Modularity of Galois representations

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

C

HRIS

S

KINNER

Modularity of Galois representations

Journal de Théorie des Nombres de Bordeaux, tome

15, n

o

_{1 (2003),}

p. 367-381

<http://www.numdam.org/item?id=JTNB_2003__15_1_367_0>

© Université Bordeaux 1, 2003, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Modularity

of Galois

_{representations}

par CHRIS SKINNER

RÉSUMÉ. Dans cet

_article,

nous donnons une

_{interprétation}

en

termes de théorie de Galois des

_{représentations galoisiennes}

p-adiques

de dimension 2 associés aux formes modulaires

holomor-phes

de Hilbert

_qui

sont des "new forms" . L’article suit pour

l’essentiel

_l’exposé

des Journées

_{Arithmétiques}

de 2001.

ABSTRACT. This paper is

essentially

the text of the author’s

lec-ture at the 2001 Journées

Arithmétiques.

It addresses the

prob-lem of

_identifying

in Galois-theoretic terms those

two-dimensional,

p-adic

Galois

_{representations}

associated to

_holomorphic

Hilbert modular newforms.

By

the work of _many mathematicians

_(Eichler,

_{Shimura, Deligne,}

Carayol

... )

it is known that

_{if f}

is a

holomorphic

Hilbert modular new-form over a

_totally

real field F of

_degree

d and if the

_weight

(k1,... , kd)

of f

is such that the

ki’s

are

_greater

than zero and all have the same

parity,

then for each rational

_prime

_p there is a continuous

_{representation}

Gal(F/F) 2013~

_{GL2 (Qp)}

such that trace

_{(frob f)}

= for almost all

primes t

of

_F,

where is the

_eigenvalue

of the action

_{on f}

of the usual

Hecke

_operator

_Tt.

_Conversely,

it is

_expected

that _{any "reasonable"} repre-sentation p :

Gal(F /F) -

GL2(Qp)

should be

isomorphic

to some

(see

the

_Conjecture

_{in 91}

_below).

The first

_significant

result of this

type

was

obtained

_by

A. Wiles

_[Wl]

for the case F =

Q.

This

paper discusses efforts

and results

_extending

those in Wiles’

_{ground-breaking}

work.

_Emphasis

is

placed

on

_{loosening hypotheses}

(e.g.,

_{replacing Q}

with an

_{arbitrary F}

or

allowing

reducible residual

_{representations).}

1. Introduction

This lecture focuses on

modularity

in the context of Galois

represen-tations. More to the

_point,

it discusses various

_{recently-proven}

results

pertaining

to the

_problem

of

_{intrinsically}

_{characterizing}

the

_p-adic

Galois

representations

associated to

_automorphic

_{representations.}

We

_{begin by recalling}

some of the

_properties

of the Galois

representa-tions associated to

_holomorphic

Hilbert modular newforms.

_Suppose

that F is a

totally

real number field of

degree

d and that

f

is a

holomorphic

Hilbert modular newform of level _n,

_{Nebentypus x,}

and

_weight

_{(kl, ..., kd)}

with

_{each ki &#x3E;}

1 and all

_having

the same

parity.

For each

prime

ideal i of

F let

_c¡,(f)

be the

_eigenvalue

of the usual

~h

Hecke

_{operator T~}

_acting

on

f

(so

Ttf

=

c, ( f ) f ) .

It is known

by

the work of _many mathematicians

(Eich-ler, Shimura,

Deligne, Carayol,

Wiles, Taylor, Blasius,

Rogawski,

Tunnell,

Jarvis...)

that for each rational

_prime

_{p, upon}

_fixing

an

_embedding

_Q

~

_Qp

(so

we _may view the as elements of

Cap),

there is a continuous

rep-resentation

such that

(i)

is

_irreducible,

(ii)

for all

_complex

_conjugations

c

(i.e.,

is

_odd),

(iii)

=

xem,

_m =

11,

(iv)

trace =

~(/),

’

In

_(iii),

we

view X

as a Galois character via class field

theory,

with the

reciprocity

map normalized so that arithmetic Frobenii

(frob l)

_correspond

to uniformizers.

_{Also, -}

is the usual

_{p-adic cyclotomic}

character.

The results described herein are

_mostly

concerned with the

_following

problem.

Problem.

_{Intrinsically}

characterzze the set

_MgFp

_of

continuous repre-sentations p :

Gal(F/F) -

such that

_for

some Hilbert

modular

_newform

_f

,

By

an intrinsic characterization we mean a characterization in terms of the

Galois-theoretic

_properties

of the

_{representation}

_(such

a characterization is found in the

_conjecture

stated

_{in 31}

_below).

