Congruences modulo $\ell $ between $\epsilon $ factors for cuspidal representations of $GL(2)$

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

M

ARIE

-F

RANCE

V

IGNÉRAS

Congruences modulo ` between ε factors for

cuspidal representations of GL(2)

Journal de Théorie des Nombres de Bordeaux, tome

12, n

o

_{2 (2000),}

p. 571-580

<http://www.numdam.org/item?id=JTNB_2000__12_2_571_0>

© Université Bordeaux 1, 2000, tous droits réservés.

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

Congruences

between 03B5 factors for

cuspidal representations

of

_GL(2)

par MARIE-FRANCE

VIGNÉRAS

Pour Jacques Martinet

RÉSUMÉ. Titre

_{français :}

_Congruences

entre facteurs ~

des

_{représentations cuspidales}

de

_GL(2)

~

p deux nombres premiers

distincts,

F un

corps local

non archimedien de

_{caractéristique}

résiduelle _p, une clôture

algébrique

_du

corps des nombres et corps

résiduel de On _{conjecture que} la

_{correspondance}

locale de

Langlands

pour

GL(n, F)

sur _respecte les _congruences

mod-entre les facteurs

_{L et}

~ de _{paires, et que} la

correspondance

locale de

_Langlands

sur est caractérisée par des identités entre

de nouveaux facteurs L et ~. Nous allons le démontrer

lorsque

n = 2.

ABSTRACT. _p be two different _{prime numbers,} let F be

a

local

non archimedean field of residual characteristic p, and let

be an

_{algebraic closure}

of the field of numbers the _ring of _integers of the residual field of We

_proved

the

_existence

and the _unicity of a

_Langlands

local

_{correspondence}

over for all n &#x3E; _2,

_compatible

with the reduction in

[V5],

without _using factors of _pairs.

We

conjecture that the

_Langlands

local

_{correspondence}

over respects congruences between L and 03B5

factors

of pairs, and that the

_Langlands

local correspondence over is characterized

_by

identities between new L and 03B5 factors. The aim

of this short _paper is _prove this when n = 2.

Introduction

The

_Langlands

local

_{correspondence}

is the

_unique

_bijection

between all irreductible

_{Q£-representations}

of

_{GL(n, F)}

and certain t-adic

which is induced

_by

the

_reciprocity

law of local class field

_theory

when n = 1

(Wpb

is the

_biggest

abelian Hausdorff

_quotient

of

_WF),

and which

_respects

L and s factors of

_pairs

[LRS], [HT], [H2].

Let,o :

F -~

Ze

be a non trivial character. We denote

by

CuspR

GL(n,

F)

the set of

_isomorphism

classes of irreducible

_{cuspidal R-representations}

of

GL(n, F).

When 7r E

_CUSPQi

_{GL(n, F),}

Henniart

_[H1]

showed that 7r is characterized

_by

the

_epsilon

factors of

_pairs

for all Q E

CuspQe

GL(m, F)

and for all m n - 1

_(note

that

_Q)

=

1),

using

the

_theory

of

_Jacquet,

_{Piatestski-Shapiro,}

and Shalika

_[JPS1].

Does this remain true for

_cuspidal

irreductible

_{Ft-representations}

of

GL(n, F) ?

We need first to define the

_epsilon

factors of

_pairs.

Let 7r E

_Cuspe

_GL(n,

_F).

It is known that the constants of the

_epsilon

factors of

_pairs

_{£( 7r, a)}

_belong

to

_Ze

for all Q E

_Cuspe

_{GL(m, F)}

and for all

m n - 1,

and that the conductor does not

_{change by}

reduction modulo t

(this

is

_proved

_{by Deligne}

_[D]

for the irreducible

_{representations}

of the Weil group, and

by

the local

Langlands correspondence

over

_(ae

is true for

cuspidal

representations).

Now let 7r E

_CusPFi

_GL(n,

_F).

Then 7r lifts to

_F)

_[VI,

III.5.10~ .

