Davenport-Hasse relations and an explicit Langlands correspondence, II : twisting conjectures

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

C

OLIN

J. B

USHNELL

G

UY

H

ENNIART

Davenport-Hasse relations and an explicit Langlands

correspondence, II : twisting conjectures

Journal de Théorie des Nombres de Bordeaux, tome

12, n

o

_{2 (2000),}

p. 309-347

<http://www.numdam.org/item?id=JTNB_2000__12_2_309_0>

© Université Bordeaux 1, 2000, tous droits réservés.

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

Davenport-Hasse

relations

and

an

explicit

Langlands correspondence,

II:

_twisting

conjectures

par COLIN J. BUSHNELL et GUY HENNIART

To _Jacques Martinet on his retirement

RÉSUMÉ. Soit F une extension finie de La

correspondance

de

Langlands

donne une

bijection

_canonique entre l’ensemble des classes

_{d’isomorphisme}

de

_{représentations}

irréductibles de

di-mension n du _groupe de Weil de

F,

et l’ensemble des classes

_{d’isomorphisme}

de

_{représentations}

irréductibles

supercus-pidales

de

_{GLn (F) .}

Nous

_regardons

le cas où n est une _puissance de p, n =

pm.

Dans _un travail

antérieur,

_nous avons construit une

bijection

03C0 de sur

grâce

à la construction d’un

changement

de base modéré

_(non

nécessairement

_galoisien).

Si

le

_changement

de base satisfait certaines relations

_{conjecturales,}

dites de

_{Davenport-Hasse,}

et si l’on admet l’existence d’une

cor-respondance

de

_Langlands

en

_degré

_pm, alors _cette

correspon-dance n’est autre que 03C0. Dans ce _papier, nous ne _{supposons pas} a

priori l’existence d’une

_{correspondance}

de

_Langlands,

mais nous

voulons _prouver

_directement,

en _supposant vérifiées les

conjec-tures de

_{Davenport-Hasse,}

_que 03C0 est une telle

_{correspondance.}

Nous réduisons le

_problème

à des

_propriétés

élémentaires des

con-stantes

_{locales 03B5(03C01}

x _pour ~

_(qui

_peuvent

d’ailleurs se déduire de l’existence de la

_{correspondance}

de

Lang-lands et de

_{propriétés analogues}

du côté

_galoisien).

Au cours de

cet

_article,

nous obtenons de nouvelles

_propriétés

inconditionnelles

pour 03C0, et décrivons

complètement

les

propriétés

de rationalité des

fonctions L et des constantes locales de _paires pour

GLn(F).

ABSTRACT. Let

_F/Qp

be a finite field extension. The

_Langlands

correspondence

gives

a canonical

_bijection

between the set of _equivalence classes of irreducible n-dimensional _{representations} of the Weil _group of F and the set of

equivalence

This paper is concerned with the case where n =

pm.

In earlier

work,

the authors constructed an

_{explicit bijection}

03C0 :

using

a non-Galois tame base

_change

map. If this tame base

_change

satisfies a certain

conjectured automorphic

Davenport-Hasse

_relation,

and there exists a

Langlands correspondence

in

p-power

degree,

then 03C0 is the

Langlands correspondence.

This

paper is concerned with the

problem

of

showing,

without

assum-ing a _priori the existence of the

Langlands

correspondence,

that

(on

the

_{Davenport-Hasse}

_{conjecture) 03C0}

preserves local constants

of

_pairs,

and so is a

Langlands correspondence.

The

principal

ob-struction is the lack of

_knowledge

of certain

_{elementary properties}

of the local constant

_03B5(03C01

for ~ We state

these _properties as _conjectures

(which

are

_certainly

_true, as

con-sequences of the existence of the

Langlands correspondence

and

analogous

properties of the

_{Langlands-Deligne}

local

_constant)

and

show that

_{they imply}

the desired result: 03C0 is a

Langlands

corre-spondence.

In the process, we _prove several new unconditional

results concerning 03C0, and

give

a

_complete

account of the

ratio-nality

properties of L-functions and local constants of _pairs for

GLn(F).

Introduction

Let _p be a

prime

number. We fix an

algebraic

closure

Qp/Qp

of the

p-adic number field

_Qp,

and consider a finite field extension

_F/Qp

contained

in We write

_1NF

for the Weil _group of We use the standard notation _oF for the discrete valuation

_ring

in

_F,

_{pF for the} maximal ideal of

oF,

Ul -

1+pF

for the _group of

_{principal units,}

and vF for the normalized

valuation Z.

The

_{Langlands Conjecture}

for

_GLn(F)

has been

_proved,

for

all n &#x3E;

1,

in

[16]

and

_[20].

The methods of these _{two papers} are

different,

but

they

both

rely

on constructions from

[14], [15].

Their

approach

is

global

in

_nature,

with a

significant geometric

_component;

they

can therefore _say

nothing

explicit

about the local

_Langlands

_{correspondence}

whose existence

_they

establish. To

_get

an

explicit

correspondence,

once has to work in a local

framework,

using

the classification of

_{representations}

from

_[11].

This _paper

is concerned with the

_problem

of

_obtaining

a local

proof

of the existence of

the

_{correspondence}

in these terms. We deal

_only

with a

_special

_case, but

one which is

_particularly

subtle and basic to the overall

_picture.

1. To

_specify

this _case, we take an

_integer

_{m &#x3E;} 0 and write for the set of

_equivalence

classes of irreducible continuous

_{representations}

a of

a 0 X ’;p a

for _any unramified

_{quasicharacter X =11}

of

’WF.

There is a

cor-responding

notion on the other side: we write for the set of

equiv-alence classes of irreducible

supercuspidal

representations

7r of

such that

_{X7r r.}

7r for _any unramified

quasicharacter x #

1 of FX.

(Here

and

throughout

X7r denotes the

representation

g H

x(det g)7r(g) .) A

spe-cial case of the

Langlands

Conjecture

for

GLn(F)

predicts

the existence of a

_bijection

which _preserves local constants

_of

_pairs:

for oi E and 7ri =

(ai),

i =

1, 2. Here,

the first E is the

lo-cal constant of

_{Langlands-Deligne}

_[13]

and the second that of

_Jacquet,

Piatetskii-Shapiro

and Shalika

_{[22], [26],}

attached to a

complex

variable s and a non-trivial character

OF

of the additive _group of F. The

family

has several other formal

_properties;

_together

with

_(1),

these

_specify

it

_uniquely.