The celebrated

_proof

of the

_modularity

of

_elliptic

curves defined over

Q

(a

consequence of the main results of

[Wl] ,

[TW]

and their extensions in

[D l] ,

[CDT],

and

_{[BCDT] )}

entailed

_showing

that the

representation

attached to

the Tate module for some

_prime

_p

_(and

hence for all

primes)

for such a

curve is contained in This result has been

amply

discussed

else-where and was a

_topic

of a lecture

_by

B. Conrad at the

_previous

Journees

Arithmétiques.

The

_organizers

of the current conference

_requested

that the author discuss other

_aspects

of efforts to resolve the above

_problem,

so this

is the last that will be said about

_elliptic

curves until the final section.

The

_organization

of the rest of this lecture is

_essentially

as follows. Since

it is often easier to couch results in terms of

_{automorphic representations}

reformulate the

_properties

_{of p f,p} in these terms _{and then go} on to state a

conjectural

resolution of the

_problem.

Then the related notion of a residual

representation

is introduced. This is done in

_§1.

In

_§2

we summarize the

strategy

introduced and

_successfully

_{implemented by}

A. Wiles to _prove

many cases of the

_conjecture

is summarized. In

_§3

we discuss results of

K.

_Fujiwara

and of Wiles and the author in the case where the residual

representation

is irreducible but F _may not be

_equal

to

_Q.

In

§4

the work of Wiles and the author in the case where the residual

_{representation}

is reducible is described.

_Finally,

in

_§5

we mention a few

things

that the

constraints of _space and time

_(and

of the author’s

_{understanding!)}

have

forced us to omit from the

_preceding

sections.

2.

_{Automorphic representations}

and Galois

_{representations}

To

_simplify

_matters,

fix once and for all an

_embedding

_{Q y}

C and

compatible embeddings Q Y

C~~

and

_{Qi y.}

C for each

_prime

R. Let F be

a

_totally

real number field of

_degree

d with

ring

of

integers

OF-

Suppose

that 1T = _01Tv is a

_{cuspidal automorphic}

representation

of

GL2/F

such that

(i)

if

_vloo

then _7rv is a discrete series

representation

with

weight

1

and central character

_’lfJ7rv

_satisfying

_{1/J7rv ( -1) =}

_(20131)~,

(ii)

the

_kv’s,

all have the same

parity,

(iii)

the central character of 7r is

X7r1 .

1~-1

where _X7r is finite and m =

1}.

There is a one-to-one

_{correspondence}

between such

_{representations}

and the

Hilbert modular newforms as in the introduction.

For each

_prime

_p there is a continuous

_{representation}

such that

{1 i i 1

(i)

is

irreducible,

where

Wv

is the Weil group at the finite

_{place v}

and is the repre-sentation of

_Wv

associated _{to 7r v} via the local

_Langlands

_{correspondence}

(cf.

[Ta,

Theorem

_4.2.1]),

which we normalize so that if 7r, =

1r(X1,

~C2)

is unramified then .

1;1/2

e

~2 1 I

. ~~ 1/2.

’ If x

corresponds

to

the newform

_{f ,}

then is

_just

_properties

_(ii),

_(iii),

and

_(iv)

of _{pj p}

following

from

_property

_(ii)

of

_(1.1).

One can also _say

_something

about

_P1r,pIDv

for where

Dv

is a

de-composition

group at v,

namely

that in many cases

(and

_{conjecturally}

in

all)

The definition of

_"potentially

semistable" is

_fairly

technical

_(cf.

_[Fo])

but

reflects

_properties

of

_{p-adic representations}

that appear in the

cohomology

of

_algebraic

varieties.

_(That

does when 2 is

_essentially

the

content of

_[BR]).

The

_problem

from the introduction is thus to characterize the continuous

representations

p :

Gal(F/F) -

GL2(Qp)

such that for some 7r. A

guess at a solution can be obtained

_by

_extracting

as _many Galois-theoretic

properties

as

possible

from

(1.1)

and

(1.2)

and then

positing

that

consists of all

_{p-adic representations}

_having

these

_properties.

This leads to

the

_following

_conjecture

_(which

is

_essentially

_Conjecture

3c of

_[FoM]).

Conjecture.

Let be the set

_of

Tate twists

_of

_{representations}

in

MgF,p.

The set consists

_of

the continuous

_{representations}

_{p :}

Gal(F/F) -+

GL2(Qp)

such that

(i)

p is

irreducible,

(ii)

p is

odd,

(iii)

p is

unramified

at all but

finitely

many

places,

(iv)

for

all

_{vlp, plDv is}

_potentially

semistable.

In order to state various theorems in the direction of the above

Con-jecture,

we need to introduce the notion of a residual

_{representation}

and

discuss some

specific

instances of

_{potential semistability.}

Suppose

p : G -

GLn(Qy)

is a continuous

_{representation}

of a

_compact,

Hausdorff group. A

simple

measure-theoretic

_argument

shows that some

conjugate

of p takes values in

GLn (O)

with 0 the

ring

of

integers

of some finite extension of

_Qp

_containing

the

_eigenvalues

of all elements of

_p(G).

