sends to and satises Also is a uniformiser for F so

Gu u

c u mo d u

u u H

The case of F Recall that corresp onds to the order homomorphism

Z GL Q

determined by

where det and but Recall also that F Q is the unique cubic extension such that

F Q is Galois but not ab elian and that F is any cubic subeld We may take F Q and F

p p p

where is a ro ot of X F X The other ro ots of X X are

p p

Thus GalF Q is generated by two elements and dened by and

p p

and

Thus and is a uniformiser for F and We have

Gu u

c u mo d u

u H

p p

u u H u

Recall that is the unique typ e such that j The case of F corresp onds to the order

Q 3) (

homomorphism

Z GL Q

determined by

with where det and but Recall also that F is the degree ab elian extension of Q

p p

We also let top ologically generated by and denote the unique norm subgroup in Q

element of I of order and we let F denote the xed eld of some Frob enius lift of order

F Q

p p

where is a ro ot of X X and a ro ot of X To verify Q We claim that F

p p

We must check that F Q is ab elian and that N F this set F Q

F Q

p p

To see that F Q is ab elian note that if is one ro ot of X X then the contains and

p p p p p p

p p p p

other ro ots are and where

p p

for deniteness we cho ose Z Note that N and N

F Q F Q

3 3

is generated by three commuting elements and of resp ective orders Note that GalF Q

and They may b e dened by

p p

and

p p

and

p p p

p p p p

and

I Cho ose an element I Then h i As we may alter our choice

F Q

3 3

p p

we see that of so that h i As so We will rename as and

supp ose it chosen so that Thus GalF Q is generated by elements and satisfying

and

of order and hence coincides with our previous denition The element is the unique element of I

F Q

The element is a Frob enius lift of order and so we may take F to b e its xed eld ie F Q

where is a uniformiser for F We are not asserting that equals the element denoted e in

section One can check that

mo d

Note also that h i pro jects isomorphically to the quotient of GalF Q by the wild inertia subgroup

We conclude

c mo d u

u H

u u mo d u H

for g g h i t f

g g g

Recall that is the unique typ e such that j The case of F corresp onds to the order

Q ( 3)

homomorphism

GL Z Q

determined by

where det and but Recall also that F is the degree ab elian extension of Q

p p

top ologically generated by and We also let denote with norm subgroup in Q

the unique element of I of order and we let F denote the xed eld of some Frob enius lift of

F Q

order

p p p

where is a ro ot of X and a ro ot of X Q We claim that F

p p p

To verify this set F Q Then F Q is ab elian and so we must check that

p p

F contains and N But note that we have the identities N

F Q F Q

3 3

p p

N and N

F Q F Q

3 3

Note that GalF Q is generated by three commuting elements and of resp ective orders

and They may b e dened by

p p

and

p p

and

p p p

and

so h i As we may alter our choice I Cho ose an element I

F Q

p p p

we see that so We will rename as h i As of so that

Thus GalF and supp ose it chosen so that Q is generated by elements and

satisfying

and

of order and hence coincides with our previous The element is the unique element of I

F Q

denition The element is a Frob enius lift of order and so we may take F to b e its xed eld ie

F Q where is a uniformiser for F We are not asserting that equals the element

denoted e in section One can check that

mo d

Note also that h i lifts tame inertia

We conclude

c mo d u

u H

u u mo d u H

t f for g g h i

g g g

The case of F Here i ZZ and we will let denote the unique lifting of i to Z with

corresp ond to the homomorphism Recall that is the unique extended typ e whose restrictions to G

i Q

Q GL Q

determined by

where det and but Recall also that F is the degree ab elian extension of Q

p p p

and with norms the subgroup of Q top ologically generated by

p p

Q which corresp ond resp ectively to We let and denote the elements of GalF

and

p p

We claim that F Q where is a ro ot of X and a ro ot of X

p p p p

To verify this set F Q so F Q is ab elian and we must check that

p p

F contains and But note that N N

F Q F Q

3 3

N and

F Q

p p p

mo d N

F Q

Note that is an element of I of order but also has order and is an element

F Q

of I of order Thus

F Q

p p

and

p p

and

p p

and

p p p

is a norm from Q b ecause has norm so Moreover

p p

where The determination of is more delicate Let b e a ro ot of X so

p p p

Y Q is unramied and is a ro ot of Y Y Thus Q

p p p

mo d Frob

p p p p

The norms from Q to Q The norms from are generated by Z and

p p p p p

Q to Q are generated by Z and The norms

p p p p p

from Q to Q are generated by Z and

Thus

p p p p

Q Q Frob GalQ GalQ

p p

Q GalQ

p p p p p p

corresp onds to Q As has norm to Q the pro duct of

p p

we conclude that Frob xes Thus Frob and

In other words

Then h i As we may alter our I Cho ose an element I

F Q

p p p

choice of so that h i As we see that and so We will rename

Thus GalF Q is generated by elements and as and supp ose it chosen so that

satisfying

and

The element is a Frob enius lift and it has xed eld F Q where is a uniformiser for F

One can check that

We conclude

c mo d u

u H

H u u mo d

u u mo d u t

for g g h i t f

g g g

1 u mo d u f 1 1 u f 1

3 3 3 3

Pr oof of Theorem

In this section we will keep the notation of x and either x or x dep ending if we are working with

or S We will set in the case of S We will write F for F and F for F If G resp M is S

a nite at O group scheme resp Breuil mo dule over O we will write G resp M for the unramied

F F

base change to O

Rank one calculations We recall from Lemma that the only O mo dels for ZZ are

and G In ZZ G and G and the only O mo dels for are G

O O F

F F

admits unique descent data over Q compatible with the each case by Lemma the base change to O

canonical descent data on the generic bre of ZZ resp over Q We will refer to this descent data

as the standard descent data on these nite at group schemes

o calculations Rank tw

Lemma The group of extensions of M by M over O is parametrised by c F The

Breuil module M c corresponding to c is free of rank two over F uu with a basis fe e g such

that

M hue u e ce i

ue e u e ce e

N e N e cu e

The standard descent data on M and M extends uniquely to descent data on M c The

corresponding representations G GL F are of the form

and are peu ramie Any such peuramie extension arises for a suitable choice of c

Proof The classication of extensions of Breuil mo dules follows from Lemma Next we compute N

on M M c We will not in fact need the result of this computation of N but the calculation

is given here as a representative sample of calculations needed later in more complicated settings By the

last part of Lemma N e and N e g e for some g F uu divisible by u In F uu

we compute

c G u u u

Using the dening prop erties of N we compute in F uu

N e N u e ce

N u e ce

u e u N e

u u e ce cu e u N e

cu e

since u N e u M u u M u M and the Frob eniussemilinear must kill u M Thus

cu c u

N e cu ue ue e cu e

c c c

To see existence and uniqueness of the descent data on M c compatible with the standard descent

data on M and M we will work on the side of nite at group schemes Because G c is

the unique extension of G by G with generic bre G c F by Lemma uniqueness

reduces to the corresp onding questions on the generic bre which follows from the injectivity of