2. In

_[5],

we constructed a

bijection

using

the

_description

of the elements of

_implied

_by

_[11]

and group-theoretic constructions which are, at least in

spirit, elementary

and

straight-forward. The

_family

has _many of the

_properties

demanded of a

Lang-lands

_{correspondence}

_(see

_especially

_{[5, §2]).}

One also knows from

_[6]

_that,

for 7 we have

where _xa is an unramified character of F~ of finite order

strictly dividing

pm.

However,

we do not know at this

_stage

whether

_enjoys

the critical

property

of

_preserving

local constants of

_pairs.

The construction of the

_{correspondence}

is based on that of a certain

tame base

_change

_map. If this tame base

_change

satisfies the

_automorphic

Davenport-Hasse

relation

_(recalled

below as

_Conjecture

A)

then indeed

7r~ = ~

for all m and

F,

~6~ .

3. In this _paper, we are more concerned with how one could

_proceed

with-out

_assuming

a

_priori

the existence of the

Langlands

correspondence.

In

particular,

we consider the

_problem

of

_showing

_directly

that the

correspon-dence

_(7r§)

preserves local constants of

_pairs.

From now _on, we make no use

of

the existence

of

the

Langlands

As we shall _see, the

principal

obstruction to

_carrying

out this _programme

is our

_present

_ignorance

of the structure and

_significance

of the local

con-stant

_6(-xl

x _ir2,

_{8,’OF) -}

In this paper, we list as

_conjectures

some basic

properties

of the function

_{(7ri , 7r2)}

_{- e (7ri}

x _2,

(the

_"twisting

conjec-tures" )

and

_investigate

their consequences. We show that

they

imply

many cases of the

_{Davenport-Hasse}

relation. If we assume the full

Davenport-Hasse

_relation,

_{they imply}

that the

_family

_{(7r§) preserves}

local constants

of

_pairs

_{up to}

_sign

_{(if p}

is

_odd)

or

_exactly

(if

_{p =}

2).

To eliminate the

sign

ambiguity,

we have to assume a rather

_deeper

_conjecture

(Conjecture

B

_below)

_concerning

the local constant. This

_conjecture

_implies

that

_{17r’ I}

has all the

_properties

demanded of a

_{Langlands correspondence.}

4. We now

_give

an abbreviated account of these

_conjectures.

_They

are all

suggested by analogous

properties

of the

_{Langlands-Deligne}

local constant

~(~1 ® ~2~ s~ ~F)~

oi E

Temporarily

let 1I’"i be an irreducible smooth

_{representation}

of

_{GLni (F),}

fori=1,2.

We have

where x _~r2,

_OF)

is an

integer

and q

=

qF is the size of the residue field

oF/pF

of F. An immediate _consequence of the definition

_[22]

of the local

constant is that

for _any unramified

_{quasicharacter X}

of FX and _any c E FX with

_{VF(C) =}

~2, ~F)~

Our first

_conjecture

is a refinement of this

_property.

Let

_K/F

be a finite tame

_extension,

and let

be the tame base

_change

_{( "tame lifting" )}

_map defined in

_{~5) .}

We write vr

for the

_{contragredient}

of 7r.

Twisting properties conjecture.

Let 7ri e

~4~.(~)~

i =

1, 2,

and

sup-pose that 7ri

~

X*2

for

any

ramified quasicharacter x

of FX.

(i)

There exists c =

c(7ri

_x

FX,

determined modulo such that

for

any

tamely ramified quasicharacter

X

of

FX.

(ii)

Let

_K/F

be a

finite

tame

_extension,

and

_put

’ØK

=

_OF

o

TrK/F.

Suppose

that

_{~rK ~}

_for

_any

_{tamely ramified}

_{quasicharacter}

_X

_of

KX. Then

-For the definitive statement of this

_conjecture,

see

_§2

below.

Remark. This

_conjecture

is

_certainly

_true,

as a _consequence of results in

[8]

and the existence of the

_Langlands

_{correspondence.}

Direct

_proofs

are

known for many cases, but we do not consider the matter here.

5. We recall from

_[6]

the

_{automorphic Davenport-Hasse}

relation. We take

an odd

_prime

number £ =I

_p and a

totally

ramified extension

K/F

of

degree

~. We need the Adams

_operation

7r H

-xt

on defined in

[6, §3]

or

[5, §10],

and also the unramified character zl =

_L1t,m

of Fx defined

_by

is the

_{Legendre symbol.}

The

_{Davenport-Hasse}

relation is then

Conjecture

A. Let 7ri E =

1, 2,

and

suppose that

for

any

unramified quasicharacter x

of KX.

Then

This

_conjecture

is known when ml or _m2 is 0

[6,

_Proposition

4.1].

In all

other _cases, we have =

1,

_so the relation then reduces to

Remark.

_Conjecture

A is a test of the "correctness" of the definition of

lK/P.

The

_Langlands

_{correspondence}

_~,Gm~

_gives

a _map

~4~(7~)

corresponding

to the _{restriction map}

_{9~(~) 2013~ ~m (I~);}

this

sat-isfies the

_analogue

of

_Conjecture

A

_[6,

Theorem

_{2.2~ .}

6. The final

_entry

in our list of

_conjectures

is more

_overtly

arithmetic in

nature. For

_this,

we need the normalized classical Gauss sum x E

_F~,

defined in

_[18]

and recalled in

_3.3,

3.4 below.

Conjecture

B. Let E i =

1, 2,

and

let w2

denote the cen-tral

_{quasicharacter}

_Suppose

that 7rl

~

X7r2,

for

any

ramified

quasicharacter x of

px. Then

Remark. In the case of _ml or _m2

_being

_zero,

Conjecture

B is

proved

in

[5,

1.4);

the

_general

case follows from

[8]

via the

correspondence

7. We

_give

a _summary of our results.

_{In ~ 1,}

we extend the domain of defini-tion of the

_{correspondence}

to a

larger

_set,

using

unramified

automor-phic

induction

_[21]

and the

_{Langlands-Zelevinsky}

classification. The

_point

is that this set is stable under the

_operations

of base

_change

_(in

the sense of

[1])

in

_{arbitrary degree}

and

_automorphic

induction _{in p-power}

_degree.

We show that the extended

_{correspondence}

commutes with such

_operations,

_up

to

_twisting

with unramified characters of finite _p-power order. The results of

₃₁

are

completely

unconditional,

depending

on no

_conjecture.

