BULLETIN (New Series) OF THE

AMERICAN MATHEMATICAL SOCIETY

Volume 40, Number 1, Pages 3953
S 0273-0979(02)00963-1
Article electronically published on October 10, 2002
THE PRINCIPLE OF FUNCTORIALITY
JAMES ARTHUR
Preface
Following the explicit instructions of the organizers, I have tried to write an arti-
cle that is suitable for a general mathematical audience. It contains some analogies
and metaphors that might even be put to nonmathematicians. I hope that experts
will be tolerant of the inevitable simplications.
The principle of functoriality is one of the central questions of present day math-
ematics. It is a far reaching, but quite precise, conjecture of Langlands that relates
fundamental arithmetic information with equally fundamental analytic informa-
tion. The arithmetic information arises from the solutions of algebraic equations.
It includes data that classify algebraic number elds, and more general algebraic
varieties. The analytic information arises from spectra of dierential equations and
group representations. It includes data that classify irreducible representations of
reductive groups.
1. Spectra
The expected relationships between arithmetic and analytic objects are through
properties of spectra. In this sense, the theory bears a resemblance to some key
notions from classical physics.
One of the great discoveries of nineteenth century physics was the recognition of
absorbtion spectra in extraterrestrial sources of light. The spectral decomposition
of starlight revealed dark bands at characteristic wavelengths across the spectrum.
This suggested that light of certain discrete frequencies was being absorbed before
it reached telescopes on earth. Physicists also observed that the missing wave-
lengths matched wavelengths of light emitted by chemical elements in laboratory
experiments. They were thereby able to deduce that stars in our galaxy contain
the very elements on which the chemistry on earth is based. Later observations of
light from other galaxies revealed a downward red shift in characteristic absorbtion
spectra. This indicated that the galaxies were moving apart at enormous velocities,
and suggested that the universe might have had a cataclysmic origin.
It was not until the advent of quantum mechanics in the twentieth century that
absorbtion spectra were given a satisfactory theoretical explanation. They were
Received by the editors October 10, 2000, and, in revised form, December 1, 2000, and Febru-
ary 21, 2002.
2000 Mathematics Subject Classication. Primary 11R39; Secondary 22E55.
Key words and phrases. Spectra, automorphic representations, Galois group, functoriality,
Langlands group, motives.
The author was supported in part by a Guggenheim Fellowship, the Institute for Advanced
Study, and an NSERC Operating Grant.
c 2002 American Mathematical Society
39
40 JAMES ARTHUR
shown to correspond with eigenvalues of appropriate Schr odinger operators. A
given atom could absorb or emit light only at certain frequencies, corresponding to
the energy levels of bound states represented by dierent eigenvalues. The mathe-
matical spectra of dierential operators thus carried fundamental information about
the physical world, which even now seems almost magical.
The analogy with number theory is through spectra of other dierential opera-
tors. These are Laplace-Beltrami operators (and variants of higher degree) attached
to certain Riemannian manifolds. The spectra of these and other operators are ex-
pected to carry fundamental information about the arithmetic world, a possibility
that also seems quite magical.
2. Automorphic representations
The spectral data that is believed to be related to number theory is framed in the
language of automorphic forms. Suppose that H is a locally compact, unimodular
topological group, and that is a discrete subgroup of H. Let R be the regular
representation of H on the Hilbert space L
2
(H), taken with respect to a Haar
measure on H. Then R is the unitary representation of H dened by
(R(y))(x) = (xy), L
2
(H), x, y H .
One is interested in the decomposition of R into irreducible unitary representations
of H. For example, = Z is a discrete subgroup of the additive group H = R. The
decomposition of R in this case amounts to the theory of Fourier series.
The theory of automorphic forms concerns a much richer family of examples. For
, one takes a group of rational points G(Q), where G is a (connected) reductive
algebraic group dened over Q. For example, one could take G to be the general
linear group GL(n) of rank n. Then GL(n, Q) is just the multiplicative group of
invertible, rational, (nn)-matrices. In general, it seems pretty clear that = G(Q)
has an interesting arithmetic structure. What may not be obvious at rst glance is
that there should be an associated locally compact group H.
The rational eld comes with the usual absolute value
v