Replacing

p with this

conjugate,

we define the residual

_{representations}

_p

of _p to be the

_{semisimplification}

of the

_{representation}

into

_{GLn(k) ,}

k the residue field of

_0,

obtained

_{by reducing}

the matrix entries of p modulo

the maximal ideal of 0. This is well-defined up to

equivalence

and up to

extension of the field k.

Suppose

p :

Gal(F/F) -+

GL2(Qp)

is continuous. For the rest of this

lecture we will be concerned with two

_{possibilities}

for

_plDv

whenever

_vlp.

The first will be that

-

( X2 ) ,

X2 I has

finite

_order,

Iv

being

an inertial group at v. In this instance _p is said to be

nearly

ordircary at

v. We will _say that such a _p is

Dv -distinguished

if

_X2~

From work of Wiles

_[W2]

_(see

also

_[H2])

it follows that if 7r =

07rv is such

that the

_k",

_{v I oc,}

are all

_equal

and that for all

v ip,

~2)

is either a

satisfying

is a unit in the

_ring

of

_integers

of

_Qp

(A,

a fixed uniformizer of

_Op,v),

then _p~,p is

_{nearly ordinary}

at

The second is that

plDv

is the extension of scalars of a

_{representation}

arising

from a

_p-divisible

_group over

_OF,v.

In this instance _p is said to be

_flat

at v. Note that _p can be both

_nearly

ordinary

and flat at v. From work of

Carayol

and

_Taylor

(see

[T]),

it follows

that if x = _07r, is such that

kv

= 2 for all

vloo

and 7rv is unramified for all

vlp,

then is flat at all

_vip

if either the

_degree

of F is odd or if is

irreducible.

’

I can now state the first

significant

results in the direction of the

Con-jecture.

These were obtained

by

Wiles

[Wl], [TW]

and

Taylor

and

subse-quently

refined in

_[Dl]

to

_yield

Theorem 1

_([Wl],[Dl]).

_Suppose

_p is an odd

_prime.

If

_{p :}

Gal(Q/Q)

-GL2(Qp)

is a continuous

_{representation}

such that

(i)

p is

irreducible and

_unramified

at all but

_finitely

_many

_primes,

(ii)

p is odd,

(iii)

det p ~

with X

finite

and

_1,

(iv)

p is

either

_flat

at _p or

nearly ordinary

_{at p} and

_{Dp-distinguished,}

(v)

is

_irreducible,

(vi)

_for

some _7r,

then _p E

(Note:

hypotheses

(iii)

and

_(iv)

ensure that _p is

_potentially

_semistable.)

Since the appearance of

[D l]

other refinements and

generalizations

of this theorem have

appeared,

most

_notably

in

_{[CDT], [BCDT],}

and

_[Di],

but to

simplify

both notation and statements we state

_only

the theorem above.

Subsequent

efforts _{have also gone} a

_long

_way towards

allowing

Q

to be

replaced by

a

_totally

real field F in Theorem 1 and towards

loosening

con-dition

_(v).

In order to

_explain

these efforts and the obstacles

_encountered,

we first recall the

_strategy

behind the

proof

of Theorem 1. 3. The

_strategy

_(very

_briefly)

Suppose

p is an odd

prime

and _{suppose p} satisfies the

hypotheses

of

Theorem 1.

_Replace

_p

_by

a

_conjugate

_taking

values in some

GL2(O)

as was

done to define

p.

Let k be the residue field of 0.

By

an 0-deformations of

_‘p

we mean an

_equivalence

class of continuous

representations

a :

Gal(Q/Q) 2013~

A a

complete

local Noetherian

0-algebras

with residue field 1~ and maximal ideal _ny, such _{that p mod} mA =

P.

Let

Eo

be the set of

_primes

at which

_p

is ramified. Given a finite set of

primes E D

Eo,

we _{say that} an 0-deformation a is

of

_provided

a is unramified _away ftom £ U

_{p},

. if _p is

nearly

ordinary

at p

( WI ;2)

with

~2

=

_X2?

. if _p is flat at _p, but not

nearly ordinary,

then for all _{n, a} mod

mn A

arises from a finite flat _group scheme over

_Zp

with an A-action.

Clearly

p is of

type-E

for

any E containing

all the

primes

at which p is

ramified.

We call an 0-deformation a minimal if

. cr is of

type-Eo

with the

slightly

_stronger

condition that

o if _p is flat at _p, then for all n, d mod

m~

arises from a finite flat _group

scheme over

_Zp

with an A-action.

~ ~ is

"minimally

ramified at each t E

_Eo."