H G H G

to exhibit a continuous representation G GL F of the form For existence it suces

which is p eu ramie but not split with restriction to G corresp onding to a lo callo cal nite at O group

F F

such that E furnishes the desired scheme G By Theorem of Man we can nd an elliptic curve E

example This also proves the nal two assertions of the lemma

Lemma Suppose that F is a total ly ramied abelian cubic extension of Q and suppose that G is a

locallocal nite at O group scheme kil led by such that G F is an extension of ZZ by Then

e e

F for some nite at Q group scheme G G F G

Q O

Proof As in the pro of of the last lemma we see that M G M c for some c F As the only

action of GalF Q on a one dimensional F vector space is trivial we see that each such c gives a class

in H G which is invariant by GalF Q But

Gal F Q

1 3

H G H G

and so the lemma follows

Lemma The group of extensions of M by M over O is isomorphic to the group of linear

polynomials c c u in F u The Breuil module M c c u corresponding to c c u is free of rank

two over F uu with a basis fe e g such that

M c c u hu e ue c c ue i

u e e ue c c ue e

M c c u admits unique descent data compatible with the standard descent data on M Each

and M As c c vary over F the corresponding descent to Q of the generic bre of G M c

c u runs over al l extensions of by ZZ The corresponding representation of G is peu ramie if

and only if c

Proof The classication of extensions of Breuil mo dules follows from Lemma The uniqueness of

the descent data on M c c u follows from Lemma and the injectivity of H G

as in the pro of of Lemma Note that Frob enius vanishes on the Dieudonne mo dule of H G

G c c u if and only if c Thus the lemma will follow if for each torsion extension G of

es ramie we can nd a nite at O group scheme G such that by ZZ over Q which is tr

the generic bre of G is isomorphic to G F

the closed bre of G is lo callo cal

and Frob enius is not identically zero on D G

for some v mo d where the three choices of v mo d The splitting eld of G contains a cub e ro ot of v

corresp ond to the three choices of tres ramie The calculations in x of Man give explicit additive

reduction elliptic curves E and E over Q with E E G where E acquires go o d sup ersingular

reduction over the nonGalois cubic ramied extension

Q X X X b

with b v and E acquires go o d sup ersingular reduction over the ab elian cubic ramied extension of

Q with norm group generated by v mo d Q The appropriate G are provided by the torsion on the

Neron mo dels of E or E over O

Corollary Suppose that G is a nite at O group scheme and that fg g is descent data on G

Q corresponds to Then G O such that G fg g

F Q

M G M c c u

for some c c F with c

Proof From the connectedetale exact sequence and its dual we see that G F must b e lo callo cal The

corollary now follows from Lemma and the discussion of x

Lemma The group of extensions of M by M over O is isomorphic to the group of linear

polynomials b b u in F u The Breuil module M b b u corresponding to b b u is free rank two

over F uu with a basis fe e g such that

M b b u hue ue b b uei

ue e ue b b ue e

This extension splits over an unramied extension if and only if b If F Q is nonabelian then any

descent data on M b b u compatible with the standard descent data on M satises

d d

1 1 1

b e e b e e e H u e e H u e h ue

3 3 3

where

h bH

Proof The classication of extensions of Breuil mo dules follows from Lemma The computation of

which of these split over an unramied extension follows from Lemma and Corollary

Since Now supp ose that F Q is nonab elian By Lemma the only issue is to compute h

mo d by evaluating the congruence H

b b mo d u M b b u

on ue b b ue and comparing constant terms of the co ecients of e on b oth sides we get

u H

b h h bH h bH j h

3 3 3 3

3 3

mo d see x in F where we have used the equality H

is a ro ot of T In other words h we must have h Since H T bH

3 3

1 1

u h u are forced by the u H u and h a for some a F Since H bH

3 3

identity we see that h bH a for the same a F The identity

b mo d u M b b u

so a h then implies h

Lemma The group of extensions of M by M over O is isomorphic to the group of quadratic

polynomials vanishing at b b uu in F u The Breuil module M b b uu corresponding to

b b uu is free rank two over F uu with a basis fe e g such that

M b b uu hu e u e b b uuei

u e e u e b b uue e

This extension splits over an unramied extension if and only if b If F Q is nonabelian then any

descent data on M b b uu compatible with the standard descent data on M satises

ue u e h u e b e H b e H

3 3 3

where

bH h

will b e very imp ortant in x The pro of of this lemma is essentially the bH The sign in h

same as that of Lemma but we rep eat it anyway

Proof The classication of extensions of Breuil mo dules follows from Lemma The computation of

which of these split over an unramied extension follows from Lemma and Corollary

Since Now supp ose that F Q is nonab elian By Lemma the only issue is to compute h

mo d by evaluating the congruence H

b mo d u M b b uu b

on ue b b ue and comparing constant terms of the co ecients of e on b oth sides we get

u H

b h h bH h bH j h

3 3 3 3

3 3

see x in F where we have used the equality H

we must have h Since H In other words h is a ro ot of T T bH

3 3

1 1

u h u are forced by the u H u and h a for some a F Since H bH

3 3

identity we see that h bH a for the same a F The identity

b mo d u M b b u

h so a then implies h

Rank three calculations

Lemma Suppose that G is a nite at group scheme over O which is kil led by Suppose that there

G G G G is a ltration by closed nite at subgroupschemes G G G such that G

F descends to Q in such a way that it is a tr G Suppose nal ly that G and GG es ramie

extension of by ZZ Then

G G GG

compatibly with the extension class structure

M GG Using Lemmas and we see that we can write Proof Let M M G and N

M F uu e F uu e F uu e

M hue u e be ue c c ue f e i

for b c c F with c and with f F uu It suces to show b Since we must have u M M

we see that

c c uu e be u ue c c ue f e u e bc bc u u f e M

The Breuil mo dule N is spanned as a F uu mo dule by e and e so by Lemma u must divide

bc bc u u f As c we must have b as desired

we get the following Combining this with Lemma and the injectivity of H G H G

corollary which is also the rst part of Theorem

Corollary The natural map

H G Ext

is zero

Conclusion of pro of of Theorem Consider rst the case of F We still have to explain why

G ad H H I F

do es not map to zero in H I F is zero Supp ose x H G ad

By our hyp othesis on x we may cho ose a totally ramied ab elian cubic extension F Q such that x

restricts to zero under the natural map H G ad H G F Then the image of x under the

H G is the image of some xe H G under the natural map natural map H G ad

e e

F F

1 1

H G H G The element xe parametrises a nite at F group scheme H which is an

e e

F F

1 1

extension of by ZZ and which is a sub quotient of the restriction to G of the extension of by itself

paramitrised by x It follows that H has a nite at mo del H see Lemma and the sp ecial bre of

by itself parametrised by x splits over F and this is H must b e lo callo cal if xe then the extension of

clear while if xe we would otherwise get a contradiction from the connectedetale sequence By Lemma

we may therefore lift xe to H G Using the commutative diagram

H G H G

res res

H G H G

e e

F F

1 1

and noting that the right hand vertical map is injective we conclude that

H G G ad x H

a contradiction with the hyp othesis that even the image of is in the image of H G H G

x in H I F is nonzero

Now consider the case F F which is nonab elian over Q We must show that

Ext H I F

is zero

An element x Ext group scheme G killed by and descent data gives rise to a nite at O

by itself corresp onds to the extension of fg g for F Q on G G F such that G fg g