(t) = [t[

, t Q ,
and its corresponding completion Q

= R. For each prime number p, there is also

a p-adic absolute value
v
p
(t) = [t[
p
, t Q ,
on Q. Its completion is the eld Q
vp
= Q
p
of p-adic numbers. We recall that if
t = (ab
1
)p
r
, (a, p) = (b, p) = 1, r Z ,
then
[t[
p
= p
r
.
In particular, t is of small p-adic absolute value if it is highly divisible by p. The
p-adic absolute value satises the strong form
[t
1
+t
2
[
p
max [t
1
[
p
, [t
2
[
p
, t
1
, t
2
Q ,
of the triangle inequality. It follows that the compact subset
Z
p
= t
p
Q
p
: [t
p
[
p
1
of Q
p
is actually a subring.
THE PRINCIPLE OF FUNCTORIALITY 41
The elds
Q
v
=
_
R, if v = v

,
Q
p
, if v = v
p
,
are all locally compact. They can be put together into a locally compact ring. Since
the elds are actually noncompact, their direct product will not in fact be locally
compact. However, one can form the restricted direct product
A =
rest

v
Q
v
= R
rest

p
Q
p
= t = (t
v
): t
p
= t
vp
Z
p
for almost all p .
Endowed with the obvious direct limit topology, A = A
Q
becomes a locally compact
ring, called the adele ring of Q. The diagonal image of Q in A is easily seen to be
discrete. One deduces from this that H = G(A) is a locally compact group, in
which = G(Q) embeds as a discrete subgroup.
The regular representation R of G(A) on L
2
(G(Q))G(A) is not generally a direct
sum of irreducible subrepresentations. It does often have a part that decomposes
discretely, like the theory of Fourier series, but it can also have a part that decom-
poses continuously, like the theory of Fourier transforms. An automorphic represen-
tation of G can be described informally as an irreducible representation of G(A)
that is a constituent of R. (The formal denition in [L4] includes representations
obtained by analytic continuation in the parameters of the continuous spectrum.)
Suppose that is an arbitrary irreducible representation of G(A). Under a weak
continuity condition, which is always satised by automorphic representations, one
can show that is a restricted tensor product
=

v
of irreducible representations of the groups G(Q
v
), in which almost every
v
is in a
natural sense unramied. Conversely, one could obviously construct an irreducible
from any such choice of representations
v
. However, the condition that be
automorphic is very rigid. It imposes deep and interesting relationships among the
components
v
of .
3. Dual groups and conjugacy classes
As an example, let us describe the unramied representations of general linear
groups. Suppose that G = GL(n). An irreducible representation
v
of G(Q
v
) =
GL(n, Q
n
) is unramied if v = v
p
is p-adic, and if the restriction of
vp
=
p
to
the maximal compact subgroup G(Z
p
) of G(Q
p
) contains the trivial representation.
The unramied representations can be classied as follows.
Let
B(Q
p
) =
_
b =
_
b11
.
.
.
0 bnn
__
be the Borel subgroup of upper triangular matrices in G(Q
p
). Suppose that u =
(u
1
, . . . , u
n
) is an n-tuple of complex numbers. Then

u
(b) = [b
11
[
u1+
n1
2
[b
22
[
u2+
n3
2
. . . [b
nn
[
un
n1
2
, b B(Q
p
) ,
42 JAMES ARTHUR
is a 1-dimensional representation of B(Q
p
). (The exponents
_
n1
2
, . . . ,
_
n1
2
__
represent a normalization that compensates for the failure of the group B(Q
p
) to
be unimodular.) Let

V
p,u
be the space of continuous functions from G(Q
p
) to C
such that
(bx) =
u
(b)(x), b B(Q
p
), x G(Q
p
) .
There is then an induced representation
p,u
of G(Q
p
) on

V
p,u
by right translation:
(
p,u
(y))(x) = (xy), x, y G(Q
p
) .
The representation
p,u
is usually irreducible. It always has a unique irreducible
constituent
p,u
that is unramied. If u

= (u

1
, . . . , u

n
) is any other n-tuple, one
can show that the representation
p,u
is equivalent to
p,u
if and only if
(u

1
, . . . , u

n
) (u
(1)
, . . . , u
(n)
)
_
mod
_
2i
log p
_
Z
n
_
,
for some permutation of (1, . . . , n). Conversely, any unramied irreducible rep-
resentation
p
of G(Q
p
) is equivalent to
p,u
, for some u.
Set