The third condition

_(which

we do not make

_precise)

is such that the

rami-fication is

_essentially

the same as that of

ptDt8

From the

_theory

of deformations of Galois

_{representations}

as introduced

by

Mazur

_{[MI], [M2]}

it follows that there are universal 0-deformations PE :

GL2(RE)

and pmin :

Gal(Q/Q) ~ GL2(Rmin)

of

_type-£

and

_{minimal, respectively.}

The

_universality

is such that if a is any 0-deformations of

type-E

into

GL2 (A)

then there exists a

_unique

_map

RE -7 A

of local

_0-algebras

such that a is obtained

_by

_combining

this _map

with _ps, and

_similarly

for minimal deformations.

Step

2. Fznd some modular

deformations

Let

_flr

and

_Hmin

be the sets of

_{cuspidal automorphic representations}

,7r such that is

isomorphic

to an (9-deformation that is of

type-£

or

minimal

_(with

scalars extended to of

_course),

_{respectively.}

Let

_T~

be the

_0-subalgebra

of

_TI7rEIIE

_Qp

_{generated by}

_Ti

=

e V E

U

_{p}.

Let

_Tmin

be defined

_similary

but with

_TIE

_{replaced by}

_Ilmin

·

Then both

_TE

and

_Tmin

are

_complete,

local Noetherian

0-algebras

with

residue field 1~. When p is

nearly ordinary

at _p but not

_flat,

then

Tr

is a

nearly-ordinary

Hecke

_ring

in the sense of Hida

[HI]

and hence

3-dimensional. Otherwise these

_rings

are finite

0-algebra.

That the set

nmin

is

_non-empty

_(and

so

Tmin

and

Ts

are

non-zero)

is a _consequence

of

_hypothesis

_(vi)

of Theorem 1 and the

_proof

of Serre’s

_{epsilon conjecture}

There are 0-deformations

of

_type-E

and

_{minimal, respectively,}

such that

and

Thus there are

_surjective

_maps

coming

from the

_universality

of

_lR

and

_Rmin

_(with

_surjectivity

_coming

from

_(2.1a,b)).

Step

3. Prove that

_Pmill

is an

isomorphisms

This is

_essentially

done

_by

_realizing

both

_Rmln

and

_Tmin

as

_quotients

of

the same _power series

_{ring by}

the same

_ideal,

at the same time

_showing

that

they

are also both

_complete

intersection

_rings.

The power series

_rings

and the

_quotient

_maps are obtained

_by

_"patching"

_together

various

_quotients

of

_auxiliary

deformation

_rings.

Step

l~.

Deduce

_{from Step 2}

that

_OE

is an

isomorphism

This is done

_{by comparing}

two numerical invariants.

_By

_extending

0 if necessary, we can assume that there exists a 7r such that _p~.~p comes from a

minimal deformation into

_{GL2 (O)}

_(i.e,

is

_isomorphic

to such a

representa-tion with scalars extended to This

_gives

rise to maps

as well as _maps

Note that

_9min

=

Wmin 0

9r

=

_{~~ ,}

and and

0r

factor

through

the natural

_surjective

_maps

To -

Tmin

and

Rr -

Rmin,

respectively.

Put

The

_isomorphism

of

_Step

_3,

_being

an

isomorphism

of

complete

intersection

rings, implies

(cf.

[DRS])

that

That

_OE

is an

isomorphism

would follow from

To establish

_(2.3)

one needs to

_give

an _upper bound on the difference of

the left hand sides of

_(2.2)

and

_(2.3)

and a lower bound on the difference

hand sides are orders of

_global

Galois

_cohomology

_{groups, but their} differ-ence is bounded from above

_by

a

_product

of orders of some local Galois

cohomology

groups that can be

_{easily computed using}

local Tate

_duality.

The

_right

hand sides

_essentially

measure _congruences between

cuspforms

and the difference between the two sides can be bounded below

by

general-izations of Ribet’s methods

_{(see [Wl,}

_Chapter

_2]).

This last

_step

_requires

some technical results on the

_injectivity

of various

_cohomology

_groups of

congruence

subgroups

and their freeness over the Hecke

_rings.

This latter

result is a _consequence of the

_{"multiplicity}

one" result in

Chapter

2 of

[Wl]

(nowadays

it can be _proven in other _{ways -} but that’s the next

_section).

Step

5. Deduce that _p is modular

Suppose £

contains all the

_primes

at which _{p is} ramified. The _map 0

_{corresponding}

to _p via the

_universality

of

_RE

sends

_Ti

to

tracep(frobt)

for

_{all £ g}

E

_by

_(2.1b).

From this one

_easily

deduces that

for some x E

_TIE

_(in

the

_{nearly ordinary}

case one needs the main

results of

4.

_Residually

irreducible

_{representations}

over

totally

real fields About a _year after the

_publication

of

[Wl],

K.