By

reduction modulo

_~,

one can define

_epsilon

factors of

_pairs

£(7r, o,)

for all a E

_CusPFi

_{GL(m, F)}

and for all m n - 1. Let _q be the order of the residual field of F. We

_expect

that 7r is characterized

by

the

_epsilon

factors

_{£( 7r, Q)}

for all Q, when the

multiplicative

order

of q

modulo f is &#x3E;

n -1;

otherwise,

7r should be characterized

by

less naive but natural

_epsilon

factors. The same should be true when 7r is

_{replaced by}

an

Fe-irreducible representation

of the Weil _group

_WF.

The existence

_[V4]

of an

integral

Kirillov model for 7r E

_CUSPQi

_GL(n,

_F)

seems to be an

_adequate

tool to solve the

_problem.

The

_description

of the

representation

7r on the Kirillov model is

_given

_by

the central character _W7r

and

_by

the action of the

_symmetric

_group

_Sn

_(the

_Weyl

_group of

_{GL(n, F)).}

The action of

_Sn

is related with the

_e(r,

_a)

for all Q as above

[GK,

see the

end of

_paragraph

_7].

When n = 2

Jacquet

and

Langlands

[JL]

described the action of

_S2

on the Kirillov model in terms of

_{£( 7r,}

_X)

=

£( 7r 0 X)

for all

(ae-characters x

of

_{F*, using}

the Fourier transform on F*.

In the case n = 2 and

only

in this

case, we will _prove that two

_integral

F)

have the same reduction modulo t if and

_only

if their central characters have the same reduction modulo t and the factors

X),

X)

have the same reduction modulo f for

_integral

Qg-characters X

of F* when t does not

_{divide q -}

1. When f

_{divides q -}

1

congruences modulo satisfied

by

the

£(1f

0

x)

for all x.

By

reduction

modulo f,

we

_get

that the local

Langlands

_{Fl-correspondence}

for n = 2 is

characterized

by

the

_equality

on L and new E factors of

_pairs.

The field

_Ft

can be

replaced by

_any

algebraically

closed field R of characteristic .~.

The case n = 3 could be treated

probably,

but the

general

case n &#x3E; 4 remains an _open and

_{interesting question.}

1.

_Integral

Kirillov model

The definition of the L and E factors of

_pairs

_[JPS1]

uses the Whittaker

model,

or what is

_equivalent

the Kirillov model. We showed

[V4]

that these

models are

compatible

with the reduction modulo .~.

We denote

_by

_OF

the

_ring

of

_integers

of F. Let R be an

_{algebraically}

closed field of

_{characteristic #}

_p, and

_{let 0 :}

: F - R* be a character such that

_OF

is the

_biggest

ideal on

_{which o}

is trivial. We

_{extend 0}

to a R-character of the _{group N} of

strictly

_upper

triangular

matrices of

G =

GL (n, F)

for n = _E N. The mirabolic

subgroup

P of G is the semi-direct

_product

of the _group

_{GL(n -}

_1,

_F)

embedded in

_{GL(n, F)}

_by

i’

and of the _group embedded in

_{GL(n, F)}

_by

The

_{representation}

TR := of the mirabolic

subgroup

P

(compact

induction)

is called mirabolic. It is irreducible

_(this

is a

corollary

of

[V4

prop. I] ) ,

but it is not admissible when n &#x3E; 2. Lemma.

EndRp

TR ^~ R.

Proof.

This is a

_general

fact: the

_{representation}

_TR is

_absolutely

irreducible

[VI,

1.6.10],

hence

_{EndRp -rR -}

R. From the Schur’s lemma

_{[VI, 1.6.9]}

EndRP

R when the cardinal of R is

_{strictly bigger}

than

diMR

TR

(countable dimension).