G equal to the complex general linear group GL(n, C). An unramied rep-
resentation
p

p,u
of G(Q
p
) can then be represented uniquely as a semisimple
conjugacy class
c(
p
) =
_
_
p
u
1
0
.
.
.
0 p
un
_
_
in

G. (The powers p
ui
eliminate the ambiguity caused by the congruence condition
above, while the identication of c(
p
) with a conjugacy class takes care of the
ambiguity caused by permutations.)
Suppose now that is an automorphic representation of G = GL(n). Then
determines a family
c() = c
p
() = c(
p
): p / S = S()
of conjugacy classes in

G, one class of almost every prime number p. An automor-
phic representation thus provides us with an explicit set of elementary data.
The construction we have described for GL(n) was carried out by Langlands
for a general reductive group G. His rst step was to dene a dual group

G. In
general,

G is a complex reductive group that is related to G in two ways. First,
any maximal torus

T in

G is isomorphic to the complex dual torus
X

(T) C

= Hom(X

(T), C

)
of any maximal torus T in G. Secondly, the Coxeter-Dynkin diagram for

G is
the dual of the associated diagram for G. The dual group provides a natural
parameter space for representations, but it has to be supplemented by a second
piece of information.
The general situation is complicated by the fact that G need not be split over
Q, in the sense that it contain a maximal torus T that is isomorphic over Q to a
product of groups GL(1). One can always choose an isomorphism over Q from
G to such a group. The map
(
1
), Gal(Q/Q) ,
THE PRINCIPLE OF FUNCTORIALITY 43
then represents the obstruction to being dened over Q. This map determines a
homomorphism

Q
= Gal(Q/Q) Out(G) = Aut(G)/Int(G)
from the absolute Galois group of Q to the group of outer automorphisms of G.
The last homomorphism does not determine the isomorphism class of G over Q
uniquely. To every such homomorphism, there does correspond a unique G over Q
that is quasisplit, in the sense that it contains a Borel subgroup (i.e. a maximal
solvable subgroup) that is dened over Q. For this reason, quasisplit groups play a
central role in the theory. In general, the group Out(G) is canonically isomorphic
to Out(

G). The group G therefore determines both a complex dual group

G and a
homomorphism from
Q
to Out(

G).
Given G, Langlands introduced what is now called the L-group of G. It is dened
as a semidirect product
L
G =

G
Q
,
relative to the homomorphism from
Q
to Out(

G), and a suitable section from

Out(

G) to Aut(

G). The action of

Q
on

G factors through a quotient Gal(E/Q) of

Q
, for a nite Galois extension E of Q. For many purposes, one can replace the
pronite group
Q
by the nite quotient Gal(E/Q). In particular, if G is a group
such as GL(n) that splits over Q, one can often work with the dual group

G instead
of the full L-group
L
G.
The notion of an unramied representation of G(Q
p
) can be dened whenever G
is quasisplit over Q
p
and is split over an unramied extension of Q
p
. Under these
conditions, which hold for almost all p, the construction for GL(n) above is easily
extended to G(Q
p
). It provides a bijection
p
c(
p
) between the unramied
irreducible representations
p
of G(Q
p
), and the conjugacy classes c(
p
) in

GGal(E/Q)
whose image in

G is semisimple, and whose image in Gal(E/Q) equals Frob
p
, the
Frobenius class of p.
At this point, we ought to recall that Frob
p
is a well dened conjugacy class in
Gal(E/Q), for every p outside a nite set S(E). For example, if E is represented
as a splitting eld of an irreducible monic polynomial with integral coecients,
the primes p with Frob
p
= 1 are the ones for which the polynomial breaks into
distinct linear factors modulo p. A rather deep theorem from algebraic number
theory asserts that these primes completely determine E. In other words, the map
E Spl
E
= p: Frob
p
= 1 ,
from nite Galois extensions E to families of prime numbers, is injective. An
independent characterization of the image of this map could be regarded as a clas-
sication of the nite Galois extensions E.
Given the parametrization of unramied representations, we see that an auto-
morphic representation of G determines a family of conjugacy classes
c() = c
p
() = c(
p
): p / S = S()
in

G Gal(E/Q). The dependence of this family on S() and E is, incidentally,
quite inessential. One can remove it by declaring two families of conjugacy classes
c = c
p
and c

