Canadian Mathematical Society
Conference Proceedings
Volume 1 (1981)
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY*
James Arthur
Department of Mathematics
University of
Toronto
Toronto, Canada
automorphic
representations and the relations with number theory is the
Corvallis proceedings, Automorphic Forms, Representations and
L-Functions, Parts 1 and 2, Proc. Sympos. Pure Math., vol. 33,
1979. Although composed mainly of survey articles, the proceedings are already rather formidable. They are a measure
of the breadth of the field. They will be most useful to
mathematicians who are already experts in some branch of the
subject.
Our purpose here is to give a modest introduction to the
Corvallis proceedings. More precisely, our goal is to describe
the Langlands functoriality conjecture, a mathematical insight
of great beauty and simplicity. We will try to show both why
it is a compelling question, and how it arose historically
from Langlands' work on Eisenstein series. We hope that
mathematicians from diverse - or at least neighboring - fields
will find these notes accessible and will be encouraged to
read other survey articles [2], [6], or to plunge directly into
the Corvallis proceedings.
An excellent introduction to the
*
Lectures
given
theory
of
at the Canadian Mathematical
Society
Summer
Seminar, Harmonic Analysis, McGill University, Aug. 4-22, 1980.
Copyright © 1981,
3
American Mathematical
Society
4
JAMES ARTHUR
We will discuss
global functoriality conjecture,
and only that part of it which corresponds to the unramified
primes. It is then a statement about families of conjugacy
classes in complex Lie groups. The point of view is essentially
that of Tate's introduction to global class field theory [38,
only
the
§1-5].
§1.
Suppose that
A
problem in number theory
f(x)
is a monic
integral coefficients.
with
f(x)
Q.
over
Let
E
polynomial of degree n
be the splitting field
f(x)
If we were to order the roots of
would obtain an
embedding
Gal(E/Q),
of
we
the Galois
group of
the symmetric group on
E over Q, into a subgroup of S ,
n letters.
There is no canonical way to do this, so we obtain
only a conjugacy class of subgroups of S . If p is a
prime number, we can reduce f(x) mod p. It decomposes into
a product of irreducible polynomials mod p of degrees
n ,
where n1 +* * +nr = n. These numbers
n1 , n2 ,
of
.
conjugacy
determine a
n
of
the
:
the set of
.
permutations
n
, n2 , ...
The intersection of this class with any of the
Gal(E/Q) in Sn may give several conjugacy classes
Gal(E/Q). However, if p does not divide the discriminant
f(x), there is a distinguished class among these, called
Frobenius class of p. Incidentally, any finite Galois
images
in
S
decompose into disjoint cycles of lengths
which
and
class in
of
extension
splitting field of such
an f(x). However, the Frobenius class in Gal(E/Q) depends
only on E and p, and not on f(x). The prime p is said
E if its Frobenius class is 1;
to split completely in
that is, if
E
can be realized as the
p does not divide the discriminant of
f(x)
and
5
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
f(x)
set
splits into linear factors mod p.
of primes that split completely.
THEOREM 1.1:
S
:
E -+
Let
S(E)
be the
is an injective, order reversing
S(E)
into subsets of prime
map from finite Galois extensions of Q
numbers.
part is the injectivity.
The difficult
It is a conse-
quence of the Tchebotarev density theorem (see [38, p. 165]).
What is the
PROBLEM:
prime
image
of this map?
S(E)?
numbers are of the form
A reasonable solution to this
nonabelian class field theory.
finite Galois extensions
kind of
E
problem
One could
would constitute
parametrize
the
by the collections S(E).
independent characterization
is often called a
i.e. what sets of
reciprocity Zaw
of one of the sets
for
Any
S(E)
E.
2
Example: f(x) = x +1, E = Q(//T), Disc f(x) = -4, and
S(E) = {p : p 1 (mod 4)}.
The problem has a solution in terms of such congruence
conditions if Gal(E/Q) is abelian. On the other hand, if
Gal(E/Q) is a simple group S(E) cannot be described by con=
gruence conditions alone.
It is convenient to embed the Galois
conjugacy classes in GL (C)
parametrize. Suppose then that
since the
to
r :
Gal(Q/Q)
-+
group in
are
GLn(C)
n
GL
(C),
particularly easy
6
JAMES ARTHUR
is a continuous
limit
homomorphism. Gal(Q/q)
lim Gal(E/Q)
is the projective
over all finite Galois extensions
E
of
E
so it is a
compact totally disconnected group. Continuity
means that r has finite image. The kernel of r equals
0,
for a finite Galois extension
Gal(Q/Er)
Gal(E /Q)
embeds
E
E
of
Q
equals Er for
conjugacy class in Gal(E /Q), which
unique semisimple conjugacy class p (r) in
(C).
Thus
S(Er)
A
r
is unramified in
If p
r.
and
we have the Frobenius
embeds into a
GL
some
Q,
finite Galois extension
GLn(C). Any
into
of
E
semisimple conjugacy
{p
=
%p(r)
:
class in
nomial deserves to have
special
Lp(z,r)
the local
L-function of
r.
If
which includes the ones that
I}.
is
completely determined
the characteristic poly-
so
notation.
det(I
=
(C)
GL
by its characteristic polynomial,
=
(r)z)1
t
-
S
In
fact,
one
defines
,
is a finite set of
ramify, one
primes
defines a global
L-function,
L
(s,r)
=
p/S P (p
n
L
It can be shown that this infinite
in some
right half plane of
series implies the factors
by the analytic function
simple conjugacy
classes
C.
An
r),
s
e
C.
product converges for s
elementary lemma on Dirichlet
Lp(p S,r)
are
uniquely determined
Ls(s,r). So,therefore,are the semi{fp(r) : p / S}. It is also known
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
7
Ls(s,r) can be analytically continued with a functional
equation relating Ls(s,r) with L (l-s,r), where
that
r(g) = r(g-1 ) t .
The location and residues of the
determined from a
poles of Ls(s,r) could be
conjecture of Artin, (see [16]. Artin's con-
jecture is referred
to on p. 225).
27Tiv
Fix
Example:
f(x)
and let
N e I,
I-
=
x-e
N.
re(Z/NZ)*
f
Then
f(x)
is irreducible and
Gal
jIe
2~i
N J/
})*
(Z/N
prime p is unramified if it does not divide N. The Frobenius
class is the image of p in (Z/NZ)*
[24]. To fit in with
discussion above we can take a character X of the (Z/N )*.
A
The functions
At this
LS(s,x)
point,
are called Dirichlet L-series.
we
perhaps
should recall the
definition of the Frobenius class.
p-adic numbers.
Gal(Qp/0p)
Then
Let
Q
precise
be the field of
is naturally embedded in
Gal(Q/Q) up to conjugacy in Gal(Q/Q). There is a normal
subgroup Tp of Gal( p/Qp) (the inertia group) such that
the quotient group is isomorphic to
Gal(Fp/Fp)
and, in
particular, has a cyclic generator 4. Fp is a field with
p elements.) Suppose that H is a closed normal subgroup of
Gal(®/Q). It corresponds to a Galois extension E of Q.
The prime p is called unramified if the images of T
in
Gal(Q/Q) are contained in H. Then 4 maps to a well
defined
conjugacy class
in
Gal(E/Q)
=
Gal(j/Q)/H.
8
JAMES ARTHUR
This is the Frobenius class of
the facts from
algebraic
number
(The reader unfamiliar with
p.
theory discussed
[38], §1,2, and then go to [7], [10]
details.)
read
or
so
far might
[24] for the
possible way to obtain collections in the image of S
is through algebraic geometry. Suppose that X is a nonsingular projective algebraic variety defined over Q. Fix a
nonnegative integer i. Grothendieck has defined for every
prime number k the k-adic cohomology group Hi(X,Q9), a
vector space over QQ whose dimension, n, equals
the
i
Betti number of X(C). It comes equipped with a continuous map
One
Pk
:
Gal(Q/Q)
GL(Hi(X, QZ)).