O Q

classied by x Let N denote the Breuil mo dule for G and let N N F According to Lemmas

and we see that we can write

N F uu e F uu e F uu e F uu e

with

N hu e ue c c ue u e au a u e ue c c ue b b ue he i

where h F uu is some p olynomial and where a a b b c c F with c as is tres ramie By

Lemma what we must show is that a

in F by x By Lemmas and the action b is determined by Note that H

u e h ue ue b e H u e g u e b e H b e H

3 3 3 3 3

ue h e u g b e H ue e G

3 3 3 3

u F uu and h and h are as in Lemmas and resp ectively u G where g

3 3 3 3

Due to the requirement b N N we must have

b ue c c ue b b ue hue N

and this element is obviously equal to

h e e H e c c uH h e G e g e H uH

3 3 3 3 3 3

3 3

e H e huH e g H b b uH

3 3 3

3 3

We now try to express this as a linear combination of the generators of N listed in while working

mo dulo hu N u e i N hu mo d u we arrive at the in F and huH Using that H

3 3

expression

u g c H

3 3

ue c c ue b b ue he H u e au a u e

h b H ue ue c c ue F

3 3 3

where

hu H g h b b uH c c uH G u uH F

3 3 3 3 3 3 3 3

u h b H c c uH u g a a uc H

3 3 3 3 3

u mo d u The condition mo d u and F uu g In particular c H in F uu

3 3 3

mo d u can b e reformulated as u g c H

3 3

cH g

cH But a direct calculation using g u mo d u we have to have F Since F

3 3 3

gives and the denition of F

ch h bH ch bH ch bg F

3 3 3 3 3 3

3 3

so the nonvanishing of c forces

h h bH

3 3

Lemmas and give us the values

h bH h aH

3 3

and so a This completes the pro of of Theorem and hence of bH Thus a bH

3 3

Theorem

Proof of Theorem

In this section we will keep the notation of x and either x or x dep ending if we are working with

so that c mo d u Note the signs We will write S or S We will set in the case of S

F for F F for F and I for I If G resp M is a nite at O group scheme resp Breuil mo dule

over O we will write G resp M for the base change to O

F F

Rank one calculations We remark that with our choice of p olynomials H u in x and x any

has an action of h i via b and b the action of b eing Frob semilinear ob ject M in DD

F Q

and H u Z u we see that b must commute with Also since and commute H by

Corollary

We recall from Lemma that the only mo dels for ZZ over O are Gr for r

and the only mo dels for over O are Gr for r ZZ with G

F O

In each case the base change to O admits unique descent data over Q such with G

that descent of the generic bre to Q is ZZ resp See Lemma We will write G resp

fg g We will also let M fg g resp Gr O G for the corresp onding pair Gr O

F F

r r

or the underlying In particular for resp M denote the corresp onding ob ject of DD

F Q

F uu mo dule has the form F uu e with e the standard generator though we write e rather

than e if is understo o d

We have the following useful lemma

Lemma Let r e be an even integer The descent data on M is determined by

r r

e e H 1 u e b e e b e

and the descent data on M is determined by

r r

u e e e H b e e b e

if and only if r or on D G if and only if on D G In particular

r r

Proof Certainly b e e We have already seen in Lemma that descent data must exist in each case

F uu so that so our task is to compute the unique units

[

e b e e e

corresp onds to generic b er descent data for the mo d cyclotomic or trivial character on G The case of

follows from Lemma

From the condition

r r

b u e b u e

so u we get

The nonzero morphisms M M are given by e u e and the nonzero morphisms M

and b e e on M In M are given by e u e Thus it suces to check that b e e on M

b oth cases we have shown that b e e and so we only need to check that on D G and D G

4 4

arising from ZZ and group scheme maps ZZ That is we have to show that the O

the canonical generic bre descent data induce the identity on the sp ecial bres This is easy

corresponding to a nite at group scheme G and let fg g Lemma Let M be an object of mo d

can be ltered so that each graded relative to Q Assume that G fgbg be descent data on G G O

Q F

piece is isomorphic to ZZ or and so that the corresponding ltration of M fgbg in DD has

F Q

with r f g and f g Then on M u M and successive quotients of the form M

j j

there exists a basis fe g of M over F uu so that for al l j

e M

e is an eigenvector of the F linear map b on M

and this surjection sends e onto e lies in the part of the ltration of M which surjects onto M

j j

over F uu the standard basis vector e of M

j j

Proof Since acts linearly on M u M and the action of must b e semisimple The eigenvalues

of are all equal to so necessarily on M u M

We now argue by induction on the numb er of JordanH older factors in the generic b er the case of length

b eing clear Thus we can assume we have a short exact sequence in DD

F Q

N M M

so the lemma is known for N We just have to nd e M mapping onto the standard basis vector

e in M such that e is an eigenvector of b Since M M is a surjective map of F

r r

vector spaces which is compatible with the semisimple F linear endomorphism b on each side we can nd

e M mapping onto e with e an eigenvector of b say b e e Since

p p

b b e b b e b e b e

the element e e b e maps to e and is an an eigenvector for b Also e M is

so e M b invariant and b commutes with

Mo dels for

corre osition There exists a unique object G fg g of FD such that G fg g Prop

F Q I

sponds to If we set M fgbg M G fg g then M fgbg is an extension of M by M

in DD Moreover Frobenius is not identical ly zero on D G

F Q

corresp onds to Proof Let G fg g b e an ob ject of FD such that G fg g and set M fgbg

F Q I

M for a Breuil M G fg g As in the discussion following Theorem we have canonically M F

mo dule M over O with b acting as By Lemma there is a short exact sequence of Breuil

mo dules over O

Ms M Mr

with r s f g and this is compatible with descent data after base change to O in the sense that we

obtain an exact sequence

M M M

s r

compatible with descent data Because is tres ramie it follows that j is nonsplit so the sequence