= c

p
to be equivalent if c
p
= c

p
for almost all p. An automorphic
44 JAMES ARTHUR
representation thus provides a well dened equivalence class c() = c
p
(). In any
case, our discussion to this point leads us to draw the following conclusion. The
rigidity inherent in the condition that be automorphic will be reected in rather
concrete objects. We should expect there to be interesting relationships among the
dierent conjugacy classes in the family c() attached to .
The conjugacy classes c
p
() attached to an automorphic representation
=

v
=
R

p
_
represent analytic data. The language of group representations is really what is
most natural, but the constructions could have been carried out in terms of the
spectral theory of dierential operators. Any irreducible representation
R
of G(R)
determines a set of eigenvalues of an algebra of Laplace-type dierential operators.
These operators act simultaneously on a tower of Riemannian locally symmetric
spaces attached to G. The p-adic representations
p
have their own spectral in-
terpretation. In particular, any family c
p
(): p / S determines a homomorphism
h
S
=

p/ S
h
p
from a certain commutative C-algebra
H
S
=

p/ S
H
p
into C. Elements in H
S
are called Hecke operators. They act on the underlying
tower of locally symmetric spaces, and commute with the Laplace operators. They
are in fact combinatorial analogues of Laplace operators, in the precise sense of
graph theory (and its higher dimensional analogues). The conjugacy classes c
p
()
thus provide eigenvalues of Hecke operators. In the language of quantum mechanics,
they represent energy levels of momentum observables.
4. Functoriality
The principle of functoriality describes deep relationships among automorphic
representations on dierent groups. In its most fundamental form, it is a remark-
ably simple conjecture that relates the associated families c
p
() of semisimple
conjugacy classes.
Suppose that G

is a second reductive group over Q, and that is an algebraic

homomorphism from
L
G

to
L
G such that the diagram
L
G

L
G

Q
is commutative. Suppose also that

is an automorphic representation of G

. Then
c
p
(

) is a family of semisimple conjugacy classes in

L
G

. Composing with , we
obtain a family of semisimple conjugacy classes (c
p
(

)) in
L
G.
Conjecture (Langlands [L1]). Suppose that G, G

and are given, and that G is

quasisplit. Then for any automorphic representation

of G

, there is an automor-
phic representation of G such that
c
p
() = (c
p
(

)), p / S() S(

) .
THE PRINCIPLE OF FUNCTORIALITY 45
The principle of functoriality, in the basic form we have just stated, is really
quite concrete. Semisimple conjugacy classes in complex groups can certainly be
described in explicit terms. Homomorphisms between L-groups are also pretty
concrete objects. For example, if G = GL(n) and G

is split, can be decomposed in

terms of irreducible representations of the nite group Gal(E/Q) and the complex
reductive group

G

. The former objects go back to Frobenius, while the latter are

classied by the highest weight theory of Weyl. The general principle of functoriality
asserts that the relationships imposed on the classes c
p
(

) by the condition that

be automorphic, whatever form they might take, are supplemented by a completely

separate set of relationships imposed by the condition that be automorphic.
Any case of functoriality that can be established thus represents a reciprocity law
between the arithmetic data implicit in the automorphic representations and

.
To get a sense of the depth of the conjecture, we consider what might at rst
glance seem to be an elementary special case. Suppose that G