Any choice of basis of H (X,I) identifies GL(Hi(X, Q9)) with
GLn(Q). Since Gal(Q/Q) is compact, it is possible to choose
a basis such that p (Gal(Q/Q))
lies in GLn (2), where ZQ
is the ring of
9-adic integers [33, p. 1]. It follows from
the work of Deligne on the Weil conjectures that the collection
{p I: 9
sentations,
p
prime}
=
is a
9-adic repre-
compatible family of
in the sense that the
following
two
properties
hold:
(i)
There is a finite set
p 4
Sp
u
{U}, p
(regarding
S
of
primes
is unramified in
this group as a Quotient
such that if
p,(Gal(Q/Q))
of Gal(/Q?)),
there is a Frobenius conjugacy class
(ii)
For
p ¢
Sp
u
{U},
det(I - p (p)z )
independent
of
the characteristic
has coefficients in
k.
p (p)
in
so
GL
(Z ).
polynomial
z
c_ T Q
and is
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
If
p 4
Sp
class in
det(I
unique semisimple conjugacy
GL n (C) whose characteristic polynomial is
let
,
p(pt)z),
-
p
be the
(p)
for any
2
p.
det(I - Pp(p)z)
,
As before, one can define
L-functions
L
P
Again,
(z,p)
=
(s,p)
L
L
=
S
P (p
sp).
global L-function converges in some right half plane.
However, it is not known to have analytic continuation. It is
expected that the conjugacy classes p(p ) are semisimple,
although this is also not known. If it were so one could
construct a number of collections S(E) simply from a knowledge
of
the
L (s,p).
For
suppose
N
N =
is a
positive integer.
Kp(N)
=
{k
P
p
P.
N
>
P
0,
Set
N
GLn(~p)
E
: k
1 (mod
t2 prime K
(N)
p
P
)},
and
K(N)
It can be shown that
K(N)
K(1)
and that
=
is a normal
.
primeGLn(
subgroup of
),
naturally isomorphic to
1.38). By composing the maps
K(1)/K(N)
(see [36], Lemma
=
is
Gal(Q/Q)
PGal >
K(l)
->
K(1)/K(N)
GL
(Z/NZ)
9
JAMES ARTHUR
10
one obtains a
homomorphism
p(N)
The
image
corresponds to a finite Galois extension
there is an injection
q;
Gal(Ep(N)/)
If the
GL n(Z/N).
,
p(N)
of
of
Ep(N)
P (N)
Gal(Q/Q)
:
conjugacy
->
GLn(Z/NZ)
semisimple, the
image of the Frobenius class in GL (Z/NZ) will be determined
by the reduction modulo N of the characteristic polynomial
off
(p). In particular,
S(EN))
{P
=
classes
¢
0
(p)
are all
Spp4N : Lp(z,P)
1 E
(-z)n
(mod N)}.
This section is meant to serve as motivation for what
number theory or
family
{p
:
§2.
p
algebraic geometry
S}
A
is any
the group of
a unit in
direct
A.
can be
G
encapsulated in a
ring
identity),
with
matrices over
One such
of
A
vQv
is the adeles,
= IR
x
pQp
pp
GLn.
G(A)
Then
is
whose determinant is
product
vv
GL ().
GL
will stand for the group
ring (commutative,
(n xn)
data from
of semisimple conjugacy class in
Automorphic representations
In this section
if
interesting
The main theme has been that
follows.
.
A,
the restricted
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
a valuation over
(We shall write v for
11
p for the
Then G(A) is a
Q,
valuation associated to a finite prime.)
and
locally compact group. Embedded diagonally, G(Q) is a discrete subgroup of G(A). One studies the coset space
G(Q)\G(A), with the quotient topology. The volume of this
space, with respect to the G(A) invariant measure, is not
finite. To rectify this, define
z
Z=
'
:
0
a central
G(A)
subgroup
of
0
G(R).
z
0
>
z
Then modulo the
does have finite invariant volume.
representation
of
G(A)
L
on the Hilbert
=
,
Let
subgroup
R
be the
*G(Q),
Z
regular
space
L2(Z ·G(Q)\G(A)),
so
(R(y)Q) (x)
=
4(xy),
E L, x,y
e
G(A)).
representation of G(A). One tries to
decompose it as a direct integral of irreducible representations.
Incidentally any irreducible representation X of G(a) can
be written as a restricted tensor product
This is a unitary
Sv
of irreducible
At first
Rv
i
p
p
representations of the local groups [8].
glance the space Z-G(Q)\G(A) might seem unduly
abstract in comparison with, say, a quotient of a real Lie
group. However, it is really a very natural object. In §1 we
JAMES ARTHUR
12
defined,for each N = n p p
N
P,
K (N)
K(N) = n
of
G(GA).
compact subgroup
the
Set
LK(N) =
L2(z.G(Q)\G(A)/K(N)),
fixed functions under
K(N)
the Hilbert space of
L
=
Then
R.
K(N)
lim
N
it is necessary and sufficient to underK(N) is
K(N)
Notice that LK(N)
stand each of the spaces LK(
so to understand
L
certainly contains SLn(3R).
Z*G(Q)G(G()/K(N) over
R
an abelian
is
N
y
P
mp)},
1 (mod p
=
(I/NZ)*.
group which is isomorphic to
SLn(3R)-invariant,
and as an
SLn(Q)\SLn(A)/(SLn(()
However, by strong approximation for
SLn(A)
=
which
G(A) ,
Now the determinant fibres
I:
M\Ideles/Hn {y E*
point is
isomorphic to
of any
in
K(N)
invariant under the normalizer of
SLn(Q)(SLn(A)
n
SL
n
SLn(R)
K(N)).
,
SLn(R)-K(N)),
The fibre
space
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
The proof in general is the same as for
(see [36, Lemma 6.15].
It follows that as an
n=2).
13
SL (R) space, the fibre is
diffeomorphic with F(N)\SL (JR), where
F(N)
Thus, as
SL
an
=
{y
e SL
n (2)
of
utility
0
<
i
to
L2 ((N) \SL (R)).
of the adele
picture
operators that we can
most
clearly.
If
define an element
n,
<
N)}.
isomorphic
is
It is in the definition of Hecke
see the
1 (mod
=
LK(N)
(IR)-module,
#[K/NZ)*] copies
y
:
11
t.
pi
p
P.
in
G(Q p ).
of
G(Qp)
p
Tp
f
Let
E
How does
be the characteristic function in
p (N)
t p,i*iK (N).
p
onL" K(N)) by
K
Tp if
for
p,ii
LK(
Ti
P'i
Define the
Pi
G(Qp)
Hecke operator,
fpi(i) (R(y).)dy,
It is clear that
.
ith
P,1i.
T
belongs
to
LK(N)
behave on the irreducible constituents of
R?
Suppose that
(r,U)
is an irreducible
=
(fv , vUv)
representation
of
G(A)
which is equivalent
JAMES ARTHUR
14
to the
L
subrepresentation of
L.
of
on a closed invariant
R
Then
LK(N)
7T
corresponds
to the
=
|UpP
[P
LK(N)
L iT n
subspace
K(N)
r K (N)
subspace
is the space of
K
® .XP
K
P
U
P
P(
(N)-invariant
U
vectors in
.
K (N) = GLn(7
np ), a maximal compact
p
on GL( (p), such as fpli
Functions
subgroup of GL n(Q).
n
n p
p,i
which are bi-invariant under GL (Zp) are p-adic analogues of
If
does not divide
p
N,
spherical functions for real groups [17], discussed in
Helgason's lectures. The theory is similar. In particular,
the
GL (Q )
has dimension at most one. If
space U p
p
K(N)
LK(N)
{O}, and p does not divide N, the dimension of
the
Kp(N)
)
K
U
must be
p
L (N
space
exactly
one.