Thus

they

have some

_independent

_interest,

and

_they

also form the first

_step

in the

proofs

of our later results.

In

_§2,

we

_give

the definitive statement of the

_{twisting properties}

conjec-ture. From

_this,

we deduce in

§3

some

_rationality

_properties

of the numbers

e(7rl

x

7T2, -

OF).

Here we

_employ

a result

(Theorem 3.2)

describing

the ac-tion of the _group

_Aut(C)

of all

_(not

_necessarily

_continuous)

_{automorphisms}

of C on L-functions and local constants of

_pairs.

This

_{again depends}

on no

conjecture,

and is

_proved

in

_§6.

These

_rationality

_properties

have

_interesting

_{consequences.}

_Always

as-suming

the

_{twisting properties conjecture,}

we

_get:

Theorem 1.

_Suppose

that

_K/F

is

_{totally ramified of}

_prime

_{degree f}

and that the normal closure

_{of K/F}

has odd

_degree

_prime

_{to p.} Then

_Conjecture

A holds

_for

_K/F.

Further: Theorem 2.

(a)

Conjecture

B holds

_{when p}

= 2.

(b)

When

_{p is}

_odd,

_Conjecture

B holds _{up to}

_sign.

8. Our main

_result,

of which the

_following

is

_only

an

_approximate

state-ment, is

_given

_{in §5:}

Theorem 3. Assume

_Conjecture

A and the

_{twisting properties conjecture.}

Then:

(a)

The

_{corresporcdence}

_compatible

with tame base

_change

in

p-prime

degree,

with

_cyclic

base

_change

in

_{arbitrary degree,}

and

_automorphic

induction in _p-power

_degree.

(b)

If p

=

2,

the

_{correspondence}

local constants

_of

_pairs,

for

oi E

9" (F)

and ~ri =

(c)

If

p is

odd,

formula

(*)

holds up to

sign.

It holds

exactly if Conjecture

Thus our

_conjectures

imply

the

Langlands

Conjecture

in p-power

dimen-sion,

via a

proof

in the

spirit

of

[23]

(the

case

pl

=

2)

and

_[17]

_(p’"‘

=

3).

We also note that the

_properties

listed in Theorem 3 determine the

corre-spondence

just

as in

[6].

1.

_Explicit

wild

_{correspondences}

In this

_introductory

_section,

we first recall from

[5]

the construction of

the

_bijections

There is a natural _way to extend the definition of

7r~

to

give

a

bijection,

at

the level of

_{equivalence classes,}

between irreducible

_{representations}

of

_’1~F

of dimension

_p"2

and irreducible

_{supercuspidal}

_{representations}

of

_{GLp. (F),}

for each and m &#x3E; 0. We extend the definition

_further,

to a set which

is stable under certain

_operations

of base

_{change, automorphic}

induction and tame

_lifting.

The main

_point

of the section is to show that this extended

correspondence

is

_compatible

with these

_operations,

_{up to}

_twisting

_by

an

unramified

character

_of

_finite

_p-power order.

All of the

_arguments

and results of this section are

_completely

uncondi-tonal.

_However,

_they

also form an

_important

first

_step

in the

proof

of the

more

_precise,

but

_conditional,

results below.

1.1. Let

_{K/ F}

be a

_{finite, tamely}

ramified field extension. In

[5],

we

con-structed a canonical _map

The main

_properties

of this "tame base

_change"

_map are listed in

[5, §1].

We

_recall,

in

_particular,

that is transitive in the field extension

_K/F,

and natural with

_respect

to

_isomorphisms

of the extension

_K/F.

If

_K/F

is a tame

_cyclic

_extension,

base

_change

in the sense of

Arthur-Clozel

_[1]

likewise

_gives

us a _map

We have

_(see

_{[5, 1.8]):}

Proposition.

(i)

Let

_K/F

be a

finite cyclic

extension

[K:F].

Then

bK/F(7r)

=

for

_E m &#x3E; 0.

(ii)

If

K/F

is

_{unramified of degree}

_p and 7r E

_A;7(F),

there exists an

unramified

quasicharacter

X1r

of KX,

of

finite

order

strictly dividing

pm,

such that =

X1r

The

_precise

relation between

_bK~F

and

_LK/F,

when

_K/F

is unramified of

_degree

_p, remains unknown.

_{This, however,}

will not trouble us.

1.2. We recall the definition of For each m ~

0,

we define a subset

of as follows. A

_{representation}

a E

_{~m (F)}

lies in

if there is a tower of fields F =

Fo

_C

Fl

_{C " - C}

Fm,

with each

cyclic

and

_totally

ramified of

_degree

_p,

_together

with a

quasicharacter x

of

Fm,

such

_(Here

and

_throughout

we write

_IndK/F

to

mean where

K / F

is a finite field

extension.)

There is an

analogous

subset of a

representation

7r E

r(F) liesinAc(F) if there is a tower of fields F = Fo C Fl

C ’" C

_Fm,

with each

_cyclic

and

_totally

ramified of

_degree

_p,

_together

with a

quasicharacter x

of

_Fm,

such that

where i denotes the

_operation

of

_{automorphic induction,}

in the sense of

~21~.

..

We know from

_[19]

that there is a canonical

bijection

as follows. If Q E is

_given

_by

the tower of fields and the

quasicharacter X

as

above,

we set

Remark. The definition we have used for ’7r is not

_apparently

that of

_[19].

However,

the two versions are

_equivalent,

as follows from

[10, 2.6~.

1.3. We fix a e we view a as a

homomorphism

_{WF -}

GLn(C) ,

where n =

pm. Composing

with the canonical

projection,

_we

get

a

projec-tive

_{representation 3 :}

_{’WF -3}

_{PGLn (C)}

with

_finite

_image.

We _may

_identify

this

_image

with

_Gal(E/F),

for some finite Galois extension

E/F.

Let P

denote the inverse

_image

in

W F

of some

_{p-Sylow subgroup}

of the

_image

of

3, and let K C E be the fixed field of P. Thus

_{p t}

_[K:F]

and P =

_WK.

The restriction = _(J

1

’WK

is

_effectively

an irreducible

representation

of a central extension of a finite _p-group and so

a K

E In this

situation,

the

_{representation}

_7r§(a)

is defined to be the

_unique

element

7r E

_.~4mr(F)

such that:

where,

as

usual,

W7r denotes the central

quasicharacter

of 7r. For a full

account of this

_definition,

and of the

_properties

of the _maps

_7r~,

see

[5,

especially

§2].