= 1 and G =
GL(n). Then

is of course trivial. The choice of amounts to that of a homo-

morphism
r : Gal(E/Q) GL(n, C) ,
for a nite Galois extension E of Q. Functoriality in this case therefore reduces to
the following assertion.
Conjecture (Langlands [L1]). For any r, there is an automorphic representation
of GL(n) such that
c
p
() = r(Frob
p
), p / S(E) .
We can assume that r is faithful. The conjecture then asserts that
Spl
E
= p / S(E): c
p
() = 1 .
This amounts to a characterization of the set Spl
E
in analytic terms and, in this
sense, represents a classication of the nite Galois extensions E of Q. The conjec-
ture remains for the most part unproved. The cases that have been established are
summarized in the following examples.
(a) n = 1. The conjecture in this case is the Kronecker-Weber theorem. An
elementary analysis of the cyclotomic elds
Q
_
e
2i
N
_
, N 1 ,
and of the automorphic representations of GL(1) converts this case of the conjecture
to the assertion that any extension E with Gal(E/Q) abelian is contained in some
Q
_
e
2i
N
_
. Langlands actually formulated the principle of functoriality for a general
global eld F in place of Q. In this more general context, the case of n = 1 is the
Artin reciprocity law, which is the essence of class eld theory. (See [H].)
(b) n = 2, Gal(E/Q) solvable. This is the theorem of Langlands [L3] and Tunnell
[T] that was an important ingredient of Wiless proof of Fermats Last Theorem. It
is a consequence of cyclic base change for GL(2), among other things, and is valid
if Q is replaced by any number eld F.
(c) n arbitrary, Gal(E/Q) nilpotent. The conjecture was established in this case
as an application of cyclic base change for GL(n) [AC]. The result is again valid if
Q is replaced by an arbitrary number eld F.
(d) Further results. Partial results for n = 2 have recently been established in
the remaining case that the image of Gal(E/Q) in PGL(2, C) is the simple group
46 JAMES ARTHUR
A
5
[BDST]. These exploit the p-adic properties of holomorphic modular forms.
In the case that n = 4 and Gal(E/Q) is solvable, the conjecture has recently
been established for representations that factor through the group GO(4, C) of
orthogonal similitudes [Ra]. The result is valid for a general ground eld F (of
characteristic 0).
5. The automorphic Langlands group
Suppose for the moment that G = GL(n). The special case of functionality we
discussed in the last section asserts that there is an automorphic representation
of G attached to any continuous, n-dimensional representation of the compact,
totally disconnected group
Q
= Gal(Q/Q). It would be natural to inquire about
the converse. Given any automorphic representation of G, is there a continuous
homomorphism
r : Gal(Q/Q) GL(n, C)
such that
c
p
() = r(Frob
p
), p / S(E) S() ?
The answer is no! For it is known that there are rather stringent necessary con-
ditions on the local component
R
of any such . To show that these are also
sucient conditions is an open problem, on which there has been progress in the
case n = 2. However, the point we are making is that there are more automorphic
representations than there are Galois representations r. To what might they
correspond?
In [L5], Langlands suggested that there ought to be a universal group L
Q
whose
n-dimensional representations parametrize automorphic representations of GL(n),
or rather, the set of families c = c
p
attached to automorphic representations.
Formulated in terms of a later suggestion of Kottwitz, L
Q
would be a locally com-
pact group, equipped with a surjective map L
Q

Q
. It should have many of the
formal properties of the absolute Galois group
Q
. In particular, it should have a
local analogue L
Qv
, for each v, that ts into a commutative diagram
L
Qv

Qv

_

_
L
Q

Q
of continuous homomorphisms. The vertical embedding on the left would be de-
termined only up to conjugacy, and would extend the familiar conjugacy class of
embeddings
Qv

Q
of the local Galois group
Qv
= Gal(Q
v
/Q
v
).
The main property of L
Q
should be its relationship to automorphic representa-
tions. In the special case of GL(n), one would ask for a bijection
r c
between equivalence classes of continuous, completely reducible, n-dimensional rep-
resentations r of L
Q
and equivalence classes of automorphic families c of conjugacy
classes for GL(n). This would be supplemented by local bijections
r
v

v
THE PRINCIPLE OF FUNCTORIALITY 47
between equivalence classes of continuous, completely reducible, n-dimensional rep-
resentations r
v
of L
Qv
and equivalence classes of continuous irreducible represen-
tations
v
of GL(n, Q
v
). The local and global bijections should be compatible in
the natural sense that for any r, there is an automorphic representation with
c() = c, such that for each v, the restriction r
v
of r to L
Qv
corresponds to the
local component
v
of .
The general correspondence would be slightly weaker. If G is quasisplit, one
would expect surjective maps c and
v

v
, the latter at least with nite
bers, in which and
v
are conjugacy classes of homomorphisms L
Q