There exists an

algebraically

closed field R’ which

contains R and of uncountable cardinal. Two

_{RP-endomorphisms}

of TR

which are

_proportional

over R’ are

_proportional

over R. D

Theorem. An irreducible

_{R-representation}

1f

o f

G is

cuspidal

i f

and

only

if

extends the mirabolic

_{representation}

TR.

Proof.

This results from

_[BZ]

and

_[VI].

_Suppose

that zr is

cuspidal.

Then

1flp

is the mirabolic

_{representation:}

when R = C _see

[BZ,

5.13 &#x26;

5.20~,

when R = _7r lifts _to

Q~

[VI, 111.5.10]

where it is true then reduce.

Conversely,

suppose

1flp

=

TR and R =

111.1.8].

The case of a

general

R is deduced from this two cases

_by

the next

lemma. D

Let G be the _group of rational

_points

of a reductive connected _group over F. We denote

_by

_IrrR

G the set of

_isomorphism

classes of irreducible

R-representations

of G.

Lemma.

₍₁₎

A non zero

_homomorphism

_{of algebraically}

closed

_{fields f :}

R ~

R’

gives

a natural

_injective

_map 7r ~

f*

(1r) :

IrrR

G -~

IrrR,

G which

respects

cuspidality.

(2)

Let 1r’ E

_CusPR’

G. Then there exists an

unramified character x of

G such that 1r’ Q9 x =

f*(1r)

with _{1r E}

CusPR

G.

Proof.

This results from

_[VI].

(1)

f *

respects

irreducibility

[VI, 11.4.5],

and commutes with the para-bolic restriction. Hence it

_respects

_cuspidality.

The linear

_independence

of characters

_{[Vl , 1.6.13]}

shows that if

_{Jr, zr’}

E

_IrrR

G are not

_isomorphic

then

are not

_isomorphic.

(2)

Let Z be the center of G. The _group of rational characters

_X(Z)

is a

subgroup

of finite index in the _group

X ( G) .

This

implies

that there

exists an unramified

_{character X}

of G such that the

_quotient

Z/Zo

of Z

_by

the kernel

_Zo

of the central character of _{7r’ ® x} is

_profinite.

Hence the values of w are roots of

_unity.

We deduce that 7r’

_{0 X}

has a model on R

[Vl, IL4.9).

D

Let 7r E

_{CuspR GL(n, F)}

of central character w. The realisation of 7r on

the mirabolic

_{representation}

TR is called the Kirillov model

K(7r)

of 7r. It is

sometimes useful to use the Whittaker model instead of the Kirillov model.

By

adjonction

and the theorem R

_(the

_unicity

of the Whittaker

_model);

the Whittaker model is the

_unique

realisation of 7r in

_By

definition

for

_{all g}

E G and for all Whittaker functions W E We denote

_by

T(j)

the

_subgroup

of matrices k E

_GL(n,OF)

of the form

for _any

_{integer j}

&#x3E; 0. The

_{smallest j}

&#x3E; 0 such that 1f contains a non-zero

vector

_transforming

under

_r(j)

_according

to the one dimensional character

Theorem. Let 7r E

_{CUSPR GL(n,}

_F)

_of

central character w = w and

con-ductor

_f.

(1)

The restriction

_from

G to P induces a

G-equivariant isomorphism

from

the Whittaker model to the Kirillov model.

(2)

Let 1f’ E

_{CUSPR GL(n,}

F).

There is a natural

isomorpltism

W - W’ :

W(1f)

-&#x3E;

W(1f’)

of

R-vector _spaces

defined by

the condition

Wlp

=

W’lp.

(3)

There is

_unique

_functions

W1r

E

_{W (~r)}

such that

The

_function

_{W7r is}

called the new vector

_of

7r and

_generates

the _space

_of

vectors

_of

7r

transforming

under

r( f )

according

to _WI.

(4) W(-x) is

contained in the

_compactly

induced

_{representation}

ind

Proof.

(1)

There exists W E with

_0,

and

_{f :}

W -

Wp

is a non zero

_{P-equivariant}

_map from 7r to

Indt

_0.