Now
; relative to the
T
pri
leaves invariant the
equivalence
K
acts
through
restriction of
Tpi
LK(N)
to
operator by a complex number
c
N
p,if
representation (-r , U)
that UK(N)
{o}. Let
such
,
and let
prime
be the set of
S
p 4 S ,
0 < i <
n}.
we obtain
with
U,
(N)
.
Therefore, the
i(r).
of
N
L.
G (A)
It follows that
there is an
be the minimal such
prime divisors of
(n+l)
N .
complex numbers
Define a semisimple conjugacy
p (i)
any
{c
i( ()
in
GL
numbers:
=
i
i
n
i(n-i)
Z(n-l)
)
i=O ( (1p
ci(ni)vp~()z.
=
N,
For
by constructing its characteristic polynomial from these
det(l - p(r)zj
it
is a multiple of the identity
The smooth vectors are dense in
for the
Up
the one-dimensional space
L
of
:
(C)
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
15
(Strong multiplicity one). The representation
is uniquely determined by the family { p(i) : p 4 S } of
conjugacy classes in GL (C).
THEOREM 2.1
T
(See [21], [32]).
{ p(m)}
We have only defined the conjugacy classes
representations
that occur
is not necessary.
In
discretely
fact, if
R,(x)
X E C,
R(x)Idet
=
in
R.
for
This restriction
define the representation
xl
,
x E
G(A).
purely imaginary, RX is unitary and it, also, has a
decomposition into a direct integral of irreducible representations of G(A). For the present we shall define an automorphic
representation informally as an irreducible representation of
G(A) which "occurs in" the direct integral decomposition of
If
RX
X
is
for some
,
A.
conjugacy classes
One can define the finite set
)
{p(D :(
p
4 ST}
S
and the
as above for any auto-
morphic representation T. We will not do so for we will give
a more general definition in §5.
We do note, howevever, that
Jacquet, Piatetski-Shipiro and Shalika have recently shown that
with certain obvious exceptions, Theorem 2.1 holds for any
automorphic representation of G(A). If fT is any automorphic
representation of G(A), define
Lp(z,
if
p
)
ST
det(l
=
,
-
p(7)z)-
and for
=
(ino(-l)
=0p, (Ti)
any finite
LS(s,)
=
n
p4s
set of
L
P
primes
(P S')
S D S
,
set
JAMES ARTHUR
16
As in the
in some
examples of §1,
the
right hand side converges for s
right half plane.
just the idle class
is
Example: Let n = 1. Then G(Q)\G(A)
aroup of Q. For any N,
N
Kp(N)
=
{y
EZp
y
:
1 (mod
P
p PZ)
P }.
just a character on
N
the idle class group and is known as a Grossencharakter.
is the smallest positive integer such that X is trivial on
K(NX). If p does not divide N there is a complex number
An
automorphic representation
vXp
X =
is
X
s
P
such that
s
Xp(V) IpP ,
v c
=
-s
s
=
1
L(Xz)
=
Cp,(x)
It is clear that
cp ,(X)
and
=
PIP
= P
Therefore
f s
-
-
l -p Pz
and
LS(Xz)
=
n
r
l -p
-
(s+s)
P
automorphic representation of G(A),
the function LS(s,7) can be analytically continued with
functional equation. Moreover, the location and residues of
all poles can be determined.
THEOREM 2.2:
If
f
is an
See [18].
We could
Suppose that
this theorem.
family of semisimple conjugacy
say
a word about the converse to
{p p
:
p
S}
is a
17
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
classes in
Suppose in addition that
GL (C).
np$S
det(l-
-1
-s
rpp)
converges in a right half plane to an analytic function which
has all the properties established in Theorem 2.2 (i.e. analytic
continuation, functional equation, predicted location of poles).
Is there an automorphic representation Tr of G( ) such that
pp (r)
a
=
pP
for each
partial solution
p
is
4
The answer is no in
S?
given
for
in [19], for
GL2
general,
GL3
but
in
Rather than give
[20] and indeed is expected for GLn.
precise statements, let us simply say that these analytic
conditions on the family { p} are very strong, if not quite
strong enough to insure the existence of
CONJECTURE 2.3
(Langlands).
r:
f.
Given a continuous
Gal(Q/2)
(C)
- GL
automorphic representation T
SW is the set of primes that ramify
(r)
cpp (r) for all primes p ¢ S
Pp
there is an
of
that
for
that
=
S(Er)
:
{p
:
representation
p(T)
.
GLn(A)
r,
In
such
and such
particular
= I}.
conjecture reduces the problem of §1 to the study of
automorphic representations of GLn(A). The collections S(Er),
which classify Galois extensions of Q, could be recovered from
data obtained analytically from the decomposition of R into
This
irreducibles.
set
S
=
S
,
Notice that
so
L
(s,r)
=
L
(s,r)
for any finite
the location and residues of the poles of any
18
JAMES ARTHUR
function
the
(s,r)
L
could be
computed by Theorem 2.2.
This is
conjecture of Artin.
It is
REMARK 1:
far with the
field
easy
everything we have done so
Q replaced by an arbitrary number
to restate
ground field
F.
any N, LK(N) is isomorphic to the space of functions on (Z/NZ) . If p does
is the identity operator while Tp
divide N, Tp
Suppose that n=l.
REMARK 2:
For
not
,
corresponds
the image of
class in
one
in
2ri
p
Gal Q
dimensional
this is
(Z/NZ)* by
multiplication in
to
e
(Z/NZ)*
/Q.
also corresponds to the Frobenius
Thus, the conjecture is true for any
representation
quite elementary.
However
p.
of
Gal Q
/Q
Ie
.
So
far,
conjecture asks more, even
for n = 1. Any one dimensional representation of Gal(Q/Q)
is to be associated to an automorphic representation of GL
by what we have just observed, this in turn corresponds to a
But the
2Tri
one
dimensional representation of
N.
Thus, for any finite abelian extension
character
group of Gal(E/Q)
is
Gal
naturally
2iri
character group of
that
Gal(E/Q)
is a
Gal Q
quotient
is contained in
words
E
and is
certainly
eN
/
e
of
E
a
,
for
some
the
Q
subgroup
of the
)
/lQ
for some
of Gal
271i
Q e
.
[(e
N
N.
/Q.
This means
In other
This is Kronecker's
theorem,
elementary. If Q is replaced by a
number field F, the conjecture for n = 1 is also known. It
is just the Artin reciprocity law which is the heart of class
not
field theory (see [38], §3, 4, 5).
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
Suppose that
Remark 3:
representations r,
extending the work
problem for GL2.
it for
n
=
closely related
GLn
2.
to
although it has
Suppose that
fields of
Then for certain irreducible
2.
Langlands solved the conjecture by
of Saito and Shintani on the base change
See [28], [12].
good point
This is a
for it is
n =
19
E D F
prime order
Z.
to describe the base
change problem
Conjecture 2.3. We shall state
been proved completely only for
is a cyclic extension of number
Let
6
generator of Gal(E/F).
be a
representation of Gal(F/F), let
R be the restriction of r to the subgroup Gal(F/E). Any
n dimensional representation R of Gal(F/E) will be of this
form if and only if R ~ R, where
If
r
is an
dimensional
n
R6(g)
=
R(6-1g6),
g e
Gal(F/E).
Now as we have remarked the notions discussed in this section
all make sense if
is
Q
suggests there should be
replaced by
a
map
r
->
+
F.
conjecture
automorphic
The
from
representations of GLn(AF) (AF being the adeles of F) to
automorphic representations of GL n (AE).