Ms M Mr

is nonsplit

is selfdual in order to prove r s we We rst show that we must have r s Since

may use Cartier duality and Lemma in order to reduce to the case where r s e We will

rst rule out cases with r s and then the case r s

By Lemmas and we can write

s r

he i M hu e u e M F uu e F uu e

for some h F uu so that

s r

he e u e e u e

and

p p

r s

e b e e b e

Recall from Lemma that the parameter h gives an isomorphism of abstract groups

s r

Mr Ms F uu fu t u t jt F uu g Ext

mo d

It is easy to see that

b M M b b on M

r r

if and only if b u e he M and b e b u e he or equivalently

p p p

r s s

u mo d u hu h

This says exactly that

j r s mo d

for any j s with a nonzero u term app earing in h

If r s this would force h mo d u yet fu t u t jt F uu g contains all multiples of

u so

M M M

is split a contradiction

When r s or r s we see that h h mo d u yet

s r

u F uu fu t u t jt F uu g

may b e changed in order to arrange that so the choice of e

h F

though making this change of basis of M may destroy the diagonal form of b Since

Ms M Mr

it can b e assumed that h Then V is nonsplit necessarily h so by rescaling e e mo d u M e

by Theorem and

r s r s

e c u e u e mo d u M

This forces r s In particular r s is ruled out

j is induced by the Breuil mo dule map For r s a splitting of the generic b er

M M

dened by

e u e u f e u u e e uf u u e

where f F uu satises f f u ie f u u and a constant c F can even b e added

to this if But j must b e nonsplit so this rules out r s

s r

The remaining case with r s is r s In this case fu t u t jt F uu g contains all

multiples of u But we have j mo d for all j s such that a nonzero u term app ears in

h so again at the exp ense of p ossibly making the b action nondiagonal we may assume

h c c u

ue the commutativity of b and amounts to e h for some c c F Writing b e

for some b F The condition b e e forces b so b still has u b so h h h

4 4

diagonal action This analysis shows that the map of F vector spaces

Ext M M Ext M M

DD mo d

1 1

F Q F

has at most a dimensional image If c c then the Breuil mo dule map

F M M

dened by

c c u e e c u e u u e

Thus the image of gives a splitting of the corresp onding representation of G

Ext M M Ext

DD F G

1 3

F Q

is nonsplit the pair c c corresp onding to a mo del of j is at most one dimensional and b ecause

satises c c

At this p oint we treat the cases separately Consider rst the case We must have

ue u e h u e b e H b e H

3 3 3

u F uu lies in F uu b ecause b commutes with b Evaluating b where h

u mo d we arrive at b mo d u M on u e c c u e M and using our knowledge of H

c c h h

3 3

F with b ecause c c c c F which is imp ossible for h

Now let us turn to the case still in the case r s In this case Ext

F G

3 3

Ext is injective and so by we see that the image of

F G

Ext M M Ext

DD F G

1 3 3

F Q

is at most one dimensional Thus to exclude the case r s and it suces to show that this

is tres ramie By Prop osition of Man there is an elliptic image contains the p eu ramie line as

which has sup ersingular reduction over Q curve E with a nonsplit p eu ramie extension

is injective in the is nonsplit again b ecause H G H G j of by The representation

F F

F case Let N b e the Breuil mo dule corresp onding to the torsion on the Neron mo del of E

so N admits descent data fgb g via the universal prop erty of Neron mo dels The ltration of induces a

short exact sequence in DD

F Q

N fgb g M M

b a

Q for some even a b with a b The Neron mo del of E has lo callo cal torsion and

p p

and j the induced lo callo cal integral mo dels G and G of the diagonal characters j

Q Q

3 3

must b e the unique lo callo cal mo dels uniqueness follows from Corollary of Ra Moreover Corollary

takes the order group schemes G and G to the of Ra makes it clear that base change to O

It follows and j integral mo dels that lie in the middle of the wellordered sets of integral mo dels of j

F F

that a b so the map

Ext M M Ext

DD F G

1 3 3

F Q

indeed hits the p eu ramie line

We next exclude the case r s As a rst step we check that there is at most one p ossibility for

the underlying Breuil mo dule M ignoring the extension class structure if r s We can write

he i M hu e u e M F uu e F uu e

for some necessarily nonzero h F uu with

he e u e e u e

and

p p

b e e b e e

The combined conditions b M M and b b on M are equivalent to

u mo d u hu h

Since fu t u t jt F uu g contains u u and all multiples of u we may change e at the exp ense

of p ossibly losing the diagonal form for b so that h cu for some c F Since h is necessarily nonzero

we may rescale to get h u so there is indeed at most one p ossibility for the underlying Breuil mo dule M

when r s

Again we treat the cases separately Consider rst the case We have seen ab ove that

there is an extension E corresp onding to a nonsplit p eu N fgb g of M by M in DD

F Q

ramie extension of by Pulling back E by a nonzero map

M M

given by e u e we get an extension E of M corresp onding by M in DD in DD

F Q F Q

3 3

to a nonsplit p eu ramie extension of by The underlying Breuil mo dule of E must b e isomorphic to

in M for our uniquely determined M with h u By the injectivity of H G H G F

F F

M cannot admit descent data giving rise to a tres ramie element the case we conclude that F

in Ext This rules out the case r s and

F G

3 3

Now turn to the case r s and We will show that with the Breuil mo dule M constructed

M do es not admit descent data relative to F Q ab ove with h u the Breuil mo dule M F

with b without loss of generality One checks that N e N e so

We must have

ue u e h u e b e H b e H

3 3 3

F uu The condition F uu As usual since b and b must commute we have h for some h

3 3

b M M is equivalent to

b u e u e M

which amounts to

mo d u H u H h

3 3

H H h

3 3

u u e u e b u e u e H u e H

As N we have

b b

u e M gives on M Evaluating this identity on u e

H H h

3 3

H h

F uu for some g u g is a cub e Thus h so h

3 3 3 3

u mo d u we compute Since H

u mo d u H H

3 3

mo d u H h u

and

g g

3 3

in F This is absurd This rules out all p ossibilities for r s aside from r s Uniqueness now

follows from Corollary

From Theorem of Man and Prop osition B of CDT we see that there is an elliptic curve E

such that E By the F over O Q and has typ e Let E denote the Neron mo del of E

F E Q

we see that E has descent data over Q As in x we see that I annihilates Neron prop erty of E

M E in DD the Dieudonne mo dule of E F Thus M and it follows that Frob enius

F Q

is nonzero on D G

Completion of pro of of Theorem

corresponds to Lemma Let G fg g be the unique object of FD such that G fg g

F Q I

Set M fgbg M G fg g The natural map of groups

Ext M fgbg M fgbg Ext M M

DD DD

1 1

F Q F Q

3 3

fgbg M and pul lback by M M fgbg is zero using pushout by M

M fgbg M fgbg and let M fgbg b e its Proof Let M fgbg represent a class in Ext

F Q

image in Ext M M By Lemma M F M with b and

F Q

M F uu e F uu e M hu e u e c c ue i

with c c F Also

u e e u e c c ue e

and

p p

e b e b e ue e h

F uu for some h

The prop erties b M M and b b on M amount to

u h c h

If c then N so b b on M From this we readily see that M fgbg is so h

split in DD as desired

F Q

Now assume c we will deduce a contradiction Consider the rank three Breuil mo dule with descent

data

N fgbg M fgbgM fgbg

M Then N has an ordered basis fe e e g with resp ect to which where M M

N hu e u e e u e he b b ue i

for some b b F and h a a u a u F uu dened mo dulo fu t u t g see Since

our base eld F has absolute ramication degree N contains

u e u u e he b b ue hu e e h u b b ue

N u From the list of generators of N it is not dicult to check that in the ab ove expression for u e

must divide the co ecient of e Thus a

Since a M has basis fe e g and We must have N he i M

M hu e u e a u a u e i

Since for M satises

e a u a u e u e e u e

which using Theorem contradicts Prop osition it follows immediately that mo d uM

Corollary The natural map

H G Ext

is zero

Theorem and hence Theorem now follow from the rst case of the following lemma We

include the second case to simplify the pro of

Lemma The maps of groups

Ext M M M M Ext Ext Ext

DD F G DD F G

1 3 3 1 3 3

F Q F Q

3 3

have images inside the line of extension classes that split over an unramied extension of Q