The tame field extension

_{K/ F}

constructed here is determined _{up to}

F-isomorphism ;

we refer to it as a

defect field

for a. The

defect of a

is the

It will be useful to record the behaviour of the defect field under various

change

of base field

_operations.

Proposition.

Let a E

_’.,‘;(F)

have

_{defect field}

_E/F,

and let

_K/F

be

tamely

ramified of prime degree.

(i)

If

the

_field

extensions

_K/F,

_E/F

are

linearly disjoint,

then

QK

has

defect field KE/K.

(ii)

Otherwise,

(T has a

defect field

E/F

which

contains K,

and then

E / K is

a

defect

field for

aK .

Proof.

The

_argument

of

_{[5, 2.7]}

_adapts

_easily

to the

_present

case, so we

omit the details. D

1.4. For an

integer

1,

we denote

by

~F(n)

the set of

equivalence

classes

of irreducible continuous

_{representations}

of

’WF

of dimension n.

_Likewise,

let be the set of

_equivalence

classes of irreducible

_{supercuspidal}

representations

of

_GLn(F).

Let a E

(p)?

and let

d &#x3E;

1 denote the number of unramified

charac-ters X

of Fx’’’ such

_{that X 0}

(T. Then d =

pr,

where

0

_{r x m.} If

E/F

is unramified of

_degree

_d,

there is a

representation T

E

_~~Ly.(~)?

_uniquely

determined _{up to} the action of

_Gal(E/F),

such that a = We define

Then,

in

_support

of a claim made in the introduction to

_[5],

we have:

Proposition.

Let a E and

_{put 7r =}

_~r,~(Q).

(i)

We have 7r E and induces a

_bijection

which is natural with

_respect

to

_{automorphisms of}

the base

_field

F.

(ii)

We have

_{7r~(a) =}

7r and W7r = det _a.

Moreover,

for

_a

quasicharacter

X

of

F’ and or E

!90 (pm),

we have

Proof.

Write a =

IndE/F(r),

T

E!9’m’-r(E)

and

E/F

unramified of

_degree

pr

as above. Set _p = As

’Y ranges over

Gal(E/F),

the represen-tations 7’ are distinct. Since is

_injective

on and natural

with

_respect

to

_{automorphisms}

of

_E,

we deduce that the

_{representations}

E are also

_pairwise

distinct. It follows that is

super-cuspidal

(cf.

[10, 2.6]

and so C

We can construct a _map

90 (pm)

as follows. For 7r E

A9F (pm),

let d be the number of unramified

_{quasicharacters X}

of F" such that

_{x7r l3£}

7r.

Then d =

pr,

for some r x m. Let

E/F

be unramified of

_degree

d. There is

then _p E such that 7r =

to the action of

_Ga,l(E/F).

We _may

_{write p}

=

~r~_T(T),

_{r E}

~~ r(E).

The

representation

Q = is

irreducible,

and

7r~(a) =

1r,

by

definition.

This _process 1r - a is then inverse to

7r~,

which is therefore

_bijective,

as

required.

The

_remaining

assertions of the

_proposition

are

straightforward.

D 1.5. Now let

_9F(n)

denote the set of

_equivalence

classes of

_{q&#x3E;-semisimple}

representations

of the

_Weil-Deligne

group of F. We define a subset

QF((p))

of as follows: a

_{representation}

_p lies in this subset

_if,

when viewed as a

_{representation}

of

_’WF,

_every

_composition

factor of _p has p-power dimension.

Similarly,

let

_AF(n)

denote the set of

_equivalence

classes of irreducible smooth

_{representations}

of There is an

analogous

subset

AF((p))

of

_{representation}

lies in this subset if its

_{supercuspidal}

support

consists of elements of for various _{r &#x3E;} 0.

We

_briefly

recall a classical idea

[19], [25], [28].

Suppose

_given,

for each

n ~

1,

a

bijection

Ao :

!90 (n) --+

A9F(n),

which takes determinants to

central

_{quasicharacters}

and is

_compatible

with

_twisting

_{by quasicharacters}

of FX. There is then a standard _way of

extending

the

family

(Ag)

to a

family

of

_bijections

_{An :}

_{,A,F(n), n ~}

1. This

_technique

enables

us to extend the

_family

to a

family

of

_bijections

Remark. Thus we have

7rF(pm)

I

7r~.

From now _on, we omit

the adornments F etc from the

_notation,

at least when there is no fear of

confusion.

We will

_only

need to know this

_{bijection explicitly}

in one case. We take Q E

_9F((p))

n

_9F(n),

and we assume that a is in fact a

semisimple

representation

of ’WF. Thus or = _(B

Qt, with Qi E for

various

_integers

_{mi ~} 0.

Set 1ri

= and let _1f be the

_{representation}

of

_GLn(F)

_{parabolically}

induced

_by

7ri Q9 ... Q9 1ft. One then

gets

easily:

Lemma.

_Suppose

that the

_{representation}

7r is irreducible. Then 7r.

If

_K/F

is

_cyclic,

then restriction induces a

_~K((p)).

If

K~F

is

_cyclic

of

_degree

_p, induction

_gives

a _map

~K ( (P) ) ~ ~F ( (p~ ) .

We

show that a similar

phenomenon

occurs on the other side.

Proposition.

(i)

Let 7 E

_AF((p)),

and let

_K/F

be

_cyclic.

Then

_bK/F(1f)

lies in

_,A.K((p)).

(ii)

Let

_K/F

be

_{cyclic of degree}

_p, and let 7r E

_AK((p)).

Then E

Proof.

In

_(i),

it is

_enough

to treat the case where

_{K / F}

is of

_prime

de-gree

and,

since base

change

respects

supercuspidal

support,

we _may take

7r E for some m. If

_bK/F(7r)

is

_{supercuspidal,}

there is

_nothing

to prove.

Otherwise,

b K / F (7r)

is

irreducibly parabolically

induced from a

rep-resentation of the form

_@’T’

p,

where _{p is}

_{supercuspidal and 7}

_ranges over

Gal(K/F) [10, 2.6].

Further,

the factors are distinct. Thus

[K:Fl

=

p

and pry

E It follows that

_AF((p)),

as

_required.