L
G and
L
Qv

L
G
v
that commute with the maps to the associated Galois groups. If
G is arbitrary, the expectation is similar, except that the local correspondence

v

v
need not be surjective, while the global correspondence c would have
to be restricted to those maps such that each
v
lies in the image of the local
correspondence. In general, the local and global correspondences would again have
to be compatible. For any , there should be an automorphic representation of
G with c() = c, such that for each v, the local component
v
of maps to the
localization
v
of .
Our choice of Q for all this discussion has been purely for simplicity. The discus-
sion is essentially the same if Q is replaced by any global eld F. In particular, there
ought to be a locally compact group L
F
, equipped with an embedding L
Fv
L
F
for each completion F
v
of F. This group should classify automorphic families of
conjugacy classes for any reductive group G over F, in the manner outlined above.
The local Langlands groups are quite elementary. They are given by
L
Fv
=
_
W
Fv
, if v is archimedean,
W
Fv
SU(2, R), if v is nonarchimedean,
where W
Fv
is the Weil group of the local eld F
v
. In general, Weil groups are
locally compact groups attached to local or global elds. They are dened in a
relatively concrete way from the corresponding Galois groups, and t together into
a commutative diagram
W
Fv

Fv

_

_
W
F

F
.
The local Langlands group L
Fv
is thus a (split) extension of W
Fv
by a compact,
simply connected Lie group (either 1 or SU(2, R)). If v is archimedean, Lang-
lands has established the local correspondence
v

v
in general. His proof [L2]
uses fundamental results of Harish-Chandra on harmonic analysis on G(F
v
). For
nonarchimedean v there has been signicant progress, but a proof of the general
local correspondence in this case still seems a long way o.
The global Langlands group L
F
will be much larger. Using the theory of auto-
morphic L-functions, and analytic properties of such functions implied by functo-
riality, one can construct a reasonably explicit candidate for L
F
. It is an innite
ber product of (nonsplit) extensions
1 K
c
L
c
W
F
1
48 JAMES ARTHUR
of W
F
by compact, semisimple, simply connected Lie groups K
c
. However, one
would have to establish something considerably stronger than functoriality in order
to show that L
F
has all the desired properties.
6. The motivic Galois group
It would be appropriate to say something at this point about the conjectural
theory of motives. I do not feel condent in doing so. However, the notion of a
motive is surely one of the great ideas of twentieth century mathematics. It ought
to have at least some mention at this meeting.
The theory is due to Grothendieck. It could be regarded in nave terms as an
attempt to classify the arithmetic data in nonsingular, projective algebraic vari-
eties over Q. In this sense, it is a kind of class eld theory for algebraic varieties
of higher dimension. Varieties, however, turn out not to be the fundamental ob-
jects. Grothendieck discovered a universe within algebraic geometry, so to speak,
whose elementary particles he called irreducible motives. The theory is quite pre-
cise. However, it depends on the so-called standard conjectures, which describe
properties of algebraic cycles and their images in the cohomology of varieties.
Consider the case of class eld theory. At its most elementary level, class eld
theory is an attempt to study solutions of polynomial equations with rational co-
ecients in one variable. Such objects of course dene algebraic varieties over Q
of dimension zero. Galois theory introduces a more intrinsic object, the absolute
Galois group
Q
. It is a pronite group that captures much of the essential in-
formation from the original family of objects. To organize the information into
manageable form, one takes continuous representations
r :
Q
GL(n, C) .
As we have seen, the conjecture of Langlands relates the arithmetic information in
any r with analytic information from some automorphic representation of GL(n).
In general, one can take the larger family of nonsingular, projective algebraic va-
rieties over Q of arbitrary dimension. Grothendiecks conjectural theory of motives
introduces a more intrinsic object, the motivic Galois group G
Q
. In its most down
to earth form, G
Q
is a reductive proalgebraic group over C, which again captures
much of the essential information in the original family of objects. A motive of rank
n is a proalgebraic representation
m: G
Q
GL(n, C) .
It is conjectured that the arithmetic information in any m is again directly related
to analytic information from some automorphic representation of GL(n). The group
G
Q
comes with a proalgebraic projection G
Q

Q
. Any continuous representation
r of
Q
therefore pulls back to a proalgebraic representation G
Q
, and can in this
way be regarded as a motive.
The conjectural theory of motives also applies to any completion Q
v
of Q. It
yields a proalgebraic group G
Qv
over C that ts into a commutative diagram
G
Qv

Qv

_

_
G
Q

Q
THE PRINCIPLE OF FUNCTORIALITY 49
of proalgebraic homomorphisms. By the 1970s, Deligne and Langlands were pre-
pared to conjecture very general relations between motivic Galois groups and the
automorphic representations of any G. Expressed in terms of the automorphic
Langlands groups of 5, they are as follows.
Conjecture (Langlands [L5, 2]). There is a commutative diagram
L
Q