_Fujiwara

released a

preprint

[Fl]

that

_generalized

much of

_Steps

1-3 of the

_preceding

strat-egy to

_totally

real fields.

Theorem 2

_((F1~).

_Suppose

_p is an odd

prime

and F is a

totally

real

field.

If p :

Gal(F /F) -

GL2(Qp) is a

continuous

_{representation}

such that

(i)

p is

irreducible and

_unramified

at all but

_finitely

_many

_places,

(ii)

p is

odd,

(iii)

det p

=

XEm with X finite

_{and m &#x3E;}

1,

(iv)

either

_{p is totally unramified}

in F and

_{p is fiat}

at all

_vlp

or _p is not

flat

at any

vip

but is

_nearly

_ordinary

and

_{Dv -distinguished}

at all

_vip,

(v)

irreducible,

(vi) p=p~,p

for

somme 7r such that _P7r,p comes

from

a minimal

deformation

of p,

(vii)

p comes

from

a minimal

_deforrnation

_{of p,}

then _p E

Actually, Fujiwara

proves a

_stronger

result. In

particular,

he _proves a

result

_analogous

to

_Step

3 above: he identifies a deformation

_ring

with a

Hecke

_ring.

Step

1 of the

_strategy

sketched in

_{§2 generalizes}

to the

_setting

of Theorem

2 without _any

_difficulty.

_Step

2 also

_generalizes,

but we also need to know that

TImin

is not

_empty.

This is ensured

_{by hypothesis}

_(vi)

of Theorem 2 which is much

_stronger

than the

_{corresponding hypothesis}

of Theorem 1.

A

_stronger

_hypothesis

is needed since we do not have a version of Serre’s

,,-conjecture

for

_totally

real fields

_(but

see

[J], [Ra],

and

[F3]

for progress

towards

_such).

One of the most

_significant

new ideas in

[F1]

comes in

_generalizing

_Step

3. The

_argument

in

_[TW]

relies on the fact that certain

_Tmin-modules

are free. This fact is _proven in

_by

a delicate

_geometric

_argument

that

does not seem to

_generalize

to a

general totally

real field. But

Fujiwara

discovered that

_by

_"patching"

both modules and

_rings

he could

simulta-neously

prove that

Rmin£fTmin

and that the

Tmin-modules

are free

_(a

fact

which is

_important

for

_Step

_4).

This was

_{independently}

discovered

_by

F. Diamond

_[D3].

Step

4 has

yet

to be

_{completely generalized,}

whence

_hypothesis

_(vii)

of

Theorem 2. However in

_[F2],

a refinement of

~F1~,

Fujiwara

has made

_great

progress towards

doing

so.

Some of the

_problems

encountered in

_attempting

to

_generalize

the strat-egy can be avoided if one is content to

_merely

_{prove p} E

_M!gFp

and not

This is thanks to

_Langlands’

results on the

_image

of solvable

base

_change

which

_imply

that _p E if _p E

_MQL,p

for some

_finite,

totally

real extension L of F

_having

solvable normal closure over F. This

can be

_exploited

as follows. The main result of

[SW3]

_essentially

asserts

that

_p

_being

irreducible combined with for some 7r such that _p~~p

comes from a deformation that is minimal _{at p}

(meaning

that it satisfies

the conditions of a minimal deformation for all but not

_necessarily

at

other

_places)

_{implies hypothesis}

_(vi)

of Theorem 2 over some extension L of F as above.

_Combining

this with

_Fujiwara’s

results and the observations

about

_base-change

should

_yield

the

_following

theorem.

"Theorem 3" .

_Suppose

_p is an odd

_prime

and F is a

_{totall y}

real

field.

_If

p :

Gal(F /F) -

GL2 (Qp) is

a continuous

_{representatzon}

such that

(i)

p is

irreducible and

_unramified

at all but

_finitely

_many

_places,

(ii)

p is odd,

(iii)

det _p =

X£m

wzth x

finite

and _{m &#x3E;}

1,

(iv)

either

_{p is}

_{totall y unrarrtified}

in F and _p is at all

_vip

or

_{p is}

not

flat

at _{an y} but is

_{nearl y ordinary}

and

_{Dv -distinguished at}

all

_vip,

(v)

irreducible,

(vi)

for

some 1f such that comes

from

a

deformation of

_p

that is minimal _{at p,}

then _p E

We use _"should" and

_{put quotes}

around the label because a

_proof

has

Exploiting

a similar observation Wiles and the author were able to loosen

hypothesis

(v)

of Theorem 1 in some cases.

Theorem 4

_([SW4]).