Proof Since

Z H G Z H G

is injective and induces an isomorphism b etween the subgroups of unramied classes it suces to check that

M M Ext Ext M M Ext Ext

mo d DD mo d DD

1 1 1 1

F Q F F Q F

3 3

have images consisting of elements split over an unramied extension of F By Cartier duality it suces to

consider only the second map

Consider a representative M fgbg of an element in Ext M M Lemma ensures

F Q

that we can write

M F uu e F uu e M hu e u e he i

for some h c c u c u with c c c F and

u e e u e he e

We have

p p

e e ue h b e b e

F uu and the condition b M M is equivalent to for some h

hu h u mo d u

so c c The Breuil mo dule extension class M over O ignoring descent data therefore only dep ends

M M determined by on the parameter c F We then have a splitting F F

F F

3 3

e ae e

F satises a a c where a

Proof of Theorems and

In this section we will keep the notation of x and x We will write F for F F for F and I for I

i i

If G resp M is a nite at O group scheme resp Breuil mo dule over O we will write G resp M

F F

for the base change to O

Rank one calculations We remark that with our choice of p olynomials H u in x any ob ject M

in DD has an action of h i via b and b The action of is Frob semilinear Since and

F Q

commute and H u Z u we see that b must commute with see Corollary

By Lemma the only mo dels for ZZ over O are Gr for r with G

and the only mo dels for over O are Gr for r with G ZZ

F O

Lemma ensures that the base changes to O admit unique descent data over Q such that

descent of the generic bre to Q is ZZ resp We will write G resp G for the corresp onding

r r

resp M denote the fg g We will also let M O fg g resp Gr O pair Gr

F O F O

F F

r r

corresp onding ob ject of DD

F Q

We have the following useful lemmas for which the pro ofs are identical to the pro ofs of Lemmas

and

Lemma Let r e be an even integer The descent data on M is determined by

r r

b e e b e e e H 1 u e

and the descent data on M is determined by

r r

e e H u e b e e b e

In particular on D G if and only if on D G if and only if r or

r r

Lemma Let M be an object of mo d corresponding to a nite at group scheme G and let fg g

can be ltered so that each graded over Q Assume that G fgbg O be descent data on G G

Q F O

3 F

piece is isomorphic to ZZ or and so that the corresponding ltration of M fgbg in DD has

F Q

with r f g and f g Then on M u M and successive quotients of the form M

j j

there exists a basis fe g of M over F uu so that for al l j

e M

e is an eigenvector of the F linear map b on M

and this surjection sends e onto e lies in the part of the ltration of M which surjects onto M

j j

over F uu the standard basis vector e of M

j j

Mo dels for Recall that we are assuming that has the tres ramie form

and is not split over F We will let DD consisting of denote the full sub category of DD

F Q I F Q

3 3

ob jects M for which the ideal I acts trivially on M uM F

F Frob

9 3

is an exten such that M fgbg Prop osition Suppose that M fgbg is an object of DD

F Q I

sion of ZZ by Then we have an exact sequence

M M M

s r

F with b Frob with r s or Moreover we can write M M

mapping to g with e the standard basis element of Ms and e where M has an F uu basis fe e

the standard basis element of Mr More precisely we have the fol lowing exhaustive list of extension class

N denotes the monodromy operator described in Lemma possibilities al l of which are wel ldened

r s The natural map

M M Ext Ext

F G

3 3

F Q I

is an isomorphism with elements parameterized by pairs c c F corresponding to

cu e e cu e i u e e u e M hu e u e

so N with

p p

e b e e b e

d d

e e c u e u e e

The pairs with c are the ones which generical ly split over F In al l cases mo d uM

The natural map r s

Ext M M Ext

DD F G

1 3 3

F Q I

is an isomorphism with elements parameterized by pairs c c F corresponding to

cu e e cu e i u e e u e M hu e u e

so N with

p p

e e b e b e

d d

e u e e e c u e

The pairs with c are the ones which generical ly split over F In al l cases mo d uM These

cases are Cartier dual to the cases above

r s The natural map

Ext M M Ext

DD F G

1 3 3

F Q I

is an isomorphism with elements parameterized by pairs c c F corresponding to

M hu e u e cu e i u e e u e cu e e

so N with

p p

e e b e b e

d d

e u e e u e c u e

The pairs with c are the ones which generical ly split over F In al l cases mo d uM

r s The natural map

M M Ext Ext

F G DD

3 3 1

F Q I

is an isomorphism with elements parameterized by pairs c c F corresponding to

M hu e u e c c u e i u e e u e c c u e e

c u e and it is easily checked that N e and N e

p p

e b e b e e

d d

e e e e c c u c u e

In particular mo d uM if and only if c

is equivalent to the vanishing In the rst three cases the peu ramie condition on a class in Ext

F G

3 3

of c In the fourth case it is equivalent to the vanishing of c

Proof By Lemma we have an exact sequence

M M M

s r

with r s f g As usual

s r

M hu e u e he i

In the cases r s and as in the pro of of Prop osition we may take h We will show

that in the case r s we also have h Following the pro of of Prop osition we may supp ose

that h F Without loss of generality we can take h and lo ok for a contradiction Again following the

pro of of Prop osition and using

e i M hu e u e

we nd that e e mo d uM Also

e u e

e u e h ue

for some h u F uu which must actually lie in F uu using as usual the fact that b and

b commute Thus

e e

b e b b

b e

h h e mo d uM

on M uM have matrices The inverse linear maps

with resp ect to the basis fe e g so that h h Thus h On the other hand evaluating

b b mo d uM on u e e and comparing co ecients of e gives h a contradiction

Thus if any case r arises the underlying Breuil mo dule must b e a split extension