If I occurs
E
discretely in the regular representation it is uniquely determined
by a family of conjugacy classes in GL (C). We need only
observe how the map r
R behaves on Frobenius conjugacy
classes.
(a)
Therefore the base
For each
automorphic representation
SI
primes in S
E
of GL
(AF)
prove
automorphic representation n of GLn (AE)
E
is the set of prime ideals that divide the
and such that for any prime i of E which
that there is an
such that
change problem is:
JAMES ARTHUR
20
divides the
prime
(b)
F,
of
(n^()
if
o
tl(70)
if
-,
[¢
v
.. 4 S
splits in E
remains
prime
automorphic representation
IE.
this form if and only if E
Show that an
is of
H
E.
in
GL n (AE)
E
of
=
problem for GL2 used the trace formula
for GL2 . For n = 3 progress has been made for Flicker [9].
The L functions of algebraic geometry should also
correspond to L-functions for GLn. Suppose that X,
The solution of this
n =
dim
Hi(XQ),
p
are as in §1.
X
for all
{p
:
such that
primes p
S(Ep())
=
Gal(Q/Q)
+
=
4 S
.
S
In
{p 4 S, pi
T
= S
n =
1,
at the end of §2 of
see
particular,
N: L
[30].
for any
(z,7)1)
(l-z)n
conjecture, in
pp
(o) =
p (X)
P
N,
(mod
more
N)}.
general
For the solution in case
[33].
§3.
The
automorphic representation
and such that
p
There is a statement of this last
form,
GL(Hi(X,VQ))
Then there should be an
GL n (A)
of
and
Eisenstein series
purpose of Eisenstein series is
to describe the
continuously
The theory was begun by Selberg and completed by Langlands
[25]. We shall give a brief description of the main results
(see also [1] and [11]).
direct integral of that part of
R
which decomposes
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
21
reductive algebraic
The reader unfamiliar with algebraic groups could
We shall state the results for a
group G.
skip
paragraph where objects defined for general
to the next
G are described for
bolic
GLn
component also defined
fixed Levi
parabolic subgroup
radical of
If
Q.
over
N
M0.
Let
X(M)
to
M
GL1
is a standard
P
which
P
(written
defined over
If
Q.
the value of
X E X(M) ,
unipotent
for the
be the abelian group
of maps from
be a
M0
for the Levi component of
M
an idle so it has an absolute value.
M(A)
and let
Q,
we shall write
G,
and
and
m E M(A)
of
P
additively)
be a fixed minimal para-
P0
defined over
subgroup of G,
contains
Let
.
X at m,
Define a map
mX,
is
HM
from
to the real vector space
Chp
=
Hom(X(M),
R)
by
e
Let
ZM
Ap
be the
<X,H (x)>
M
K = II v K
be
,
component of 1 in
G
X
E X(M)
m e M()
,
Ap(R)0
.
and let
Then
MA%)
product of the kernel of HM with Z . Finally, let
a maximal compact subgroup of G(^), admissible
Mo.
relative to
If
ImX
split component of the center of M,
be the connected
is the direct
=
=
GL ,
jRo·e
we can take
*I
22
JAMES ARTHUR
*
*..
}nn
*
*
*
p
:
=
:
M = ·
,
..
|
,
r'.
M={e1}
Lj}nr
and
i,
=|,*
:'*
I
N={.
*
where
n + **+nr = n.
Then
1
*
X|*
is the group of maps
X(M)
ri=1 (det mi)
+
lo X
where
v =
isomorphic
(vl,
..., Vr)
'mll
<XV ,HM
Zr.
over
The space
tp
and
R ,
to
ranges
01
vi logidet mil,
=
.
im
Also
{.Lj.
ZM=
=
0
.·'.
i
zI
I
:
z.
>
1
0
v e
Zr
is
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
Finally,
23
we can take
K
If
OR)
0
=
npGLn ().
x
G(A),
is any element in
x
x =
nmk,
we can
N(A),
n e
m E
write
M(A), k
E K.
Define
(x)
H
If
p
)
= pp
the roots of
ip
(P,
Ap)
to
tp
+
e
projection
P(A).
simple
root
of the co-root
A=
(2p(H (p)) ,
of
Finally
6
onto
S
,0,1-1,0
ai = X0
P(A),
P
(P,Ap). Every
roots of
of a unique
multiplicity)
then
is the modular function of
simple
(m).
is one half the sum (with
p
be the
H
=
let
E Ap
a
in
Ap
\
s-
:
...0)
c
X(M)
be the
a
G = GL ,
n
1 <i<
n}
i
and for each
i,
V
is the element in
t.
v
Xv (i)X
Let
RM,disc
be the
representation M(R)
on
vin.--1
1 11
Cp
subrepresentation
L
such that
-1
- v 1i+ lni
i+1
(ZM.M(Q)\M(.f)))
c
p
is the restriction
.Let
(If
'p.
Ap
.
of the
regular
that decomposes
24
JAMES ARTHUR
discretely.
L
It acts on a closed invariant
(ZMM() \M(A))disc,
of
and
RM,disc,
which we can
of
P(A)
o(m)e
=
is any
a
p¢
to
(the
X[HM(m)) ,m
is a
e
M(A).
representation of M(A) P(A)/N(A),
lift to P(A). Let Ip(A) be this representation
induced to
=
G(A).
It acts on the Hilbert
complex valued functions, 4,
(i)
belongs
X
If
set
(a(m)
Then
(ZMM(Q)\M(wA).
L
representation of M(A),
complexification of *p),
subspace,
i+
m
the function
2
N(A)ZMM(Q)\G(GA)
on
(mx),
(ZMM(Q)\M(A))disc
m E M(A),
for each
E
x
space of
such that
belongs to
G(A),
and
(ii)
11112 j
K
I
=
I|l(mk) 2dmdk
<
o.
Z M() \M(^)
Then
(Ip(X,y)))(x)
If
A
is purely
=
f(xy)e
imaginary,
(1+p)
(Hp(xy))e -(A+p) [Hp(x))
Ip(X)
is
unitary.
expect to find intertwining operators between
these induced representations. Let Q be the restricted Weyl
A A
and also on Z' = p .
group of G. It acts on
P0
P00
We would
=
If
P
and
P'
are standard
are both contained in
Z0.
isomorphisms from
restricting elements in
distinct
Let
Q(ip , fp,)
onto
Vp,
to
Yp.
The groups
are said to be associated if
(7p, 'p,)
and
be the set of
Znp
2
T"p
parabolic subgroups,
obtained by
is not
P
and
empty.
P'
VZp,
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
If
G = GL
isomorphic
is
Q
,
symmetric
to the
25
S
group
,
by
al
aa(1)
0
·
0
The
groups
*
a
n)
r
.,
For each
s
=
=
J
(M(sX)f()(x)
w
s
G(Q).
in
A
for
some
the
and
T
E
S.
r
representative
be a
of
Define
wnx)e
s
'
only if
r = r'
are such that
(nr T(1) , ..., n T(r)J }
s E Q(pi, DS,) let
p
P
in the normalizer of
Sn
a(n)
are associated if and
P'
£
a
l
n
corresponding partitions
(n 1 ,h
C
*
0
and
P
o
,
·
--
e
dn,
-1
N' () nw 5 N(A)w1
\N' (A)
s
E Hp,
for
X E
,i*
and
p'
=
pp,
subspace Hp of Hp (the space
of functions in Hp whose right translates by K and left
translates by the center of the universal enveloping algebra
of M(R) span a finite dimensional space) such that if
H° and
LEMMA 3.1:
There is a dense
P
(Re(X)-p) (ov)
>
0,
E
Ap,
integral defining M(s,X)) converges absolutely.