G
Q

Q
together with a compatible commutative diagram
L
Qv
v
G
Qv

Qv
for each completion Q
v
of Q, in which and
v
are continuous homomorphisms.
The conjecture attaches to any proalgebraic homomorphism from G
Q
to
L
G
over
F
, an associated automorphic representation of G. In particular, it implies
that if m is any motive of rank n, there is an automorphic representation of
GL(n) with the following property. The family of conjugacy classes c() = c
p
()
in GL(n, C) associated to is equal to the family of conjugacy classes c(m) =
c
p
(m) obtained from m, and the local homomorphisms G
Qp
G
Q
at primes p
that are unramied for m. The importance of the arithmetic information in c(m)
will unfortunately not be clear from this brief discussion. We can only remark
that c
p
(m) is the image of the Frobenius class Frob
p
under a dierent kind of
representation of
Q
, namely a compatible family

/ S(m){p}
GL(n, Q

)
of -adic representations attached to m. This object comes from the piece of the
cohomology of some variety that is the ultimate origin of m.
Suppose that E is an elliptic curve dened over Q. Then E is a direct sum of a
motive m
E
of rank 2 and two simpler motives of rank 1. The family of conjugacy
classes
c(E) = c(m
E
) = c
p
(m
E
)
has an elementary description in terms of the Tate module for E. The conjecture
above asserts in this case that there is an automorphic representation of GL(2)
such that the family c() = c
p
() equals c(E). This is the celebrated conjecture
of Shimura, Taniyama, and Weil. It has now been proved in complete generality,
in a paper [BCDT] that extends the special case established by Wiles in his proof
of Fermats Last Theorem.
The general discussion of this section makes sense if Q is replaced by any global
eld F. The conjectural theory of motives over F yields a reductive, proalgebraic
group G
F
over C and a map G
F

F
. It is natural to inquire about the struc-
ture of this group [S2]. As in the case of the group L
F
, one ought to be able
to construct a fairly explicit candidate for G
F
, at least up to conjugation by ele-
ments in the connected component G
0
F
. The conjectural properties of automorphic
50 JAMES ARTHUR
L-functions, and the conjectural characterization of those automorphic representa-
tions that come from motives, suggest taking G
F
to be a proalgebraic ber product
of certain extensions
1 D
c
G
c
T
F
1
of a xed group T
F
by complex, semisimple, simply connected algebraic groups D
c
.
The group T
F
is an extension
1 S
F
T
F

F
1
of
F
by a complex, proalgebraic torus S
F
. (See [S1, chapter II], [L5, 5], [S2,
7].) The contribution to G
F
of any G
c
is required to match the contribution to
L
F
of a corresponding group L
c
, in which K
c
is a compact real form of D
c
. This
construction of G
F
, along with that of L
F
at the end of 5, comes with a continuous
homomorphism
L
F
G
F
,
up to conjugation by G
0
F
. However, it would not solve any of the problems. One
would still have to show that G
F
was indeed isomorphic to the motivic Galois group
of Grothendieck.
7. Recent progress
We shall conclude with a very brief description of four recent advances on the
problem of functoriality, and the related questions discussed in 5 and 6. These
results have come in the past couple of years. For the most part, they were quite
unexpected.
(a) Suppose that F
p
is a nite extension of Q
p
, and that G = GL(n). This
is the setting for Langlands nonabelian generalization of local class eld theory.
There are now two separate constructions of the conjectured local correspondence
r
p

p
. One is by M. Harris and R. Taylor [HT], and uses global methods. It
generalizes results by Langlands, Deligne and Carayol in the case n = 2. The other,
by G. Henniart [He], is of a more local character, but still uses global information.
It generalizes results by Kutzko for n = 2. In each case, the authors characterize
the correspondence by showing that r
p
and
p
satisfy certain natural conditions of
compatibility. (See [Ro] and [C].)
(b) Suppose that F is a nite extension of the global function eld F
p
(t), and
that G = GL(n). It is in this setting that L. Laorgue [Laf] has established a
form of the global correspondence r c. The groups L
F
and G
F
are of course
not known to exist. However, in the function eld case, one can replace either
L
F
or G
F
by the Galois group
F
, provided that one takes -adic representations
:
F
GL(n, Q

