SMR/1852-9

Summer School and Conference on Automorphic Forms and

Shimura Varieties

9 - 27 July 2007

A short introduction to the trace formula

Pierre-Henri Chaudouard
Université de Paris-Sud
Laboratoire de Mathématiques
91405 Orsay Cedex, France
 A short introduction to the trace formula
Pierre-Henri Chaudouard

1 Introduction
The aim of this note is to give a quick introduction to the Arthur-Selberg trace for-
mula through the case of GL(2). First we recall the “cocompact case” which is easy
and serves as a model. Then we introduce Arthur’s truncation operator (section 4). The
“truncated” regular representation of the group of adelic points GL(2, A)1 on the space
L2 (GL(2, Q)\GL(2, A)1 ) is of trace class. Its trace has two expressions : the first one of spec-
tral nature is given by Langlands’ spectral decomposition of the space L2 (GL(2, Q)\GL(2, A)1 )
(cf. section 5) and the second one of geometric nature is related to conjugacy classes in
GL(2, Q) (sections 6, 7 and 8) : this is roughly the trace formula. In section 11, we sketch
an application of the trace formula to the Jacquet-Langlands correspondence. In the final
section, we give some bibliographical references and give the state of the art of the stable
trace formula.
During the text, we often refer the reader to the introductions [21] by Knapp and [19]
by Gelbart to Arthur’s work. We recommend the nice reports [26] and [27] of Labesse and
the synthesis [13] that Arthur made himself of his work.

2 The cocompact case

Let G be a Lie group and a discrete subgroup such that the quotient \G is compact.
Let us consider the space
Z
L ( \G) = { : \G ! C |
2
| (g)|2 dġ < 1},
\G

where dg is a Haar measure on G and dġ is the quotient measure on \G. The group
G acts on L2 ( \G) via the right regular representation R and the algebra Cc1 (G) of
smooth and compactly supported functions on G acts on L2 ( \G) by right convolution :
for 2 L2 ( \G) and f 2 Cc1 (G), the function R(f ) 2 L2 ( \G) is defined by
Z
R(f ) (x) = (xy)f (y) dy,
G

for x 2 \G.

1
 One can see that R(f ) is an integral operator i.e. one can write
Z
R(f ) (x) = K(x, y) (y) dẏ
\G

with the kernel X

1
K(x, y) = Kf (x, y) = f (x y).
2

Since f is compactly supported, it is easy to see that for each (x, y) the sum over is of
finite support and that the kernel K is a continuous function on \G ⇥ \G. Thus K is
square-integrable and R(f ) is a Hilbert-Schmidt operator. Moreover, the Hilbert-Schmidt
norm of R(f ) is nothing else than the L2 -norm of K. Thanks to a theorem of Dixmier-
Malliavin [18], we can write f as a finite sum of convolution products of functions fi and
hi in Cc1 (G) : X
f= fi ⇤ hi
i

As a consequence, R(f ) is a finite sum of products of Hilbert-Schmidt operators and thus

it is a trace class operator. The trace of R(f ) is the finite sum of the Hilbert-Schmidt scalar
products of R(hi ) and R(fi )⇤ . We denote by a ⇤ the adjoint operator. It is easy to see that
R(fi )⇤ = R(fi_ ) where fi_ is defined by fi_ (x) = fi (x 1 ) for any x 2 G. One can show that
it is also the integral of the kernel K over the diagonal :
Z
trace(R(f )) = K(x, x) dẋ.
\G

Then one can expand the kernel K according to the set O of conjugacy classes in . For
2 , let G and be the centralizers of in G and respectively. We get
XZ
trace(R(f )) = f (g 1 g) dġ.
2O \G

Let 2 . Let us choose a Haar measure on G . We denote dḡ the quotient measure on
G \G. Let us introduce the coefficient

a ( ) = vol( \G )

and the orbital integral Z

1
J (f ) = f (g g) dḡ.
G \G

The geometric expansion of the trace formula is given (in the cocompact case) by
X
(2.1) trace(R(f )) = a ( ) J (f ).
2O

For the spectral expansion, we need the following proposition.