In

_(ii),

we _may

again

take 7r E If

_iK/F(7r)

is

_{supercuspidal,}

there is

_nothing

to prove.

Otherwise,

_iK/F(1r)

is

_{irreducibly parabolically}

induced from a

representation

~x

_Xr, where T E

_{and X}

ranges over the _group of characters of Fx trivial on norms from K

(see [10, 2.6]

again).

This

_surely

lies in

_AF((p)).

D

1.6. We are now in a

position

to formulate and prove the main result of

this section.

Theorem.

_(i)

Let a E and

_put

1f =

7r~((j).

Let

KIF

be _a

finite

cyclic

extension. There is an

unramified

character _X,

of KX, of finite

order

dividing

p’’n,

such that

If

p

~’

[K:F],

then X, = 1.

(ii)

Let a E and

put

1r =

7r~(a).

Let

K/F

be a

finite

tame

extension. There is an

_unrarnified

character _X,

_{of KX, of finite}

order

dividing p’n,

such that

(iii)

Let

_K/F

be

_{cyclic of degree}

_p, let T E and

_put

_p

_{= 7r;ae(T).}

There is an

unramified

character _X,

of px, of finite

order

dividing

such that

-The character _{xa is} trivial when

_{K/F is}

_unramified.

Proof.

In both

_(i)

and

_(ii),

_{by transitivity}

it is

_enough

to treat the case

where

_K/F

has

_prime

_degree

~.

Step

1. In

_(ii),

if

_{K/ F}

is

_cyclic,

the result is

_given

_{by Proposition}

_1.1,

[5, 2.3],

and

_{[6, 1.8.1].}

We therefore _prove

_(ii)

under the

_assumption

that

K/ F

is not

_cyclic.

Thus the

_{prime i =}

_[K:F]

is odd. Let

_L/F

be the normal closure of

_{K / F,}

and

_E/F

the maximal unramified subextension of

L~F.

Thus

_{L~E, E/F}

are

cyclic

_and,

_by

the first _case, =

Xl1r(aL),

for Xi unramified.

Thus,

since

_L/K

is

_unramified,

there is an unramified

character _x2 of K" such that

_{X21r(aK), lK/F(1r)}

have the same

_image

under

lL/K.

By

[5,

Theorem

_1.3],

this means that =

X31r(aK),

with

unramified. The

_{representations}

have the same central

quasicharacter

det

_QK,

so

XP3m

_{= 1,}

as

required.

Step

2. We now _prove

(i)

in the case (1 E If _p

_{~’ [K : F],}

the result is

given

by

[5,

Th.

_2.3(vi)~.

If

_K/F

is unramified of

_degree

_p, it is

_{given by Step}

1. We therefore need

_only

consider the case where

K/F

is

_totally

ramified of

_degree

_p. If we in fact have (1 E then

_~19~.

In

_general,

let

_E/F

be the normal closure of a defect field for (1. Then

E/F

is tame

_Galois,

and

(1E

E The

_metacyclic

base

_change

_operations

bE/p, bKE/F

etc. are well-defined

(cf.

[4,

16.4,

16.6])

with the obvious

transitivity

properties. Thus, using

the tame case

(ii),

with _X, unramified. Since we have

and

_this,

_by

the tame case

_again,

is

X2bKE/K(7r(aK)),

where _{X2 is}

unram-ified. In other

_words,

there is an unramified character _x3 of KX such that

7 have the same base

change

to

KE/K.

When

UK

is

irreducible

_(that

_is,

_7r(aK)

is

_{supercuspidal),}

it follows from

_[5,

Theorem

1.3]

that the

_{representations}

differ

_by

a tame character of .K~. That

_is,

_bK/F(1f)

=

Q7r(aK),

for _some tame character _a of KX.

Comparing

central

_{quasicharacters,}

we

_get

=

1,

whence a is unramified

and the assertion follows.

We therefore assume that

QK

is not irreducible. Thus there is a rep-resentation T E such that

a K

=

7’)’, with, ranging

_over

Gal(K/F)

(and

the 7’)’ are

_pairwise

inequivalent).

Thus

7r(aK)

is

paraboli-cally

induced from

_{~,~ p~,}

where _p

_{= 7r(7).}

_{Similarly, by}

_{[10, 2.6],}

_bK/F(7r)

is

_{parabolically}

induced

_by

_{(&#x26;7 p~ly ,}

for some _pi E

_{A;:"-1 (K).}

_{By choosing}

pi

properly

within its

Gal(K/F)-orbit,

we

_get

_bKE/x(pl)

=

bxE/x(P),

and

we are reduced to the

_{supercuspidal}

_case, which has

_already

been settled.

Step

3. We now treat

_(i)

for

_general

Q E Thus we have an

unramified extension

_E/F

of

_degree

_pr,

r ~

1,

and T E

_{gwr (E)}

such that

Q =

IndE/F(T).

If we

_put

_p =

7r(T),

then _7r =

iEIF(P)-

Suppose

first that

_K/F

is not unramified of

_degree

p. Then

a K

=

IndKE/K(7KE)

and =

iKE/KbKE/E(P).

The result now follows from the case of

_Step

2 and the definition of 7r. If

K/F

is unramified of

degree

_p, the result is

immediate.

Step

4.

In

_(iii),

the case of

K/F

unramified is

straightforward.

We

there-fore assume that

K/F

is

_totally

ramified. We set Q =

IndK/F(T),

7r =

7r E We write 7r =

7r(al).

The

representation

is

irre-ducibly

parabolically

induced from with distinct factors

_{p~7 and q}

ranging

over

Gal(K/F).

By

_part

(i),

is an unramified twist of this

representation.

Therefore some unramified twist of ~1 contains T on

_{W K.}

Therefore X17a,

where X

is unramified

and 77

is a character of F’

vanishing

on norms from K. Since a is induced from

K,

we have _17a =

~,

and the result follows in this case.

This leaves the case where is not irreducible. We have T = 71 for

every q E

_{Gal(K/ F).}

The

_{representation}

_p has the

_{corresponding}

_property

and 7r is

irreducibly parabolically

induced from

®x

_X7ro, for some 1ro E

with X ranging

over the _group of characters of F’ trivial on norms from K. The

_{representation}

_7ro satisfies

bKIF(,7ro)

=

p;

by

part

(i),

it is therefore of the form 1r0

= 1r ( aD),

where

o~o

= _T. The assertion therefore

follows in this case, with xa = 1. D

2. Local constants under

_twisting

We fix a non-trivial continuous character

_’OF

of the additive _group of F. As a matter of

notation,

if

E/F

is a finite

extension,

we

_put

V)E

=

OF

o

_TrK~F.