) (for some prime number ,= p) instead of complex represen-

tations r : L
F
GL(n, C), or proalgebraic representations m: G
F
GL(n, C).
Generalizing results of Drinfeld for n = 2, Laorgue establishes a bijection c
between the set of equivalence classes of continuous -adic representations and
a set of automorphic families of conjugacy classes c whose complement is easy to
characterize. The results include a bijection m c between motives of rank
n over F and this set of automorphic families. They also yield Langlands non-
abelian generalization of global class eld theory for F, namely, the function eld
version of the special case of functoriality that applies to complex n-dimensional
representations of
F
.
THE PRINCIPLE OF FUNCTORIALITY 51
(c) Suppose that F is a number eld, that G

is the split form of the special

orthogonal group SO(2n + 1), and that G = GL(2n). Then

G

is the symplectic
group Sp(2n, C), which comes with a standard representation
:

G

= Sp(2n, C) GL(2n, C) =

G .
J. Cogdell, H. Kim, I. Piatetskii-Shapiro and F. Shahidi [CKPS] have established
the functoriality conjecture in this situation for generic representations

. Generic
representations are expected to comprise a signicant subset of the families of all
cuspidal automorphic representations of SO(2n + 1). The proof is based on the
converse theorems of Cogdell and Piatetskii-Shapiro, which incidentally also play a
role in the results (b) of Laorgue.
(d) Suppose that F is a number eld, that G

= GL(2) GL(3) and that G =

GL(6). H. Kim and F. Shahidi [KS] have established functoriality for the tensor
product representation
:

G

= GL(2, C) GL(3, C) GL(6, C) =

G .
They have also used this result to establish a second new case of functoriality, in
which G

= GL(2), G = GL(4), and

:

G

= GL(2, C) GL(4, C) =

G
is irreducible. In addition, the arguments have recently been extended by Kim
[K] to the case of 5-dimensional representations of GL(2). These results lead to a
signicant improvement of estimates relating to a classical spectral theory problem,
as well as providing new reciprocity laws between automorphic representations.
They yield new bounds for the eigenvalues of both the Laplacian and associated
Hecke operators for arithmetic quotients of the upper half plane.
Postscript
The results we have mentioned, in 6 and earlier, represent a good sample of what
is known. When measured against the general statement of functoriality, they must
seem rather fragmentary. Indeed, they are. However, they have been hard won.
Each has been the end product of much thought, sometimes by a succession of
mathematicians over a period of years. Moreover, future progress is as likely to
build on existing results as it is to subsume them in more general methods. New
ideas will certainly be needed. But to judge from the past, a deciding factor will
be the mathematical power to bring ideas to fruition. In this light, the principle of
functoriality seems all the more remarkable for its ultimate simplicity. Its secrets
will challenge the imagination and energy of mathematicians for a long time to
come.
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representations, Duke Math. J. 109 (2001), 283318.
52 JAMES ARTHUR
[C] H. Carayol, Preuve de la conjecture de Langlands locale pour GLn: Travaux de Harris-
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4
and symmetric fourth power of
GL
2
, preprint.
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2
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3
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2
,
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The references in the article have been updated, but the original text remains unchanged
otherwise.
Given the general nature of this article, I have tried to keep the list of formal references to a
minimum. Elementary introductions to various aspects of the Langlands program can be found
in [Ro] and the following two articles:
J. Arthur, Harmonic analysis and group representations, Notices Amer. Math. Soc.
47 (2000), 2634.
S. Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. Soc.
10 (1984), 177219. MR 85e:11094
Other introductory articles on representation theory and automorphic forms are contained in
the proceedings of the Edinburgh instructional conference:
Representation Theory and Automorphic Forms (T. N. Bailey and A. W. Knapp, eds.),
Proc. Sympos. Pure Math., 61, Amer. Math. Soc., 1996. MR 98i:22001.
THE PRINCIPLE OF FUNCTORIALITY 53
For more advanced articles that are still of a general nature, the reader can turn to the Corvallis
proceedings of [L4] and [L5], and the Washington proceedings of [S2].
Department of Mathematics, University of Toronto, Toronto, M5S 3G3, Canada
E-mail address: arthur@math.toronto.edu