_Suppose

p is art odd

_prime

and F is a

totally

real

field. If p :

Gal(F /F) -

GL2(Qp) is

a continuous

_{representation.}

such that

(i)

p is

irreducible and

_unramified

at all but

_finitedy

_many

_places,

(ii) p

is

_odd,

(iii)

det p

=

XE’ with X finite

and 7rt &#x3E;

1,

(iv)

p is

nearly ordinary

at v and

Dv-distinguished

for

all

vip,

(v)

p is irreducible,

(vi)

for

soyne 7r such that is

_nearly

_ordinary

at each

_vip

and

X

then _p E

_MgF,p.

The condition of

_{being X2-good}

is that if

_{X2*,~, )}

then

with

_{’l/J2,v =}

_X2,v

for all

_{vlp. (This}

is an _unnecessary

_hypothesis

if F =

Q

or if no is

_split.)

The

_proof

of Theorem 4 is an

adaptation

of the methods

_employed

in

[SW2]

to handle cases where

_p

is

reducible,

so we will reserve comments for that _case, which is discussed in the next section.

5.

_Residually

reducible

_{representations}

So far all the results discussed have included the condition that

_p

be ir-reducible. What about the case where

_p

is reducible? Such

_{representations}

were the focus of

_[SW1]

and

[SW2].

In

_[SW1]

some

_special

cases where one could

_identify

the deformation

_rings

with Hecke

_rings

were

_examined;

roughly

the cases where

_p

= 1

_{EÐ X}

and there is _a

unique

extension of 1

by

X. A different tack was taken in

[SW2]

that

_yielded

the

_following

theorem.

Theorem 5

_([SW2]).

_{Suppose p is}

an odd

_prime

and F is a

totally

real

field.

Suppose

also that _{p :}

_{Gal(F/F) -}

is a continuous

repre-sentation.

_If

(i)

p is

irreducible and

_{unramified at}

all but

_finitely

many

places,

(ii)

p is

odd,

(iii)

det _p =

_{with X finite}

and _{7n &#x3E;}

1,

_

(iv)

p

=

_{Xl EÐ}

_X2 and

(X1/X2)BDv

~-‘ 1

for all

vip,

(v)

for

dll

_vlp,

I ’~i," * J,

_X2, dnd has

_finite

order,

(vi)

F(xi /x2) is

abelian over

Q,

_

This

_actually

follows from a theorem for

general

F that

requires

the

existence of solvable extension of

_(which

is the

_splitting

field over

F of the character

_Xl/X2)

whose class _group has

_p-rank

_relatively

small in

comparison

to its

_degree.

When

_F(Xl/X2)

is abelian this is known thanks

to a theorem of

Washington

[Wa].

We now sketch the main

_ingredients

that _go into the

_proof

of Theorem 5.

We assume _p satisfies the

hypotheses

of Theorem 5.

_Replacing

_p

_by

its

twist

_by

the Teichmuller lift of

_{x21, we}

can assume

1.

_{Deformation rings}

For _any finite set E of finite

_places

of F

_containing

all those

_dividingp

and all those at

_{which X}

is

_ramified,

let

_G~

be the Galois _group of the maximal extension of F unramified outside of E and the

_places

above

_infinity.

Let

where the maps are the restriction _maps. For each

0 ~

c E

_Hj

there is a

non-split representation

p,

Gal(F/F) -+ GL2 (k)

such that

~ _{Pc is} unramified at all finite

_places

not in

_E,

~ _PC

( 1

*)

with c

_being

the

cohomology

class associated to *. ~

v

= _{aC E}

GL2 (k)

for all

We will say that an 0-deformations cr of _Pc is

of

type-(c, ~)

if

is unramified at all finite

_places

not in

_E,

,v *

with 1 v

=

x for all

v 1 p.

2,v ,

There is a universal deformation

Gal(F /F) -

of

type-(c, E).

2. Hecke

_rings

Define

TE

and

~’v

(v

a finite

_place

not in

_E)

as in

Step

2 of

§2,

but now

let

_Ms

be the set of 7r’s such that _p,p comes from a deformation of

type-(c, E)

for some

0

c E

Unfortunately,

in

_general

there is no

longer

a

natural

_{representation}

into

_Essentially

this is because

_dimk

_HE

can be

_bigger

than one.

_However,

one does have a

_{pseudo-representation}

into

Ts.

3.

_{Pseudo-deformations}

By

an

(C~-)

pseudo-deformation of type-E

we mean a

_complete

local Noe-therian

_(9-algebra

A with residue field k and a

_{triple 4&#x3E; =}

_(a,

_d,

_x)

of

and some other

_equally

_illuminating

relations

(see

Section 2.4 of

_[SW2]).

Essentially

these

_capture

the traces of 0-deformations of _any pc. If

is an 0-deformations of some _Pc

(suitably

normalized with

_respect

to some fixed

_complex

_{conjugation),}

then

defines a

pseudo-deformation

of

_type-E.