For X in this range, M(s,X) is an analytic function with
values in Hom(Hp, Hp,) which intertwines Ip(X) and Ip,(sX).
then the
In other
words, if
functions such that
f
belongs
f(k-xk)
to
=
C(G(A))
f(x)
,
for each
the space of
k E K,
and if
JAMES ARTHUR
26
Ip(A,f)
=
f(x)Ip(X,x)dx,
|
G(A)
then
M(s,X)Ip(A,f)
Ip,(sAf)M(s,X).
=
See
[25].
Next, define
i
E(x,4,X) =
f(6x)e
(X+p)
(Hp(6x))
6EP(_)\G(Q)
for
q
E
H
x E
,
If
LEMMA 3.2:
G(A)
and
E H
and
X E 'p C.
(Re(X) -p)(a
then the series converges
function of
X
)
>
absolutely.
0,
a
Ap
,
It defines an analytic
in this range.
[See [25].
We can now state the fundamental theorem of Eisenstein
series.
Suppose that f e Hp . Then E(x,cp,X) and
M(s,X)4 can be analytically continued as meromorphic functions
.
On icp,
to .p
E(x,4,X) is regular and M(s,A) is
THEOREM 3.3:
(a)
Iff
E
unitary.
Cc(G(A))K
functional equations hold:
functional equations hold:
and
t
E(p,
,
)
the following
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
E(x, Ip(X,f),X)
(i)
j
=
27
f(y)E(xy,4,X)dy
C (A)
E(x, M(s,X)4,sX)
(ii)
M(ts,X)4
(iii)
=
=
E(x,(,X)
M(t,sX)M(s,X) .
equivalence class of associated standard
parabolic subgroups. Let Lp be the set of collections
(b)
{FP:
Let
P
be an
of measurable functions
P E P}
Fp
:
ip
+
Hp
such that
(i)
(ii)
Fp,(sX) = M(s,X)Fp(X),
f Fp (X) 112 dX
||F||112 =
Then the map which sends
a
P
extends to
closed G(A)-invariant subspace
in a certain dense subspace of
unitary map from Lp
Lp(G(Q)\G(A))
.
E(x, Fp(X),X)dX,
i'
F
<
to the function
F
Ep,. P
defined for
of
((prP),
s E
onto a
L2(G(Q)\G(A)).
Lp
,
Moreover there is
an
ortho-
gonal decomposition
L2 (G(Q)\G^))
=
pL(G()\G(A))
proof of this theorem is very difficult. See [25, §7
and Appendix II].
The theorem implies that the regular representation of
G-A) on L2(G(Q)\G(A)) decomposes as the direct integral over
The
28
all
JAMES ARTHUR
(P,X), where
belongs
to the
Ip(X).
These
P
is a standard
parabolic subgroup and A
i§'p , of the representations
positive chamber in
representations, remember,
induction from the discrete spectrum of
spectrum of G(A)
corresponds
were obtained
M VA).
by
The discrete
to the case that
P = G.
We have not burdened the reader with a discussion of
normalizations of Haar measures.
section (and the
following ones)
All the measures used in this
are Haar measures
which have
unspecified) ways.
We can now give a precise definition of automorphic representation. An irreducible representation of G(A) is said to
be automorphic if it is equivalent to an irreducible subquotient
of one of the representations
been normalized in natural (but
Ip(~),
(See [4], [29].
equivalent to
[29, Proposition 2]).
Our definition is
equivalent conditions
§4.
X
of
Global intertwining
P,
the two
operators
Langlands proved the fundamental results on Eisenstein
series before defining the functions L (s,7) of §2. In fact
the definition was suggested by the properties of Eisenstein
series, and in particular the global intertwining operators
M(s,X). A careful examination of these operators revealed a
whole family of new L-functions, some of which were seen to
have analytic continuation and functional equations [26].
Fix a standard parabolic P. Then RM disc,
regarded
as a representation of M(A), is equivalent to a direct sum
( (oU)
-
,
,
(vav VUgv)
29
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
where each
M(V )
Hilbert space
on the
(Ip(X), Hp)
Ip(Ov
where
unitary representation
is an irreducible
o
is the
,)
Then
Uv.
S(vIP (oavx,),'
-
of
v
P())
representation G(Q)
induced from the
representation
a, (nm)
P(Q ).
of
= a
X(HM(m)) ,
(m)e
N(v),
E
m
E
M(Q ),
the Hilbert space of measurable
Hp(av),
It acts on
n
functions
'v
v)\G(Qv)
N(
-
Uv
such that
Tv(mx)
(i)
=
^
(m)Y(x)
dk
= iK IIYv(k)1t2
I|I'VTI2
v~
v
v
(ii)
< a.
v
Suppose that
Tv
vector
T(x)
e
U
1.
Let
Then for each
x E
G(A),
the
i(T(x))
be the value of this function
Then
to
H.
=
Suppose that
i(T(x)),
s E
Q(
x E
,
that
G()
w
=
,
w
(M(s,X)) (x) equals
i(T(w lnx))e (X+p)(Hp(w-lnx)) e -(st+p') (Hp (x)) dn
x =
and that
{i
N'
TVV
in SHp(a ) is right
is right K v -finite and almost
to a smooth function in
corresponds
(x)
belongs
vP
T =
K v -invariant.
right
are
L2 (MM(Q) \M(A) )disc.
at
function
This means that each
K-finite.
all
a
vx
.
Then
(6)nwN(wA)w \N'(A)
=
i[^(R,(w,YX) V) (x,),
JAMES ARTHUR
30
(Rv(w,)v) (xv)
where
equals
v(w-
-1 nv)e
N' (Q)nwN (v)w
IN' (v)
w
e
Y v is right
If
THEOREM 4.1:
(v))
(
)) -(sA+p') IHp
+((w'nx
dn
K V -finite and
(Re(A)-p) (a
)
0,
>
a E A
,
integral defining R (w,X)T converges absolutely. For
X in this range, R (w,X) is an analytic function which
It can be analytically
intertwines I P (Oa v)
p' vX
vXA and Ip,(wvX).
the
continued as a meromorphic function to
w'
= w , ,
lp.
Q('pi,,jp,,), there
function p v(w, w', X, o )'
for
scalar valued
s'
c
length(s's)
length(s')
=
+
Moreover, if
.
is a
meromorphic
which equals 1 if
length(s),
such that
R
[15].
=
p
(W,
w', X,
o
)R (W', sX)R
(W,X).
of these statements see [35], [22], [23], [14]
proofs
For
and
(w'w, X)
Hv
The functions
can be
in terms of
expressed
Plancherel densities.
described above.
HP(oa)
K
v
the
suggested
of real
4(x)
belongs
space Hp is spanned by the functions
The
For almost all
space
of
K
v,
T
(xv)
invariant functions.
=
i(T(x))
to
As we
in §2, the structure of this space is similar to that
groups.
It has dimension at most one.
from the fact that the convolution
algebra of
(This follows
KV
bi-invariant
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
functions in
Cc(G(Qv))
conditions in
[31].)
is abelian, proved under general
There is a finite set of valuations,
including
the real one, such that if
Hp(o)P
has dimension
Kr
Pp~K
clearly maps
the map which
npmk
- T
exactly
one.
is not in
p
For such a
K
Hp(o ) P tov~P
Hp,(wap)p KP.
p e H p,(wa ) P
sends
The
^
(wmpkp), np
E
Hp(o P ) P gives
a scalar
m
P
S
p,
S,
,
(w,)
R
composition with
to the function
N(Qp), mp
K
in
31
E
M(Qp), kp
(w,X,a).