In this

_section,

we state some

conjectural

properties

of the local constant

6 (7r,

x _{7r2, 8,}

_OF),

for 7rj E

(F).

These form the definitive version of the

twisting properties conjecture

of the Introduction.

2.1. We first recall a

simple

_consequence of the definition

[22]

of the local

constant:

Lemma. Let 7ri E

.4;7 (F),

f or i

=

l, 2.

We have

for

any

unramified quasicharacter

X

of

F’ and any element c E FX such that

_{v F ( c) = - f (7fl}

X _71’2,

We need to refine this result. For

_this,

it will be convenient to divide

into two cases.

Conjecture

1. Let 7r E There exists E

_Fx,

well

_defined

modulo

_UF"

such that

for

all tame

_{quasicharacters X}

Here and in the

_following

_{l F}

denotes the trivial character of F~

_(or

of

WF) .

We recall

_[7]

that

_{c( 7r}

x

~r, 2, ~F)

= while

f( 7r

_x

There is a

complementary

relation:

Conjecture

2. For i

_{=1, 2,}

let 7ri E

,A~2

(F) .

Assume

that 7rl

is not

of

the

form

for

any tame

quasicharacter X of

Fx. There is then an element

E

_Fx,

_well-defined

modulo

_Up,

such that

for

all tame

_{quasicharacters X of}

Fx.

Note. In both statements the element c

depends

on the choice of

1/JF.

When necessary we shall use the more

_precise

notation

c(7ri

x _7r2,

In

_(2.1.2)

we

obviously

have

for all tame

_{quasicharacters X}

of Fx. If on the

_contrary

we have 71-1 =

x*2

for a tame

quasicharacter X

of

FX,

we

_define

In both cases

therefore,

we have

(2.1.5)

cX7rl X 7r2 = X 7r2 I for X

tamely

ramified.

2.2. We now consider how the element ~l _X7r2 behaves under extension of

the base field.

Conjecture

3. Let

_{K / F}

be a

finite

tame extension. For i =

1,

2 let

7ri E

~(F),

and let

_7rf

= Then

The

_identity

_(2.2.1)

still holds if we

replace,

for

example,

1rf

by

for an unramified

quasicharacter x

of KX.

Thus,

if

K / F

is a

_cyclic

tame

extension

_{( 2.2.1 )}

_implies

since and differ at most

_by

an unramified character

(Proposition 1.1).

Remark.

Let Q2

E =

1, 2.

The

arguments

of

_[8]

then

_yield

an

element E

Fx IUF’

with

_properties

_analogous

to those

_above,

relative

to the local constant

_{~(~l ® Q2,}

_s,

_{OF) -}

If 1ri = _we then have

since £ preserves local constants of

_pairs.

Thus

_(2.1.1),

_(2.1.2)

follow from the

_{Langlands Conjecture}

and

_[8,

Corollaire

_2.3,

Th6or6me

_1.3]

respectively.

The

_analogue

of

_(2.2.1)

for follows from the

_defining

property

of c and the

inductivity

of _6;; this

implies

(2.2.1)

via the

Langlands

correspondence.

Of course, the

point

for us is to obtain direct

_proofs

of these

_results,

without recourse to the existence of

_~m.

Known cases. A direct

proof

of

Conjecture

1 is known. Likewise

Conjec-ture 2 in the case _m2.

Conjecture

3 is known for

c(7r

x

_{fr,’ØF). (We}

do not prove these assertions

here.)

2.3. If we consider the

correspondences

~~ (F) -~

recalled

in §1,

we obtain:

° °

Theorem. For i =

1, 2, let Ui

_E and

put

7ri =

Then,

_on

Conjectures

1-3,

we have

° °

Proof.

If ði E i =

1, 2,

then the assertion is immediate: the

maps

’7r _preserve local constants of

_pairs

_[19].

The

_general

case then follows from

(2.2.1)

and the definition of

_7r~.

D

2.4. For the

_moment,

let ~r2 be an irreducible smooth

_{representation}

of

for i =

l, 2.

We consider

briefly

the variation of

£(7r1

_x

7r2, 8,

OF)

with the additive character

OF-Let

_{0’ F}

be some other non-trivial character of

F;

there is then a

_unique

a E Fx such that for x E F. We have

Write wi

for the central character of 1rl and

put Q

=

(,cJl 2 (,cJ21.

We have:

Both identities follow

_readily

from the definitions in

_[22].

As a _consequence of these

identities,

we see that all of the

_conjectures

above are

essentially independent

of the choice of

V)F:

if

they

hold for one

choice,

then

_they

hold for all.

3. Values of local constants for

_pairs

In this

_section,

we

investigate

the arithmetic

properties

of the value of

the local constant

_e(7ri

x _-X2,

1, V)F),

_{for 1Ti E}

_{AZ, (F).}

We prove, on the

conjectures

listed in

_§2,

a

_slightly

weakened version of

_Conjecture

B

_(stated

as Theorem 2 in the

introduction).

This

depends

on

knowledge

of the action

of

_Aut(C)

on the local constant. We

_give

a

_general

statement of the

_required

result in

_3.2,

and _prove it in

_§6.

3.1. We

_proved

in

_[7]

that

We shall next

_explore

the value

_e(7ri

_{X ~r2, 2,}

_OF)

when7r2

is "distant from"

Proposition.

For i =

1, 2,

let

7ri E

~4~.(F)~

and assume:

(i)

W1f i has

finite

order,

and

(ii)

7ri is not

of

the

form

X7r2

for

any tame

quasicharacter x

Then x

7r2, ~~F) ~ ~

root

_of

_unité.

Proof.

We first show that the

_analogous

statement is true for Weil _group

representations.

Indeed

_if,

for i =

1, 2, O’i

_E has determinant of finite

_order,

then ~1 0 72 is in fact a

representation

of a finite Galois _group.

If 0’1 is not of the form

x~2

for any

tame x

then ~1 0 a2 has no tame

component

hence 0

_T2?0,~)

is a real number times a root of

unity

[13, Appendix].

0

cr2, ~~p)

is a

_complex

number of modulus 1

and

with an

integral

_exponent

It follows that is indeed a root of

_unity.