In

particular,

the universal

defor-mation _Pc,E determines a

pseudo-deformation

0,,E

of

type-E.

It turns out that there is a universal

pseudo-deformation

of

type-E,

call it

(RPZ’, OP,’).

Also,

there is a

_{pseudo-deformation}

iod

dmod,

xffiod)

into

_TE

such that

Thus there are _maps

coming

from the

_universality

of

4. Pro-modular

_primes

We define a

prime p

of

Rc,E

to be

pro-modular

if there exists a _map

0 :

Tj -

Rc,E/P

such that

where _7rp is the natural

_surjection

- In

_[SW2]

we show that under suitable

_hypotheses

_every

_prime

of each

_Rc,’E

is

_promodular.

This is done

_by

a delicate induction

_argument

including

a

_{generalization}

of the

strategy

sketched in

_§2.

5.

_Finishing

_up

To show that _p E we first note that we _may assume that _p is a

deformation of

_{type-(c, ~)}

for some c. Then it follows from

Step

4 that the

kernel of the

_{corresponding}

_map -7 O is

_pro-modular.

From this and

(4.1)

and

_(4.2)

one can deduce that there is a _map

_{T~ ~ ~}

such that

fiv

maps to trace

p(frob v )

for And from this it follows that

6. Odds and Ends

In addition to the results

_{high-lighted above,}

there have been a number

of other

_{refinements, generalizations,}

and additions to the circle of ideas

that started with

_(W1~.

We

_briefly

discuss a few.

p = 2. M. Dickinson

[Di]

has

begun

to _remove the condition that

p be odd.

Artin’s

_Conjecture.

This Artin’s

_Conjecture

is that the Artin L-function attached to an irreducible Galois

_{representations}

Gal(Q/Q) - GL2(C)

has an

_analytic

continuation to the whole

_{complex plane.}

The

ground-breaking

work of

_Langlands

and Tunnell in the case where the

_image

of this

representation

is solvable

_played

a fundamental role in

[Wl]

and

subsequent

work. This in turn has led to some

_partial

results for the last

remaining

case

(when

the

_{projective image}

is

_{icosahedral).}

See

_[BDST],

_[BT]

for

_example.

The most

_{spectacular application}

of the results

_stemming

from

[Wl]

has been the

_proof

that all

_elliptic

curves over

Q

are modular. While

not all

_elliptic

curves

(over

a number

field)

are

expected

to be modular

(in

the sense of

being

a

quotient

of the

jacobian

of a modular

curve),

there is a class that is: the

Q-curves.

An

elliptic

curve over a Galois extension is called a

Q-curve

if it is

isogeneous

to all of it’s Galois

conjugates.

In

[ES]

results mentioned in

_previous

sections are used _{to prove} the

_modularity

of

many such curves.

n &#x3E; 2. There has also been effort to

_generalize

the aforementioned

theo-rems

by

replacing

GL2

with

GLn

and 7r with an

_{automorphic representation}

on Some

partial

success has been achieved

_by

M. Harris and R.

Taylor.

Generalizing

all the

_steps

in

_§2

_poses series technical

_{difficulties,}

not the least of which is

_finding

the Galois

_{representations}

associated to

the 1r’S on

Alternate

_approaches.

C. Khare and R. Ramakrishna

_{(unpublished)}

have

_developed

an alternate

_approach

to

_identifying

deformation

_rings

with

Hecke

_rings

_(and

_thereby

_establishing

_modularity)

that

_yields

many cases

of Theorem 1. Their

_approach

is

_{fairly different,}

and among other

things

avoids the

_{"patching arguments" usually employed}

in

_Step

3 of

_§2.

To do

justice

to their novel ideas would

_really

_require

another lecture. References

[BR] D. BLASIUS, J. _ROGAWSKI, Motives _for Hilbert _{modular forms.} Invent. Math. 114 _(1993),

no. 1, 55-87.

[BCDT]

C. BREUIL, B. _CONRAD, F. _DIAMOND, R. _TAYLOR, On the _{modularity of elliptic} curves over _Q: wild 3-adic exercises. J. Amer. Math. Soc. 14 _(2001), no. 4, 843-939.

[BDST]

K. BUZZARD, M. DICKINSON, N. SHEPHERD-BARRON, R. _TAYLOR, On icosahedral Artin

[BT] K. _BUZZARD, R. TAYLOR, Companion forms and _weight one _forms. Ann. of Math. 149

(1999), no. 3, 905-919.

[CDT] B. CONRAD, F. DIAMOND, R. _{TAYLOR, Modularity of} certain _potentially Barsotti-Tate

Galois representations. J. Amer. Math. Soc. 12 _(1999), no. _2, 521-567.

[DRS] B. DE SMIT, K. RUBIN, R. SCHOOF, Criteria for complete intersections. In: Modular

forms and Fermat’s Last Theorem, 343-356, Springer, 1997.