E
Kp
Define
p mp(w,X,o)
mS(w,X,o)
=
nS
PS(ww' X ao)
=
I vS Pv(w,w'
and
Then Theorems 3.3 and 4.1
THEOREM 4.2:
there is
R
di
m,disc
immediately yield
representation
If the
finite set
a
IX,).
a
occurs in
such that if
S
(Re(X)- p) (a
M(A)
of
)
>
0,
a
E
p
,
the infinite
It can be
product defining mS(w,X,a) converges absolutely.
analytically continued to a meromorphic complex
valued function of
in
X
-
PCC
If
w'
=
w , ,
s
'
P
'(,)
s E(C,
P
then
mS(w'wXa)
Thus the
theory
P(www',aXo)m
=
(w',sXso)ms(w,X,o)
of Eisenstein series leads to some
meromorphic functions.
.
interesting
Let us show how to express them more
32
JAMES ARTHUR
They are already quite intriguing for their analytic
continuation and functional equations rely on the very deep
Theorem 3.3. For simplicity, assume that G splits over Q.
Then B = P0 is a Borel subgroup of G. Also
explicitly.
MO
is a maximal torus in
=
T
GL1)1
Hom(T,
=
= L
(Recall that an algebraic torus is an
is the dual module.
isomorphic
which is
anti-isomorphism
tori defined over a field
(GL1)
to
category of algebraic
category of Gal(F/F)-
between the
and the
F
modules which are finite and free over Z .
isomorphic
simpler since T splits
is
Gal(F/F)
The functor
.
Hom(T, GL1)
L =
T
defines an
AO
and
G
X(MO)
0
algebraic group
=
Hom(L,F*).
to
0,
over
so
Given
L,
T(F)
The case here is even
Gal(Q/Q)
acts
trivially
"splits over Q". We need only
know that (after having fixed a Chevalley lattice) it is
possible to speak of the group of integral points; as in the
on
L.)
case of
any
p.
We will not define
GLn
the groups
We can take
K
Ho
For
any prime
p,
=
=
G(Z),
G(Z).
Hp 0
:
T(Qp)
etc. are
Consider the
defined for
map
+
G(A)).
set
= log
Hop(xp)
p H0(xp),
x
E
G(Qp).
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
33
Then
ItXp
P
=
<XX,H
p
|p
The range of the valuation
so
a
(t)>
t E
T(Qp),
It follows
L
onto
L
easily
=
Hom(L,2).
that
X(T) .
is the set of powers of
<X, H 0,p (t)> is always an integer. Therefore
homomorphism from the multiplicative group T( p)
additive group
X
isomorphically
T(pp )/T(Zp)
p
Hp
01p maps
p is
0 ,p
to the
is compact.
T(Zp)
The group
H
p,
V
Suppose that a = vav is as above and that p ¢ S .
K
Then H (a ) P . {0}. This means that I (a ) is a constituent
p p
p p
of a so-called class one
principle
series - a
B(
induced from the pull-back to
G(Qp)
character on
T(Qp)
p)
which is trivial on
representation
of a quasi-
T(ip).
of
Such a
quasi-
a e
T(p),
character can be written
X(Hoop((a))
a - p P
X
where
p
,
is an element in
Hom(L ,C).
p,X instead of
It follows that
To construct
X
+ X.
m
a
(W,A,o)
J
=
N'
for
A
p
a
p
,
we simply replace
(Xp++p0)
(H0
p(W'
X
p
by
n)) dn,
(p) nwN(Qp)w \N' (p)
in the domain of absolute
convergence
of the
integral,
and
34
JAMES ARTHUR
PO
Incidentally, the
intersection with
Consider the
and
of
GL
x
for both
and
for which the
1
GL2 , w
I
automorphic representation
G =
=
GL;
and
X2
X1
and
°
a0
=
a2
xi (a)X2(a2)
Gr6ssencharakters.
If
there will be
X2,
p
is unramified
complex numbers slp
such that
s2 p
xi(a)
Identify
wN(Qp)w
n
will be an
a
X1
is the quotient of those Haar
dn
case that
special
a
+
of each group has volume 1.
G(Zp)
[aI
0
with
PP nM
=
N' (Q )
and
Then
.
P = P
P
measure
N' (p)
measures on
=
L
Z2
with
|al|
p
in the canonical
a c
way.
Q
Then
°
a
Ho0
=
l
=
(logplapa1 logpla2lp)
quasi-character X P corresponds to the pair (s 1ip , s 2,p )
which acts on LV by the dot product. The dual space 'p*P,C
is identified with
Set
, and p0 becomes (2
The
A =
sp0
for a fixed
(X(p p +X+P)) [HOp
0
complex number s.
a°H
a2
l2
Then
la
S2,p+22
'+IHSs
If
1
n(x)
x
=
x c
0
1
Q,
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
35
let
r
wn(x)
0
k,
n
=
as a
function of
r E
p
k
integral it is necessary
In order to evaluate the
r|p
N(p),
n
E
GL2 (2p)
to express
Notice that if
x.
(u1 , U2)
v =
1,
U2
E
Qp
and
lvil
then
iIvkll
=
i|v||
Max{lu lp, 21p},
=
u
for each
Irlp1
k E
GL2(2p).
(0,1)wn(x)i|
=
Therefore
Max{l,
=
x|p }.
Thus,
f
mp(w,X,o) =
J
e
(p+X+p0)
p (H0,p(wn))
Hpw
dn
N(<p)
(Max{, xlp })-(t+l)dx
=
p
=
I
1 dx+
I Ix-(t+l)
dx
p-p
p
where
t =
sp
S2,p
+ s
vol(Zp)
is the disjoint union of the sets {pnU :
pP
We want the Haar measure on
one.
-
Qp
such that
equals
-
<n< co},
JAMES ARTHUR
36
where
Up
xlp
{x E p:
=
=
1},
so
1-=
= O vol(pnp)
vol(Zp) vlUp
In=
U
p
=vol(Up) n=(0
=
In other words,
m
(w,X,a)
.
It follows that
1 -p.
equals
vol(U )
equals
1+
If
vol(Up) |1-
Re(t)
> 0
X1
p-n(t+l) pnl 11.
converges
this
1
-1 p (Sp
1
1-p
and
equals
[r 1 p-tp
P+
-t
2,p P -(s+l)
p (1,p
2,pp
-1
_
L
X'
'
V(s'
(s, X1X2
Lp(l+s,
XX21
mS(w,X,o) equals
We have shown that the scalar
Theorem 4.2
implies
LS (s, X1X2-1 )
that
-11
LS(l+s, X1X2
mermorphic function.
functional equation
continued as a
LS(s, X1X21)
--1L
(l+s, X1X2 )
=
Is(W,
w
-
It also
spo
)
)
LS(s, x1X2
1
12-
LS (l+s, X1X2 )
can be
)
yields
analyticall
analytical5
the
Ls(1-s, X2X1)
-1
LS(-s, X2X1
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
is a
(pS
product
densities,
of Plancherel
It follows that
continued.
This was well known as was
and is an
elementary
) can also be analytically
the functional equation.
LS(s,X1x2
function.)
37
(The usual function equation is of course stronger, for it
relates the functions Ls(s, X1X2 ) and Ls(1-s, X2X1 ). It
is the case
in Theorem 2.2, which was established
n=1
years ago by Hecke.)
G
However, Theorem 3.3 applies to
certainly leads to nonclassical
general, the integral
where it
In
N'
results.
dn
'
P
many
arbitrary
(Up)nwN( p)W \N' (Up
Karpelevic ([13], [26]).
with the unique element in Q
Identify each s E Q(Z'L,
P ,)
P
which maps each simple root of (M,A ) to a simple root. We
can certainly assume that w acts on A0 by this element in
is evaluated by the method of Gindinkin and
Q.
Then
N' (
p)
n
For almost
wN(Qp)w \1N' (Qp )
p,
N
each element
fore suppose this is so for
w
p
B
n
wN
(Qp)w
belongs
to
K .