The _map

_f

is

_injective

of

image

indp

1/;,

because

_{End R TR =}

R. We

_get

also

_(2).

(3)

The _space of TR is

isomorphic by

restriction to G’ =

GL(n -

1,

F),

to

_{the space}

of where N’ = N n G’.

As V)

is trivial _on

OF,

the characteristic function of

_{GL(n -}

_{1, OF)}

_belongs

to For the conductor

_[JPS2].

(4)

Let W E The function z -

_{W (xg)}

on the

parabolic

standard

subgroup

PZ is

_locally

constant of

_{compact support}

modulo NZ for all

g E G.

As G =

PZGL(n, OF),

the function W is of

_{compact support}

modulo NZ. D

Let 7r E

_{Irr Ql}

G. Let

_E/(aP

be an extension contained in a finite extension of the maximal unramified extension of

_{Qi. Example:}

the extension

generated

by

the values of

_0.

The

_ring

of

_{integers OE}

is

_principal.

An

OE-free module L with an action of G such that L is a finite

_type

_OEG-module

and such that

_Qi

_{00 E}

L - 7r is called an

OE-integral

structure of 7r. If such

an L is

_exists,

7r is called

_integral,

the

_{representation rtL}

= L

00E Fe

is of finite

_length.

One calls

_{ZP OOE}

L an

integral

structure of 7r. When

_L,

L’

are two

_integral

structures of 7r, then the

semi-simplifications

of

rf L,

rgL’

are

isomorphic

(see

[VI, II.5.1l.b]

when is

finite,

and

[Vig4,

_proof

of theorem

_2,

_page

_416]

in

_general).

When 7r E

_CuspQe

G is

_integral,

rgL

= L

OOE Fe

is

_irreducible;

the

_isomorphism

class rt-x of

rEL

is called the reduction of _{7r; any} irreducible

_cuspidal

_{Ft-representation}

of G is the reduction of an

integral

irreducible

cuspidal

_{Q£-representation}

of G. For all these facts see

~Vl, 111.5.10].

A function with values in

_Qt

is called

_integral,

when its values

_belong

to

Zt .

We denote

_by

_Zt),

_resp.

_W(1r,

_Ze),

the set of

_integral

functions

in the Kirillov

_model,

_resp. Whittaker

_model,

of 7r E G. Let A be

the maximal ideal of

Zi.

The reduction modulo t of an

integral

function

f

is the fonction

_rtf

with values in

_Ft

deduced from

_{f .}

Theorem.

_(A)

Let 1r E G with central characters úJ1f. Then the

following

properties

are

equivalent:

(A.1)

W1f is

integral.

(A.2)

7r is

integral.

(A.3) K(,7r, Zf)

is a

ZP-structure

of

_7r, called the

integral

Kirillov model.

(A.4)

Zt)

is a

Zi -structure

of 7r,

called the

integral

LVjcittaker model.

(B)

When 7r is

_integral,

we have

(B.1)

The restriction to P

_frorrc

_{W(1r, Zt)}

to

_Zt)

is an

isomorphism.

(B.2)

The

_integral

Kirillov model is

_ZtP-

_{generated by}

_any

_{functions f}

with

_{f (1)}

= 1. The

integral

Whittaker model

W(1r, Zt)

is

_ZfG

_{generated by}

the new vector.

(B.3)

Ft

_0zl

K(ir, Zt)

=

K(rt7r, Ff)

is the Kirillov

model,

and

Fe

0zi

W(1r, Zt)

=

W(re1r, Ft)

is the Whittaker model

_of

rf1r.

Proof.

The

_equivalence

of

_(Al)

_(A2)

_{[VI, 11.4.12];}

for the rest

_[V4

_th.2]

and

the last theorem. 0

Corollary.

Let

_{1r, 1r’}

E

_CUSPQl

G

_integral,

with central character _W1f, W1f1 .

Then re1r =

re1r’ if

and

only if

for

all w E and

_for

all

_f

in the

_integral

Kirillov model.