M F uu e F uu e M hu e u e i

e u e e u e

so N with

e b e e b e

p p

e e b e b e

We also have

d d

1 1

e H u e e H u e h ue

3 3

d d

and on M Evaluating this on u e for some h F uu Since N we have

r r

comparing co ecients of e gives h u u h u H u This forces h u if r If

r it forces h u c u for some c F We will show c c Indeed evaluating the

congruence

b b b mo d uM

gives on e

e c c e mo d uM

so that c c On the other hand the congruence

b b b mo d uM

gives

p p

c c

p p

F so c c Thus c c and h for r as well Thus for r and the in M

is split a contradiction Breuil mo dule with descent data M is split so

This rules out the p ossibilities and Using Cartier duality we can also rule out

and We are left with the four p ossible pairs r s as asserted in the prop osition and must determine

which p ossibilities arise in each case

Next consider the case r s Using the same analysis as in the r s case in Prop osition

we nd that the p ossibilities for the Breuil mo dule M are the ones in the statement of the prop osition

and N is easy to check though we only know that

p p

e e ue b e h b e

u F uu The conditions for some h

b M M b b on M

are equivalent to

mo d u h h

4 4

a The solutions to this are h u for a F Replacing e by e au e preserves our standardized

form but makes h

p p

e e b e b e

The wild descent data must have the form

d d

e e e u e h ue

for some h F uu The conditions

d d d

M M on M

recall N are equivalent to

h mo d u h u

whose solutions are

h c u u

for some c F Since N we have

d d

c c

Using Lemma and Corollary we see that all of these p ossibilities are welldened We also

see that I annihilates MuM F It is straightforward to check that generic splitting over F which is

equivalent to generic splitting over F is equivalent to c and that such splitting is compatible with

descent data ie descends to Q if and only if c c For dimension reasons the map on Ext s is

therefore an isomorphism

Now consider the case r s Here we have

M hu e u e he i

for some h F uu with

u e e u e he e

and

p p

b e e b e e

In order that

b M M b b on M

it is necessary and sucient that

u mo d u h h

But fu t u t jt F uu g is spanned by u u u u u u u u and all multiples of

u so we may supp ose

h c c u cu

for some c c c F at the exp ense of p ossibly losing the diagonal form of b

The mono dromy op erator satises

c u c u c u e N e N e

Since the wild descent data must take the form

d d

e e u e u e h e

for some h F uu we compute

u e he H u u u e he f ue

where

f u hH u u u H h u huH 1

1 1

3 3

M Thus in order that M it is necessary and sucient that f satises

f mo d u

Using and H u mo d this amounts to

h c u mo d u

However N M u M so

d d

mo d u M

when evaluated on M This gives

h mo d u

Since h is a cub e mo dulo u by we must have c and so N mo d u M Thus

commute mo dulo u M when evaluated on M and so we get

h mo d u

and h is a cub e mo dulo u

On the other hand with c we see from that

h mo d u

Because h is a cub e mo dulo u so we get the slight improvement

h mo d u

Combining this with the vanishing of c we deduce from that f c u mo d u so by

h c mo d u

This forces c so N Thus and commute on M so

in F uu Using h cu this b ecomes via

h u

h c u u

for some c F As b efore we get c c

Now we diagonalise b Since we have

p p

e e ue b e h b e

F uu the conditions for some h

b M M b b on M

are equivalent to

mo d u h h

4 4

which is to say

u a h

for some a F Replacing e by e au e then puts us in a setting with a Thus all extensions

have the form asserted in the prop osition It is easy to check that in each case I annihilates MuM F

Pushout by the nonzero map M M in DD induced by e u e takes our examples

F Q

to our examples compatibly with the lab elling of parameters c c as in the statement of the prop osi

tion Thus all p ossibilities for c c do o ccur and we get an isomorphism of Ext s as asserted Moreover

generic splitting over F which is equivalent to generic splitting over F is equivalent to c and such

splitting is compatible with descent data ie descends to Q if and only if c c

Using Cartier duality and the case r s we see that in the case r s the map of Ext s

is an isomorphism It is easy to check that the ob jects in our asserted list of p ossibilities for r s

are welldened and that pullback by the nonzero map M M induced by e u e takes these to

our examples compatibly with the lab elling of parameters c c

Finally we turn to the case r s Cho osing a basis with resp ect to which b has a diagonal

action the conditions

b M M b b on M

are equivalent to

u mo d u hu h

to get Since fu t u t jt F uu g consists of multiples of u we may change e

h c c u

for some c c F with

p p

e ue b e h e b e

so h h F uu Feeding this into we get h for some h a for some a F

4 4 4

Replacing e by e ae returns us to the setting with diagonal b action and preserves the standardizations

we have made so far

It is easy to compute N e c u e and we know N e The wild descent data is

d d

e e e e h e

for some h F uu Using the congruence for t in x the identity

d d

t N

on M amounts to the condition

h h c u c u

whose solutions are

h c c u c u

for some c F The identity

d d

mo d u M

implies c c Thus

V e e ce mo d u M b b b e c e mo d u M

F if and only if c c Thus I annihilates MuM

By Lemma and Corollary it is easy to see that all of these ob jects are welldened The kernel

Ext M M Ext

F G

3 F

F Q I

consists of pairs c c where generic splittings are induced by any of the nonzero Breuil mo dule maps

M M

dened by

e u e cu cu e uc u cu e u u e c cu e

which are split over F or equivalently split over with c F Thus the pairs c c corresp onding to the

F are exactly those for which c c The map

Ext M M Ext

DD F G

1 3 3

F Q I

is therefore injective b ecause the splitting given ab ove resp ects descent data if and only if c c

It remains to establish which of the given extensions of Breuil mo dules corresp ond to p eu ramie extensions

s in the of ZZ by over Q We noted ab ove that the maps among the Ext

F Q

cases induced by pushoutpullbac k along e u e are compatible with the parameterization by pairs c c

With a little more care one checks that the maps

M M Ext M M Ext M M Ext

DD DD DD

1 1 1

F Q I F Q I F Q I

3 3 3

induced by e u e send the pair c c in the middle to the pair c c c on either end to construct the

necessary commutative diagrams of short exact sequences in the two cases use the maps

e e u e c e e e e e c e u e

resp ectively This reduces us to checking the case

and of by which are nonsplit over F arise By Corollary the two tres ramie extensions

from elliptic curves E and E over Q for which is p otentially BarsottiTate with extended typ e

see x Let G denote the torsion in the Neron mo del of E over O From the universal prop erty of

j j F

inherits descent data fg g over Q By the same argument used O Neron mo dels we see that G G

F j O

Moreover we see that F on D G at the end of x we see that G fg g is an ob ject of DD

F Q I

Since all non cases ab ove have mo d uM by the parts of Prop osition which we have already

proved we see that M G fg g is an extension of M by M and corresp ond to a pair c c with c

fg g is notsplit over F Hence M G since F and c c by our analysis of since

and M G fg g must corresp ond in some order to the lines c and c c in F

As a nonsplit p eu ramie extension of by remains nonsplit over F we see that the p eu ramie line

Ext M M Ext

F G DD

3 3 1

F Q I

cannot corresp ond to c c By the ab ove analysis it cannot corresp ond to c or c c Thus it

must corresp ond to the remaining line c

The prop erties of in the cases listed in Prop osition make it clear that the case there is

dierent We will see further manifestations of this dierence later

Further rank two calculations

Lemma For r s and we have

M M Ext

s r DD

F Q I

Proof The case follows from the case by Cartier duality Thus we assume r s f g

Let M fgbg b e such an extension By Lemma M fgbg arises from a Breuil mo dule over O of the

form

M F uu e F uu e M hu e u e he i

with

u e e u e he e

and

p p

e b e e b e

where h F uu

The combined conditions

b M M b b on M

are equivalent to

p p p

h u mo d u hu

Treating the cases s and s separately we conclude from Lemma that we may change e so

that h when s and h F when s As a result of this change we only have

p p

b e ue e b e e h

However with h F when s and h when s the condition

b b on M

after all so that in fact h u h forces h

4 4

When h so M is split in mo d compatibly with b on M and it is easy to check using N

that the wild descent data must also b e diagonal so we have the desired splitting in DD