We write 7ri = _{cr, E}

Suppose

first that _{~Z E}

for i =

1, 2.

We then have

ê( 7rl

x

7r2~ ~) = ê( al 0

_{~2, s,} whence the

result in this case.

In

_general,

we can find a

finite,

_tame, Galois extension

E/F

_containing

the defect fields of both In

_particular,

_af

E

_(E),

and

_{(af) =}

Xi for an unramified character _Xi of EX of finite _p-power order

(Theorem 1.6).

From the first case we have

where _q is a root of

_unity

of _p-power

_order,

whence

is a root of

_unity.

The conclusion of the last

_paragraph

still holds if we

replace

_by

the Arthur-Clozel base

_change

_by

_Proposition

1.1.

Now let

_L/F

be a finite

_cyclic

extension and set

_7rf

= We then have

_[1,

_{I, Prop.}

_6.9]:

where

_AL/F

is the

_Langlands

constant

and X

runs

through

the _group of characters of FX trivial on

One knows

_from,

for

_example,

the calculations in

_{[3, §10],}

that

_AL/F

is a

then, by

(2.1.2),

a root of

_unity

times

_{e (7ri}

x _{7r2, s,} the situation

above,

the extension

_E/F

is of the form L D

_F,

with

_L/F

and

_E/L

both tame and

_cyclic;

in fact

_L/F

is unramified. The result will therefore follow if we can _prove that

e(7rf

x

7r~, ~~L)

is a root of

_unity.

We next

observe:

Lemma. In the situation

_above,

_7rL

is not

_of

the

_form

_xirf

_for

_any tame

quasicharacter X

of

L~.

Proof.

This is

_only

an issue in the case ml = _m2. The fact that _vri is not a

tame twist of -k2 is

_equivalent

to the

_simple

characters in the 7rj not

being

conjugate

in

_GLP-I

_(F).

This

_implies

_{[5, 1.3]}

that the

_simple

characters

underlying

the

_{representations}

_7rf

are not

_conjugate

in

_{GLpml (L),}

and the

Lemma follows. 0

Because of this

_lemma,

we can

_apply

the above

_procedure

twice to

_get

the

_proposition.

D

3.2. We can in fact be a little more

_precise

than in

3.1,

by

taking

into

account the action of

_Aut(C)

on local constants for

_pairs.

We first state

a

completely general

result on this matter

(which

does not

depend

on _any

conjectures) .

We write

_AF(n)

for the set of

_equivalence

classes of irreducible smooth

representations

of

_GLn(F).

As

_explained

in

_{[6, §7], Aut(C)}

acts on

AF(n);

we write this action as

(T, 7r)

H _r7r, for T E

_{Aut(C), 7r}

E

_,.4F(n).

We extend

T E

_Aut(C)

to an

automorphism

of the field of rational functions in

q-S by

treating q-S

as an indeterminate on which T acts

trivially.

Theorem. For i =

1, 2,

let

71~ E

AF(ni).

Then

where

and

3.3. We are now

ready

to

_{investigate Conjecture}

B. When some _{m2 -}

_0,

the

_conjecture

is

_proved

in

_[5,

Theorem

_1.4~,

so we can assume

throughout

that 0.

We write for the _group

_{of pr-th}

roots of

_unity

in C _{and ppao} =

Ur

We first recall

_that,

when _p =

2,

the function

GF

of

[18]

is

identically

1. Theorem. Let _p = 2. For i =

1, 2,

let 7ri E

.r~Z (F) .

Assume that 7ri is not

_of

the

_form

_for

_{any tame}

_{quasicharacter x of}

FX.

_Then,

on the

conjectures

of §2,

we have

In

_particular,

_Conjecture

B holds

_{when p}

= 2.

Proof.

Assume first that and _W1r2 are of finite

_2-power

order. Then

the central

_type

of =

1, 2,

factorizes

through

a finite

_2-group

hence is

definable over for _any

sufficiently large

_integer

r. The same is true

of 7r2. If T E

_Aut(C)

acts

_trivially

on _{it2- it} also acts

_trivially

on

Jq

and

7/J F

and we have

by

Theorem 3.2. Since we know that

6(7rl

x

7r2, l,’OF)

is a root of

_unity

(Proposition

3.1),

it follows that it is a root of

_unity

in some

cyclotomic

field

_{Q(¡¿2r ),}

hence a root of

unity

in In the

general

_case, there are

tame

_{quasicharacters}

_x2, i =

_{1, 2,}

such that

r2 =

xj7r§

with

i of finite

2-power

order. As = the theorem follows from the

special

71 _i

case and

(2.1.2),

_noting

that

x 2

_{=C7r/ 1}

X7r’ . D

3.4.

_Suppose

in this

_paragraph

that _p is odd. The _map

GF :

is

defined as follows. Set a = _as in

3.2,

and take x E Fx . There are

two cases.

_First,

we set

Otherwise,

we choose a

_prime

element zu of _oF and let b be the

_integer

for

which

_{P’-l -}

We then define

Theorem.

_{Let p}

be odd. For i =

1, 2,

let _{7ri E}

.~4,~~ (F).

Assume that 7ri

is not

_of

the

_form

_for

any tame

quasicharacter

X

of

FX.

Then,

on the

Conjectures

of

§2,

we have

That

_is,

_Conjecture

B holds _{up to}

_sign.

Proof.

As above we reduce

(using

the

twisting

properties)

to the case where

the W7ri have finite p-power order. We can also assume that the mi are non-zero.

We remark that if T E

_Aut(C)

acts

_trivially

on and

lq

then it

also acts

_trivially

on x

~r2, 2, ~F)

(Theorem 3.2).

More

precisely

if

( ql) = 1

then if T E

_Aut(C)

acts

_trivially

on it also fixes

Vq

and

then

_{e (7ri}

x

7T2, ~

1/JF)

is a root of

_unity

_belonging

to for

_{large enough}

r. This

implies

+e(7ri

x

1l"2,!, ’ØF) E

and

hence, arguing

as in

3.3,

we

get

the result.

Let now

(~l) = -1.

The

_exponent

of

q-’

in

e(7ri

x _{7r2, s,}

_OF)

is

_f

= and we deduce from Theorem 3.2 that if T E

_Aut(C)

acts

trivially

on we have

But from

_(3.4.1)

above we have

for x E F with Since T acts

trivially

on we have

(-) =

T

V"-

_1;

SO E

(-Xl

X _7r2,

1,

_V)F)

and

_GF

_(Cr,

X lr2

)p""1+1"2

have the

q

;

so x and have the same

behaviour under T. Their

_quotient

_belongs

to for r

sufficiently large

and the square of the

quotient belongs

to D

3.5. We add some more _consequences of Theorem

3.2,

to be used in the next section.