Proof.

Use

_(B.3)

and

_End-p.

D

Questions.

Can one define an

integral

Kirillov or Whittaker model for

7r E

_{Irr Qt G}

_integral

and not

_{cuspidal ?}

What is the action of

Sn

in the Kirillov model ?

2. The case n = 2

We can _go further in the case n = 2. Let _{7r E}

CusPQt

G where G =

GL(2, F).

The restriction of

_{GL(2, F)}

_{to GL ( 1, F)}

= F*

gives

_an

isomor-phism

from to the _space of

_locally

constant functions F* --~

_Qe

with

_{compact support,}

which

_respects

the natural

_{ZE-structures}

C°°(F*, Zt).

The

_unique

non trivial element of

₅₂

is

The action of

_ir(w)

on the Kirillov model was described

by

Jacquet

and

Langlands

[JL,

Prop.2.10

p.46~,

using

Fourier transform for

_complex

rep-resentations.

We choose a

(ae-Haar

measure dx on F*. The Fourier transform of

f

E

C°°(F*, Qt)

with

respect

to dx is

for _any

_{character X :}

F* -~

_Qi.

We choose a

uniformizing

_parameter

_PF of F. A function

f

E

_{C,’ (F*, Qi)}

is determined

_by

the set of functions

_fn

E

_Qt)

defined

_by

_fn(x)

:=

f (pFnx)

for all n E Z. The functions

_fn

_depend

on the choice

of pF.

Exten-sion

_by

zero allows to consider as a

_subspace

of

_Qt),

because

_OF

_{is open in} F* . We have

For a

_given

_{character x,}

the sum is finite. The functions

_{depend only}

on the restriction

of x

to

OF.

Set

6*

:=

Hom(OF’

One introduces the formal series

for all x E

_OF

and for

_{all x}

E

_0*

Jacquet

and

_Langlands

_[JL

_Prop.

_{2.10 page}

_46]

_proved

that the action

of

_{7r( w)}

on the Kirillov model is

_given

_by:

for

_{all X}

_{E6* ,}

all

_{integers n}

E

_Z,

where m = _{-n -}

0 ~ ~

for _some

constant

_c(?)

E

(~e

and some

_integer

f (?)

E Z. The formula and

_c(1f0

are

independent

of the choice of dx. The formula is

equivalent

to

for all

_{Qg-characters x}

of

_OF,

where the

_epsilon

factor is

On calls

_c(7r)

the constant and

_{f (~r)}

the conductor of the

_epsilon

factor

_c(7r).

They

both

_depend

on the choice of the non trivial

ZP

which was

fixed,

but not on the choice of dx or on _pF.

_Jacquet

and

Lang-lands used

_{complex representations}

but their method is valid when the field of

_complex

numbers is

_replaced

_{by Qt,}

because one uses

only integrals

of

lo-cally

constant functions on

_compact

sets. There is no

_problem

of

_vanishing

We suppose that dx is a

Zi-Haar

measure on F* wich is not divisible

by

£. Let

£ = the Fourier transform of

_{C~ ° (OF, Ze)}

We have ,C and ,C =

C~ ° (OF,

1 Zf)

if and

_{only if q fl}

1 mod £

(V2~ .

In

_general,

we

_separate

the

£-regular

_part

X of

OF

from the

.~-part

Y of

_0*

which is a

_cyclic

_group of order m = The volume of X for

dx should be a unit in

_{Z~ ;}

we can _{suppose it is}

equal

to 1. The _group of

_{Qf-characters}

_satisfy

X X

Y.

A

_general

character

in OF

is now

written as _XM

where X

E

X

and _p E

V,

and a function v :

bh -

is

thought

as a function v :

C(Y,

with

The

_Zt-module

,C consists of all functions v : 5l - L with

_compact

support,

where

is the free

_Zi-module

with basis the characters _{y : p -} of

Y

for all

We need some

_elementary

linear

algebra.