F Q

It remains to consider the case r s with h c F It is easy to compute

N e N e cu e

The wild descent data must have the form

d d

e u e e u e h e

with h F uu

It is straightforward to check that M M and then the condition

d d

N t

on M gives

u u cu h cu h

The unique solution to this is

h cu u

Thus b b b mo d u while e ce mo d u This forces c With c we obviously have

only the split extension class

Lemma The natural map

Ext M Ext M

F G

3 3

F Q I

is an isomorphism with elements parametrised by pairs c c F corresponding to

c c u e i M hu e u e M F uu e F uu e

where

c u e N e N e c c u e e u e e u e

and the descent data is

p p

e b e b e e

d d

e e e e c c u c u e

is Proof The pro of is identical to the pro of of the case r s in Prop osition except

and when we study splitting we give M the descent data for the trivial everywhere replaced by

mo d character which amounts to using b e e rather than b e e

Lemma For r f g the maps

Ext M M Ext

DD r r F G

1 3 3

F Q

and

Ext M M Ext

DD r r F G

1 3 3

F Q

are injective and have image consisting of the dimensional space of classes which split over an unramied

extension of Q

Proof The cases r follow from the cases r using Cartier duality Thus we supp ose r We treat

p p

replaces everywhere only the case of M the case M b eing exactly the same except that

Let M fgbg represents an element in Ext M M Lemma ensures the existence of

F Q

an ordered F uu basis e e of M such that

M hu e u e he i u e e u e he e

with

p p

e b e e b e

Carrying out the usual calculation

b M M b b on M

if and only if

u mo d u h h

so that h cu with c F at the exp ense of Combining this with Lemma we may change e

p ossibly losing the diagonal form of b But with h cu and b e ue the conditions e h

a for some a F Then b forces a so b still u and so h h imply h

4 4

has diagonal form

so N Thus we must have It is easy to check that N e

d d

on M Since the wild descent data has to b e of the form

d d

e u e h e u e e

for some h F uu evaluation of on u e cu e M gives h u h so

d d

h c u for some c F The relation b e e forces c

We now have describ ed all p ossibilities in terms of the single parameter c F and it is straightforward

to use Corollary to check that all of these examples are in fact welldened Generic splittings over

unramied extension of Q corresp ond to the maps

M M F F

F F

3 3

given by

cu e e au e uu e

F satises a a c Such generic splittings can b e dened over Q ie without extending the where a

residue eld if and only if c

Lemma The map of groups

Ext M M Ext

DD F G

1 3 3

F Q

is an isomorphism

Explicitly the group Ext M M is parameterized by pairs c c F corresponding

F Q

M F uu e F uu e M hu e u e cu c u e i

with

u e e u e cu c u e e N e N e cu e

and descent data

p p

e e b e b e

d d

e e e e cu u u u e

The classes in Ext which split over an unramied extension of Q correspond to the pairs

F G

3 3

with c

The map of groups

Ext M M Ext

DD F G

1 3 3

F Q

is an isomorphism

corresponding M M is parameterized by pairs c c F Explicitly the group Ext

F Q

M F uu e F uu e M hu e u e cu c u e i

with

u e e u e cu c u e e N e N e cu e

and descent data

p p

b e e e b e

d d

e e e e cu u u u e

The classes in Ext which split over an unramied extension of Q correspond to the pairs

F G

3 3

with c

p p

with throughout gives the pro of of the Proof We treat the rst part of the lemma replacing

second part

As usual we can nd an ordered F uu basis e e of M so that

he e he i u e e u e M hu e u e

p p

e b e e and b e The conditions b M M and b b on M

amount to

u mo d u hu h

Since fu t u t jt F uu g consists of multiples of u we can change the choice of e so that

h cu c u

for some c c F where we may a priori lose the diagonal form of b But the same kind of calculation as

p p

in Lemma shows b e e a e for some a F so the condition b forces a

ie b still has diagonal action

It is straightforward to compute the asserted formula for N and then the wild descent data can b e

computed exactly as in our previous computations of wild descent data this yields the formulas

d d

e e c cu u u u e e e

where c F Mo dulo u the linear action of sends e to e c e but so

c for Thus we obtain the asserted list of p ossibilities The welldenedness of these examples

follows from Lemma and Corollary

M M if and only if c in which It is easy to see that there is a nonzero map F F

F F

3 3

case such nonzero maps are precisely those induced by

e au e u u e c u e

F satises a a c The verication that c c corresp onds to b eing in the kernel of our where a

map of Ext s is now clear since X X c has a solution in F if and only if c

Completion of the pro of of Theorem Everything in Theorem is now clear except for

the third assertion to the pro of of which we now turn Let G fg g b e as in the third part of that theorem

k gives a ltration The ltration on for some G O We may supp ose that G G

F F O

3 F

G G G

which is compatible with the descent data over Q According to Lemma we have M G

Mk r f and M G Mk r f for some r r and some f f k uu We will

M G for From this one can conclude let denote either or In particular M G u

M H Quite that if H is an sub quotient of G then M H u generally for any Breuil mo dule M

over O with M u M and any short exact sequence of Breuil mo dules

M M M

we must also have

r r

M u M M u M

Indeed M M is a surjection taking M onto M so the assertion for M is clear Since M is a

F uu mo dule direct summand of M and

M M M M u M

the assertion for M is likewise clear We conclude that M G fgbg admits a ltration with successive

quotients M Thus r f g

Consider a xed surjection of F G mo dules

over Q corresp onding This gives rise to a nite at O group scheme H with descent data on H H O

F F

to and an epimorphism

G H

compatible with descent data Consider the commutative diagram

M H M H M H

M G M G M G

where the top row corresp onds to the nonsplit ltration of The middle vertical map is an isomorphism

of the source onto a F uu mo dule direct summand of the target so the left vertical map is as well

b ecause an injection of F uu into a free F uu mo dule must b e an identication with such a direct

summand consider torsion This forces M H M and so by Prop osition we see that r

k one sees that r Rep eating the analogous argument applied to a submo dule

Thus G fg g is weakly ltered by fG G g for r s or as desired

s r

Completion of the pro of of Theorem Write A for F T T For r s

and we will dene a Breuil mo dule M over O and descent data fgbg for GalF Q

N rs

F such that M and M fgbg have compatible actions of A and on M M

N F

N rs N rs

b Frob More sp ecically set t or according as r s or Viewing

as an extension class it corresp onds to a particular pair c c F in Prop osition Fix these values

Motivated by the idea of deforming the formulae in Prop osition we are led to dene

s r t

M A uu e A uu e M hu e u e c T u e i

N N

N rs N rs

with

s r t

u e e u e c T u e e

It is straightforward to check that N on M We may dene A linear descent data on M