(p)
Then
S.
m
\N(up )
We can there-
(w,X,a)
equals
(Xp +X+pO) (Ho0dp)(n))
N(s)
where
the
N(s)
is the intersection of
unipotent radical
P'. Suppose
that
in
s
Q,
(G, Ao)
and
s
of the
s1
wN
(Qp)w
parabolic subgroup opposite
sy,
where
s1
is the reflection about a
such that
(p) ,
with N
to
is some other element
simple
root
y
of
JAMES ARTHUR
38
length(s)
simple change
product of
A
=
length(s
) + 1.
of variables exhibits the
integral
as the
(sy( Xp+X)+po)(Hp(n)) _
N(sI)
with
(X
Pp
+X+po0 )(H O'p (n)) dn.
N(sy)
Since
N(S
the one on
is one dimensional, this last integral reduces to
)
GL2,
which we have
1
-
1
-
just
Z
It follows
positive
{a
0}
:
<
s
by induction
1)
on
=
sy
roots of
{
E
(G,Ao),
+: sl
the length of
s
-((A
1 - Pp
m
P (w,X,o).
+X)
<
O}.
that the
equals
This therefore equals
equals
p
is the set of
eZ+
It
P
Pp
-(X +X) (y
Now if
calculated.
(BV)+1)
integral
39
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
L-groups and functoriality
§5.
history of representations of reductive groups has
often been of phenomena which generalize from GL2 to arbitrary
groups. The functions mg(w,X,o) could hardly be otherwise.
They must surely be quotients of L-functions which generalize
The
Ls(s,
the functions
them to arise from a
-1
X1X2
obtained for
)
family
of
We would expect
GL2.
semisimple conjugacy classes in
complex general linear group. The connection is made
through the notion of an L-group, introduced by Langlands in
[27].
some
For the moment, we shall continue to take
reductive group defined and split over
Q.
G
to be a
Suppose that
automorphic representation of G. It is said
to be unramified at a prime p if
p has a K -fixed vector.
Then
p is a constituent of the class one principle series
associated to a quasi-character
=
('T v
is an
X
a +
The
quasi-character is
p
not
(H
p
(a))
uniquely
a E
determined
tion is similar to that of real groups.
by
T(Qp)/T(r
p.
The situa-
The unramified
representations
p of G(Vp) are in bijective correspondence with
the orbits of the Weyl group Q acting on quasi-characters of
T(Qp)/T(Zp)
[3].
associated to a
Now
L
is a finite free
unique algebraic
LT0
complex points, denoted by
Hom(L , C*). The map
H
+
p
torus over
is
-X (H)
P
.:--module. It
C whose group
is
of
canonically isomorphic to
H
L
3P~~HEL,
JAMES ARTHUR
40
belongs
point
so it defines a
Hom(L , ¢)
to
LT
in
t
that is
tHP -H
p
=
(H)
Hc L
,
LT
through its action on L
There is thus a bijective correspondence between irreducible
class one representations of G( p) and orbits of Q in LT0
The
Weyl group
Q
acts on
The set of roots,
Similarly
groups
Associated to the triple
(L, A, L , A ),
datum
V
roots and
regarded only
(G,T),
Ap =
A
V
the
A
algebraic
as
Z )
,
.
there is the based root
(G,B,T)
where
L.
Thus
L .
(L, E, L
there is the root datum
C,
over
are contained in
Z ,
pair
associated to the
is contained in
(G,T)
of
Z,
the co-roots,
.
=
ApP0
simple
is the set of
simple co-roots, defined by
B.
(For formal definitions of root data and based root datum see
[37].)
Conversely any
any based
root datum comes from a
is a reductive
group
pair
root datum comes from a
over
C,
T
(G ,T)
a:gd
where
G
(G ,B , T),
triple
is a maximal torus, and
(G , B, T)
subgroup containing T [37]. Both (G ,T)
are uniquely determined up to isomorphism.
(LV
,
is a Borel
,A
,L
triple
and
LT
of
GO.
and the set of
correspondence
G
By convention,
and of
Now
The set of
simple
P
LG.
-->
and can be
simple
co-roots is
LpO
LG
roots of
as a Cartan sub-
( G , T )
is
A,
bijective
parabolic subgroups
There is a
between standard
The Weyl group of
over
As above,
G.
regarded
A.
will denote
algebraic group
of a reductive
will be a Borel subgroup of
equals Hom(L, C*)
group
of
LT).
complex points
B
and
is a second based root datum. It therefore comes from
(LGO LB
the set of
C,
A)
B
(LG ,
on
isomorphic to Q,
Q2, with its natural action on
isomorphic
LT)
T
is
.
.
Some
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
examples
are
following brief
in the
table
LGO
G
GL n (C)
GL
n
PGL
n
SL
n(C)
PGL (C)
SL
SP2n
2n+l (C)
PSP2n
Spin2n+
SO2n+l
SP2n (C)
Spin2n+l
PSP2n (C)
Spin2
SO2 (C)/{+1}
S02n /{±1}
Spin2n (C)
semisimple conjugacy
The
(C)
LG
classes in
correspondence with the orbits of
for
41
L O
in
Q
every automorphic representation
are in one-to-one
T .
of the
T
It follows that
split group
primes, and for every p ~ S
there is a semisimple conjugacy class c (7) in LG. How
p
does this compare to the conjugacy classes defined in an ad
manner for GL
in §2? The Hecke algebra for G( p),
there is a finite set
p)),
H(G(
G(Q p)
P
of
S
is the space of
which are left and
is an
algebra
(p , Up)
G(Q ),
G
hoc
compactly supported functions on
riqht invariant by K = G(Z ). It
p
P
under convolution.
If
H(G(Q p)),
f
and
is an (irreducible) unramified representation of
the operator
ITp(f)
f(x)7 (x)dx
=
G(Qp)
on
the
U
P
G(Z)
leaves invariant the
complement
subspace
)
G(
of
U
P P
in
U
P
.
U
P
Since
.
It vanishes on
G(
dim U
P
)
=
1,
the
42
JAMES ARTHUR
restriction of
Cpf(-p).
to this
(f)
T
subspace is
a
scalar, say
It is clear that
CP fl*f2 Pfl2
We
as a function on the
Cpf
LG.
regard
classes in
It can be shown that the map
H(G(Qp))
an isomorphism from
functions
cp
f
is
f
onto the algebra of class
LG generated by
on
semisimple conjugacy
the characters of finite
dimensional representations [see [3] and either [5] or [31]).
In the case of
', defined
pDi
f.
does
just
of
G = GL
the
GL n
multiple by
(C)
in §2,
p
th
th
on the
to what class function on
,
GL (C)
correspond? It turns out to be
-i(n
i)
of the character of the action
exterior power of
C
n
[39].
These
characters, evaluated on a conjugacy class in GL (C),
coefficients of the characteristic
p (T)
formula for
polynomial.
m
that occurs in
M(A)
of
(w,AX,)
Then
1c
p+:sS<O}
=
1-p
=
Let
Lp
=
LM
N
of M
ofLO
on
N .
T
L
Av
{13EC
+:sa<O}
1
-
1
-
Then
P.
Let
The
arThe
Ap
LM
iovis
a =
RMdisc If
-((X +X) (V)+l)
p
a
S
,
p(+-)(v
P
(tp)
v
V
be the standard
which corresponds to
algebra of
contained in
We obtain the
given in §2.
Now, return to the problem of §4.
representation
are the
)'
(tp)apx
parabolic subgroup
acts on
be the center of
L
,
M .
of
LG
the
Lie
ie
i
It is
of the representation
A representation
weights on the
V
, to LA
are just the restrictions,
of
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
Bv
roots
(LG0 LT0);
of
43
that is,
L 0
-
L 0
'
=
,'a v
>0
v
a
where
L 0
L.0
{X
=
E
L
:L
Ad(a)X
a
=
(Y. L
for all
X
01.