[D1] F. DIAMOND, On deformation rings and Hecke _rings. Ann. of Math. ₍₂₎ 144 _(1996), no.

1, 137-166.

[D2] F. _DIAMOND, The refined conjecture of Serre. In: Elliptic Curves, Modular forms, and

Fermat’s Last Theorem _(ed. J. _Coates), International Press, Cambridge, MA, 1995. [D3] F. _DIAMOND, The _Taylor-Wiles construction and multiplicity one. Invent. Math. 128

(1997), no. _2, 379-391.

[Di] M. DICKINSON, On the modularity of certain 2-adic Galois representations. Duke Math.

J. 109 _(2001), no. _2, 319-382.

[ES] J. ELLENBORG, C. SKINNER, On the modularity of Q-curves. Duke Math. J. 109 _(2001), no. 1, 97-122.

[Fo] J.-M. FONTAINE, Représentations potentiellement semi-stables. In: Périodes

p-adiques, Asterisque 223 ₍₁₉₉₄₎ 321-247.

[FoM] J.-M. FONTAINE, B. _MAZUR, Geometric Galois _{representations.} In: _{Elliptic Curves,}

modular forms, and Fermat’s Last Theorem _{(Hong Kong, 1993),} pp. 41-78, Internat.

Press, 1995.

[F1] K. _{FUJIWARA, Deformation rings} and Hecke _algebras in the _totally real case, preprint (1996).

[F2] K. FUJIWARA, Deformation rings and Hecke _algebras in the _{totally real case, preprint}

(1999).

[F3] K. FUJIWARA, it Level optimazation in the _totally real case, preprint (1999).

[H1] H. _HIDA, On _{nearly ordinary} Hecke algebras for GL(2) over _totally real fields. In:

Al-gebraic number theory, Adv. Stud. Pure Math. 17, 139-169, Academic Press, 1989. [H2] H. _{HIDA, Nearly ordinary} Hecke _algebras and Galois representations of several variables.

In: Algebraic analysis, geometry, and number theory (Baltimore, MD _1988), 115-138,

John _Hopkins Univ. _Press, 1989.

[J] F. JARVIS, Level lowering for modular representations over _totally real fields.

Math. Ann. 313 _(1999), 141-160.

[M1] B. _{MAZUR, Deforming} Galois representations. In: Galois Groups over _Q, vol. 16, MSRI

Publications, Springer, 1989.

[M2] B. _MAZUR, An introduction to the deformation theory of Galois representations. In:

Modular Forms and Fermat’s Last Theorem _(eds. G. Cornell et _{al.), Springer-Verlag,}

New _York, 1997.

[Ra] A. RAJAEI, On _lowering the levels in modular Galois representations of totally

real fields. Thesis, Princeton _University, 1998.

[SW1] C. SKINNER, A. _{WILES, Ordinary representations} and modular _forms. Proc. Nat. Acad.

Sci. U.S.A. 94 _(1997), no. _20, 10520-10527.

[SW2] C. SKINNER, A. WILES, Modular forms and _residually reducible representations. Publ. Math. IHES 89 _(1999), 5-126.

[SW3] C. _SKINNER, A. _WILES, Base _change and a _{problem of} Serre. Duke Math. J. 107 (2001), no. _1, 15-25.

[SW4] C. _SKINNER, A. _{WILES, Nearly ordinary deformations of} irreducible residual represen-tations. Ann. Fac. Sci. Toulouse Math. ₍₆₎ 10 _(2001), no. _1, 185-215.

[Ta]

J. _TATE, Number theoretic background. In: _{Automorphic forms, representations} and

L-functions, Part _2, Proc. Sympos. Pure Math., XXXIII, pp. 3-26, Amer. Math. Soc.,

Providence, R.I., 1979.

[T] R. _TAYLOR, On Galois representations associated to Hilbert modular forms. In: _Elliptic

Curves, Modular _forms, and Fermat’s Last Theorem _(ed. J. _Coates), International Press,

[TW] R. TAYLOR, A. WILES, Ring-theoretic properties of certain Hecke algebras. Ann. of

Math. ₍₂₎ 141 _(1995), no. 3, 553-572.

[Wa] L. WASHINGTON, The _{non-p-part of} the class number in a _{cyclotomic Zp-extension.} Invent. Math. 49 _(1978), no. _1, 87-97.

[W1] A. _WILES, Modular elliptic curves and Fermat’s Last Theorem. Ann. of Math. ₍₂₎ 142

(1995), 443-551.

[W2] A. WILES On ordinary 03BB-adic representations associated to modular forms. Invent. Math. 94 _(1988), no. _3, 529-573. Chris SKINNER Department of Mathematics University of _Michigan Ann Arbor, MI 48109 USA cskinnerCDumich. edu