N rs

by setting b Frob and using the following formulae

When r s set

p p

e e b e b e

d d

e e e u e c u e

When r s set

p p

b e e b e e

d d

e u e e e c u e

When r s set

p p

e e b e b e

d d

e u e e u e c u e

It is readily checked that this denes an ob ject of DD with an action of A Let G and

N rs

F Q I

G fg g b e the corresp onding nite at O group scheme and nite at O group scheme with

F F

N rs

descent data

If M N then we have a short exact sequence in DD

F Q I

M M M

rs N rs N M rs M

N M

where the rst map is induced by multiplication by T The case M shows that

T G fgbg G fg g

Q Q

N rs N rs

3 3

corresp onds to G Thus we get a surjection of A G mo dules A Q which must in fact b e an

N rs

denes a deformation of isomorphism count orders Thus G fg g to A For N

N rs N rs

we have mo d T

rs N rs

We also have an exact sequence

M A M fgbg M A

N F N F

3 3

s N rs r

from which we obtain an exact sequence of A G mo dules in DD

F Q

X X

N N

F as F G mo dules Moreover this sequence must split as a sequence F and X Note that X

of A mo dules Use for instance the kernel of for any G G A Thus X

N N N

A as A G mo dules so det and X

N N N

Finally we must check that the exact sequence

is not split We have maps of Breuil mo dules

f M M

e e

and

f M M

T e

T e e

compatible with descent data These give rise to maps

and

such that the comp osites

and

are nonzero

To check that

is nonsplit it suces to check that

ker f Im f

satisfying corresp onds to an ob ject N fgbg of DD Im f is nonsplit However ker f

F Q I

t s r

u T e i N hu T e u e N F uu T e F uu e

with

t s r

u T e e u T e T e u e

By Lemma the sequence of Breuil mo dules with descent data

N M M

s r

is not split This sequence recovers under generic bre descent so by Prop osition

ker f Im f

is not split

Completion of the pro of of Theorem Supp ose rst that r s or By

Lemma

Ext H G

i(rs)

is the zero map Lemma then tells us that if r then

Ext H I F

i(rs)

is the zero map while if s then

Ext H I F

i(rs)

is the zero map Thus Theorem and hence Theorem follows in these cases

Now consider the case r s Cho ose x H G ad Let G denote the corresp onding

i(6 6)

Set M M G Let O rank nite at O group scheme with descent data fg g on G G

F F O

H G denote the closed subgroup scheme with descent data corresp onding to the kernel of the map

G fg g F and let N M H Then N has F uu basis e e e with resp ect to which

N hu e u e b b u e u e c c u e f e i

where b b c c F f F uu and sends the indicated generators of N to e e e resp ectively

Also the descent data has the form

p p p

e e ue b e h e b e b e

F uu and for some h

d d d

e e e e b b u u e e e c c u u e h e

where h F uu Also as is tres ramie we see that c by Prop osition and Lemma

The requirement that u N N forces N to contain

u e u u e c c u e f e c c u u e b b u e b b u c c u f u e

so N must contain b b u c c u e As c we get b b u e N and since e u e N we

must have b b We conclude that the natural map

H G Ext

i(66)

is the zero map

Let us further analyse N Replacing e by e t e for t F uu causes f to b e replaced by

and h may change f u t u t and otherwise leaves our standardized form unchanged except that h

Using a suitable choice of such t we may assume f has degree at most On the other hand

p p p

b u e c c u e f e ue e c c u e f e h u f u f u u

so b N N if and only if

u mo d u f u f

which forces

f a u a u

for some a a F From the wild descent data formulae derived in the pro of of Lemma we also see

that h mo d u

Now M has an ordered basis e e e e with resp ect to which

a u a u e g e i c c u e he u e M hu e u e c c u e u e

where g h F uu and sends the indicated generators of M to e e e e If we try to expand

out u e as a linear combination of the indicated generators of M we nd that

c c u h ca u e mo d M u e

It follows that u e M and only if if

u c c u h ca mo d u

Since is tres ramie the last part of Prop osition tells us that c Thus u e M if and only if

h a u mo d u We can now use Lemma to see that the wild descent data action is determined by

d d d

e e e e c c u u e e e f e

with f mo d u and

e e e c c u u e h e g

where g F uu and h mo d u

e We must hav

u e c c u e a u a u e g ue M

and this expression is easily computed to equal

u H e c c u u e h e g e c c u H e f e

1 1

3 3

a u H a u H e e c c u u e g uH

1 1

3 3

b ering that hu e u M i M b ecomes Remem

u e ce c c u H e a u H a u e a cu e g ue M

1 1

3 3

the explicit generators of M given in and recalling that h a u mo d u this simplies to Using

a cu e M

Thus a cu is divisible by u so a

The image of the class x in Ext under O group scheme with corresp onds to a nite at

F G

3 3

Breuil mo dule M free of rank two over F uu with basis e e and with

M hu e u e a u e i

sends the indicated generators of M to e and e resp ectively According where to the pro of of Lemma

is split over an unramied extension of Q this implies that the image of the class x in Ext

F G

3 3

Thus

Ext H I F

i(66)

map This completes the pro of of Theorem and hence of Theorem is the zero

Corrigend a for CDT

We would like to take this opp ortunity to record a few corrections to CDT

Page line The semisimplicity of follows from that of is false and should b e deleted

This assertion was not used anywhere in the rest of the pap er

by GL Page line replace GL C R

Page line replace of typ e S by such that j is of typ e and is of typ e S

Page lines replace each by and each by

Page line replace each of the three o ccurences of A by A

Page line the discrete top ology on V should read the adic top ology on M

p p

Page part of Lemma V should b e assumed to b e a normal subgroup of V

Page line We should have noted that the key comp onent of this argument is very similar to the

main idea of Kh

x There are two signicant errors in this section The assertion SL Z U satises the

hyp otheses of Theorem is false and HomL k should b e L k The argument of this section

n n

can b e repaired by making the following changes

Page lines and Replace Setting S T fr g we nd that the group SL Z U

satises the hyp otheses of Theorem by Set S T fr g U U where U

S Sp Sp

c c

p p

if p T V where V U p if p T and U U otherwise V U p

Sp Sp Sp S

0 0 0

Hom V otherwise and L and V Then SL Z U OI M H X

O U V

S Sp S S

S S S

satises the hyp otheses of Theorem

by Y HomL k by L k M by M F by F Page lines Replace Y

S n n S S

Hom O M

and L by L

is nonempty by Using the fact that lemma holds Page line Replace and N

replacing U and replacing and the discussion on page we conclude that N is with U

S S S

nonempty

Page line Replace U by U

f rr gp fr gp

by V V U Page line Replace U

S S

Page line The assertion is false in the case It can b e corrected by adding and

j E mo d which is true if for instance E has p otentially sup ersingular reduction and

mo d after if

by E is isogenous to an elliptic curve with j invariant in the set Page line replace j

Page line replace by

Page line replace the paremthetical comment and j by and isogenous to

one with j invariant

and q by p Page line replace p by q

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