Proposition.

For i =

1, 2,

let

7ri E

~4~(F).

Then

for

any T E

_Aut(C)

we

have

°

This is an immediate _consequence of Theorem 3.2 and the

defining

prop-erties of

_c(7ri

x _~r2,

Next,

we need the Adams

_operation

7r ~

7rt

on _see, for

example,

Corollary.

Lett be a

_prime

_{number, f}

_~

_p. Then

Proof.

Let T E

_Aut(C)

have

_{restriction (}

~

(~

_{to p.} Then

= 1/J~

and

_by

construction

_[5]

differs from T7ri

by

a tame

_{quasicharacter.}

The

corollary

now follows from

(2.1.3), (2.1.4)

and the

_proposition.

D 4.

_{Davenport-Hasse}

relations

In this

_section,

we _prove some weak versions of

Conjecture A, including

that stated as Theorem 1 in the Introduction. Most results in this section

depend

on the

conjectures

of

§2.

4.1. We let

_{K/ F}

be a

totally

ramified extension of odd

_prime

_p.

As

_usual,

we abbreviate

7r K

= for _{7r E} Lemma. For i =

1, 2,

let

7ri E

Suppose

that 7rl

~

for

any

tamely ramified

quasi character

X

of

FX. Then

7rÍ

(resp.

7rf

X*2K)

f or

any

tamel y rami fied quasicharacters

X

of

Fx

(resp.

KX).

Proof.

The

_hypothesis

means that the

simple

characters

(in

the sense of

[11])

attached to 7ri and

ir2

in G =

GLpm (F)

are not

_conjugate.

It follows that the

_simple

characters attached to

1T"Í

and

_ir

are not

_{conjugate either,}

and hence that

_Jrf

_~

for _any tame

_{quasicharacter X}

of FX.

_{Similarly by}

[5,

Theorem

_1.3]

the

_simple

characters attached to

_7rF

and

_{-k Kin}

_{GL mj}

_(K)

are not

_conjugate.

Thus

_{7rF §#}

for _any tame

_{quasicharacter X}

of

K" . D

4.2.

_Conjecture

A can be

_proved

_by

_taking

the values of the local constants at _any

_{given s}

E

_C,

for

_{example s}

= ~,

because of:

Proposition.

In the

_setting

_of

_Conjectures

A we have

The absolute values

_{o f}

the two sides

_{o f f ormula}

_(*)

in

_Conjecture

A are

equal.

’

Proof. Remembering

that

_7rf

is not an unramified twist of

~r2,

the second

formula follows from

_[5,

Theorem

_{1.7~ .}

To

_get

the first

_formula,

we

_apply

the

_procedures

of

_{[9, §6].}

It is immediate that a best common

approxima-tion to the

_{simple characters 0j}

_underlying

the 7ri is also a best common

approximation

to the

_simple

characters

_01.

Since the conductor

_depends

only

on this best common

approximation,

the result follows. D

Remark. If we admit the

_conjectures

listed in

§2,

the second formula of the

4.3. Let us assume now that _Tri =

q*2

for some tame

_{quasicharacter q}

of

_Fx;

this relation

_{determines q uniquely.}

We have

_1rf

= _so the condition

7r~

~

for X

unramified means that

?/

is not unramified. We

now _prove

Conjecture

A for

(7ri,7r2)

in this situation.

Theorem. For i =

1, 2,

let _{1ri E} and assume that _7r2 =

""1rl

for

somme tame

quasicharacter"., of

Fx such not

unramified.

Then

(on

the

_conjectures

_{of §2)}

_Conjecture

A

Proof.

We can assume m &#x3E; 0

_(since

otherwise we know the

conjecture

is

true

_[6,

_4.1]).

_Also,

the assertion is

_independent

of the choice of additive character

_(2.4),

so

(to

simplify

calculations)

we take =

1,

in the

notation of 3.2.

We have

_{(by (2.1.1) )}

Taking

values

_{at .1,}

we

_get

where

This is obtained as follows.

First,

=

(and

does not

depend

on

s),

while

ê(lF, f/2,

is

_equal

to

q(1-t)/2 .

_Next,

by

3.5.

_Thirdly,

Finally,

by

[7,

Th6or6me

_1].

by

[9,

Theorem

_6.5].

Similarly,

with _{qK = q} o

_NK/F,

we

_get

Moreover,

-where

The formula

_(*)

in

_Conjecture

A therefore holds at s

_{= 2,}

then for all s

_by

Proposition

4.2. D

Remark.

_Conjecture

A holds when rrzl or _m2 is zero

[6, 4.1].

In

_light

of

the

_foregoing

_therefore,

when

_considering

further cases of the

conjecture

we can assume that _{ml, m2} &#x3E; 0 and 7ri is not of the form for any tame

quasicharacter

of F~.

Remark. If

_7y7T2

with

_qt

_unramified,

the

_conjecture

does not

_hold,

but

it is easy to

_compute

both sides

_using

_[7]

and the

_{twisting properties}

of

_§2.

4.4. As the next

_step

in the

_proof

of Theorem

_1,

we recall from

(3.1.2):

Proposition.

Let

_K/F

be a

cyclic

extension. For i

= 1,

2

let 7ri

E

_,~4mi

(F)

and

_put

_7rr

= Then

where X

ranges

through

the characters

of

Fx trivial on

NKIF(Kx)-Here,

_{A K/ F}

is the

_Langlands

constant,

as in

(3.1.3).

Combining

this with

(2.1.2),

we

_get

Corollary.

Assume moreover that

K/F

is tame

of

and 1rl is not

of

the

_form

_X1f2

_for

_any tame

_{quasichaTacter X of}

Then

where

with _X

_ranging

_through

the characters

_of

F’ trivial on

NK/P(KX).

4.5. We can now deal with a

_special

case of Theorem I.

Proposition.

Let

_K/F

be a

_{cyclic, totally ramified}

extension

_of

odd

_prime

degree f =1=

p. For i =

1, 2,

let

7ri E

.~4,~i(F)

and

put

7rf

= Then

(on

the

_conjectures

_of

_{§ 2)}

we have