The

_Zg-module

L is the set of functions v E

C§fl~ (9,

_Ql)

such that

belongs

to

_{C(Y, Ze).}

The

_{orthogonality}

formula of characters

_gives

for all v E

C(Y,

For the usual

_product,

_{C§° (9,}

is an

_algebra.

Lemma. Let v E

jQe)

-(i)

The inclusion vL C L is

_{equivalent to v}

E L.

(ii)

The

_equality

vL = L is

equivalent

to v E L and

v(p)

_E

all /-I

_E

Y.

(iii)

The inclusion vL C AL is

_equivalent

to _{v, y} &#x3E;E A

_{for all y}

E Y

(A

is the maximal ideal

_of

Proof.

(i)

The inclusion vL c L is

_equivalent

to

_{vz, z’}

&#x3E;= _v, &#x3E;E

Z~

for all _z, z’ E

_Y,

which is

_equivalent

to v E L.

(ii)

vL = L _means that _vz for z _E Y is _a basis of L. We have _vz =

¿z’EY

v, z lz’

&#x3E;

z’,

hence vL = L _means that

The Dedekind determinant

_det(

_{v, z-lz’}

is

_equal

to

_ITp,EY

_(see

[L]

exercise 28 page

495).

Let 7r E

_Cuspo

G

_integral.

As

_{7r( w)}

is an

_isomorphism

of the

_integral

Kirillov

_model,

the function

satisfies

_{c(7r © x)L}

= L for all

character X

We

apply

the lemma to

c(7r

Q9

_X).

We define new

epsilon

factors

for all _y E Y. As have &#x3E; 2 for 7r E

_G,

we have =

f(7r

0

_Jl)

&#x3E; 2 for all _Jl E

Y.

When Y is trivial

_(i.e.

_{q 0}

1

_they

are

simply

the usual ones.

Theorem.

₍₁₎

Let 7r E

_CusPQt

G

_integral.

Then the constant

_of

the

_epsilon

factor

is a unit

c(7r)

E

Ze

and the new constants

_c(7r),

_y &#x3E;E

_Zt

are

integral, for

all _y E Y.

(2)

Let 7r, 7r’ E

_CusPQt

G

_integral

with central characters W7r’ . Then

=

rf1T’ if

and

only

if

= and their _new

epsilon factors

have

the same reduction modulo t: the conductors

f(1T0X)

=

f(7r’0X)

_are

equal,

and the new constants have the same reduction modulo t :

for

all y

E

_Y,

and all

_{Qg-characters x}

E X .

Proof.

With the last

_corollary

of the

_paragraph

_(1),

rg7r =

rl!7r’

if and

only

if = and

for all

_In

E

_Zt)

and all n E Z. With the

_lemma,

we deduce the

theorem. D

We

_apply

now the theorem to

_{representations}

over

F~.

Any

7r E

_Cuspp

G lifts to

_Qg

and we can define

_epsilon

factors

for

_{all y}

E Y and

_{all X}

E

_Hom(O*,

Ft)

=

Hom(X,

F~ ),

by

reduction

modulo .~.

_They

are not zero for _any

(y, x) .

Corollary.

7r, 7r’ E

_Cuspp

G are

isomorphic

if

and

only if they

have the same central character and the same

epsilon

factors

Final remarks.

_a)

When n &#x3E;

_2,

the _groups

_{GL(m, OF)*}

for m n -1

replace

OF.

b)

Using

the

_{explicit description}

for the irreducible

_{representations}

of di-mension n of

WF

_[V3],

one could

_try

to _prove a similar theorem for the

irreducible

_integral

_{Q -representations}

of

_WF

of dimension n. To _my

know-ledge

this is a known and harder

problem,

which is not solved in the

_complex

case.

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Marie-France VIGNGRAS

Institut de _{Math6matiques} de Jussieu

Universite Denis Diderot - Paris 7 - Case 7012

2, place Jussieu 75251 Paris Cedex 05

France