LAO}
E
a
a
Each subspace
to
is invariant under
be the restriction of
r v
of
L
of
L
r
Lv
A
on
T
are
is
a.
Now
ifiB V restricts to
to this
M
0
the set of
subspace.
in
Let
The
weights
whose restriction
Z+
is a linear function
X
v
a
M .
p .
Therefore,
,
e -X(U
)
_
-
)
e -X(
It follows that
m
(wX,o)
=
n
{
'>0,s
<0}
detI(- r (t
p )p-k(X( )+1]
))
det(I -r (t )pdetI
-r
v( (T))p(((V) +1))
a=1
<0I}ar nall
det
{ct >Osc
We
of a
are
finally in
Langlands'
a
position
L-function.
to
I-r
aap
p-()pJ
V(
iat
appreciate
Suppose that
T
=
=
det[I r[(p(7())zJ
-
v
t
is an
v
automorphic representation of G(,A). Then if r
dimensional, complex analytic representation of
Lp(z,T,r)
the definition
,
is
G
any finite
define
p
S
JAMES ARTHUR
44
and
LS(s,7,r)
for
any finite
G = GL n
of GL
and
set of
r
=
npS Lp(p ,,1r),
valuations,
is the standard
S,
which contains S
.
If
n-dimensional representation
just those defined in §2.
They are called standard (or principal) L-functions.
Returning once again to the case that o is a representawe see that
tion of M(A) which occurs in RM disc
(C),
m
According
L-functions are
the
(w,X,o)
S(w,)
=
LS([(a ), a, r J
n
{a
la>OsaV<0}
L$(X(av)+1, , r a v
a
analytic continuation
maximal parabolic, and
to Theorem 4.2 this function has
and functional
equation.
If
P
is a
there is
only one root of (P,A), there will then be only one
in the product. It will follow that LS(s, a, r v) has
a
analytic continuation. It also will satisfy a functional
equation, albeit weaker than the ones satisfied by classical
L-functions.
CONJECTURE 5.1
(Langlands):
Suppose that
T
is an
automorphic
representation of G(A) and that r is a finite dimensional
analytic representation of G . Then L (s,7,r) can be
analytically continued with a functional equation relating
Ls(s,T,r)
with
LS(1-s,
See
This
handed a
, , r).
[27, Question 1].
conjecture suggests another
split group G and a family
simple conjugacy classes in
G .
one.
{p
Suppose
: p
S}
one were
of semi-
Suppose it happened that for
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
every finite dimensional representation
r
LG
of
45
the function
-1
s
+
TpIS det(l
-
r(p)p)
plane and could be analytically continued
with functional equation. With the converse theorems for
GL2 and GL3 in mind, one would strongly suspect that the
family {p} arose from an automorphic representation of G.
was defined in a half
split group, and let
be an analytic homomorphism. If r is a finite
analytic representation of G
Now, let
p :
G
+
dimensional
be another
G
,
r
=
r op
analytic representation of G . Take
an automorphic representation 7 of G, and for each p 4 S
let
p be the semisimple conjugacy class in L
which
p
is a finite dimensional
contains
p(ip()).
It is clear that
-1
L
=
analytic continuation
The
on the
LG
there is
=
and functional
)
equation
of the function
S-iT
of
G.
(Langlands): Given an analytic homomorphism
L
and an automorphic representation Tf of G
G
rT of G such that
an automorphic representation
and such that for each p 4 S ,
p() is the
CONJECTURE 5.2
p :
Hp4s det(I-r(,p)p
right would follow from Conjecture 5.1. The family
surely ought to be associated with an automorphic
{Ip}
representation
S
(s,T,r)
JAMES ARTHUR
46
conjugacy
class in
which contains
G
See
p(p
(7)).
[27], [2].
we could take
to be the standard
representation of GLn(C). Then r equals p and Ls(s,7,r) clearly
equals L (s,7T), a standard L-function for GLn. Its analytic
continuation and functional equation would be assured by
Theorem 2.2. Therefore Conjecture 5.2 implies Conjecture 5.1.
If
G = GLn
For all of
r
§5 and much of §4 we have taken
to be a
G
split group. This was only for simplicity. What happens if
G is an arbitrary reductive group defined over Q? We can
always choose a maximal torus T defined over 4 and a Borel
subgroup (not necessarily defined over 0) such that
A0
c
T
B
c
c
P.a
(L, A, L, A ),
The based root datum,
associated as it is to the group
is defined as above.
G(C),
triple (LG0C LB LT0) The theory of algebraic
groups yields in addition a homomorphism from Gal(Q/Q) to
Aut(G)/Int(G), the group of automorphisms of G defined over
Q, modulo the group of inner automorphisms. It follows from
a theorem of Chevalley that this quotient group is isomorphic
to the group of automorphisms of the based root datum
(L, A, L , A ). Every automorphism of (L, A, L ,A ) is clearly
V
V
an automorphism of
(L , A , L, A), so therefore gives an
element in Aut( G )/Int( G ). It is easily seen that the exact
So is the
sequence
1 - Int(
splits.
This
LG)
gives
InIn
other
other
wwords,
ords,
-
a
Aut(G0)
-+ Aut(
homomorphism
Gal(Q/Q)
Gal(Q/)
acts on
of
LG)/Int(G)
Gal(Q/Q)
LG.
Define
-
into
LG
1
Aut( G ).
to be
AUTOMORPHIC REPRESENTATIONS AND NUMBER THEORY
the semidirect
product
LGO
It is a
X Gal (/).
locally compact group, called the L-group of
LB
and
Gal(i./Q)
LT.
in such a way that
split
exact sequence above can be
normalizes
47
Then
B =
B X Gal(/Q)
The
G.
Gal(&/Q)
and
subgroups of G.
Suppose that T = vr'vv is an automorphic representation
of G. Then r is said to be unramified at a prime p if
ip contains a K -fixed vector and if in addition, G is
T X
T =
are closed
P
P
quasi-split over
p.
of
qp
and splits over an unramified extension
One shows that for each unramified
p there
corres-
conjugacy class D p (i) in LG whose projection
onto LG0 contains only semisimple elements [3]. For finite
dimensional representations r of LG, the L-functions
LS(s,',r) are defined exactly as above. The computation of
ponds
a natural
the functions
m
(w,X,a)
can be carried out for
The resulting formula is similar to the one for
general G.
split G (see
[34]).
Suppose that
An
L-homomorphi.m
G
G
and
is a continuous
p
which is
Gal (Q/Q)
are reductive
compatible with
the
LG
-+
homomorphism of LG
Conjecture 5.2 is
to
GL
g.
homomorphism
G
projections
and whose restriction to
groups over
LG
([3]).
of each
group onto
is a complex analytic
The
generalization of
JAMES ARTHUR
48
(Langlands): Suppose G and G are reductive
groups over q which are quasi-split. Suppose that
LLG
G
G is an L-homomorphism. Then for every automorphic
p :
representation X of G there is an automorphic representation
E of G such that S = S- and such that for each p 4 S,
G which contains p(p (7r)).
p(r) is the conjugacy class in
CONJECTURE 5.3
See
[27], [2].
conjecture is known as the functoriality principle,
and is very far from being solved. It implies that all the
L-functions LS(s,7,r) can be analytically continued with
functional equation, and that the location and residues of all
poles can be determined. For an important special case, take
This
G
=
{1}
and
continuous
G
=
Then an
GL .
is
just a
homomorphism
:
The
L-homomorphism
Gal(Q/Q)
functoriality principle
GL (C).
in this case reduces to
Conjecture
2.3.
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~notes~.~r
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