2
Proposition 2.1. — The regular representation R is isomorphic to the completion of a
discrete sum of unitary irreducible representations of G
M
\
(2.2) L2 ( \G) = V⇡m (⇡)

⇡2⇧(G)

where ⇧(G) is the set of equivalence classes of unitary irreducible representations (⇡, V⇡ )
of G. Moreover, the multiplicity m (⇡) of ⇡ in L2 ( \G) is finite.
Proof. — (Sketch) It relies on some elementary facts about the spectral theory of com-
pact adjoint operators in Hilbert spaces : a non-zero compact adjoint operator in a Hilbert
space has always a non-zero eigenvalue. Moreover, the associated eigenvector space is finite
dimensional. To prove the existence of the discrete decomposition, it suffices to show that
any closed and G-invariant subspace V admits an irreducible closed and G-invariant sub-
space (the irreducibility is taken in the topological sense). One can approximate the Dirac
measure supported in {1} by a suitable f 2 Cc1 (G) such that f _ = f . In particular, R(f )
induces on V a compact (since Hilbert-Schmidt) self-adjoint (since f _ = f ) and non-zero
(provided that f is enough close to the Dirac measure). So we have an eigenvector, say
2 V , associated to a non-zero eigenvalue of R(f ). Then we can replace V by V 0 the clo-
sure of the space generated (under G) by . Now, by using orthogonal projectors, one can
see that any orthogonal decomposition of V 0 has at most d non-zero summands where d is
the (finite) dimension of the eigenvector space which contains . Thus, any component of
a maximal orthogonal decomposition of V 0 is irreducible. In the same manner, one proves
that the multiplicities are finite (for more details, see e.g. the proof of theorem 1.5 of [21]).
⇤
From the proposition, we get the spectral expansion of the trace formula
X
trace(R(f )) = m (⇡) trace(⇡(f )).
⇡2⇧(G)

In conclusion, the trace formula in the cocompact case is the identity for all f 2 Cc1 (G)
X X
a ( ) J (f ) = m (⇡) trace(⇡(f )).
2O ⇡2⇧(G)

It is a generalization for non-abelian groups of the Poisson summation formula (take

G = R and = Z). A fundamental problem is to obtain some informations about the
multiplicities m (⇡). The trace formula is one of the most powerful tool to study them :
the trace formula “converts” spectral data into (more concrete) geometric data.

3 The GL(2) case

In the previous section, we studied the cocompact case but this setting is too restrictive :
for example, if G = GL(2, R) and is GL(2, Z) or more generally a congruence subgroup

3
the quotient \G is not compact. From now on, we will work in an adélic setting : on the
one hand, the literature on the Arthur-Selberg trace formula is written for adélic groups
and on the other hand it is more convenient to consider GL(2, Q)-conjugacy classes rather
than GL(2, Z)-conjugacy classes.
Let A be the ring of adèles of Q (more generally of a number field), A⇥ the multiplicative
group of idèles and A1 the subgroup of idèles of adélic norm 1

A1 = {x 2 A | |x|A = 1}.

These groups carry a natural topology for which they are locally compact. Moreover, Q,
resp. Q⇥ , is a discrete subgroup of A, resp. A1 , and the quotient Q\A, resp. Q⇥ \A1 , is
compact.
Let us denote G = GL(2, A) and = GL(2, Q). The topology on A induces a topology
on G for which G is locally compact. Thus G carry a Haar measure dg. Moreover is a
discrete subgroup of G but the quotient \G is not compact : it is not even of finite volume
(for the quotient measure). One can improve this by considering the subgroup G1 defined
by
G1 = {g 2 G | | det(g)|A = 1}.
It is well-known that \G1 is of finite volume (cf. [16]) : nevertheless it is still not compact.
Let K0 be an open compact subgroup of the group GL(2, Af ) of finite adèles. The
quotient K0 \G1 /K0 is a discrete union of countably many copies of GL(2, R)1 = GL(2, R)\
G1 which is a Lie group. We can therefore define the space C 1 (K0 \G1 /K0 ) of smooth
functions on K0 \G1 /K0 . Let C 1 (G1 ) be the direct limit

C 1 (G1 ) = lim C 1 (K0 \G1 /K0 )

!

where the limit is taken over all open compact subgroups K0 of GL(2, Af ). Let Cc1 (G1 )
be the subspace of smooth and compactly supported functions on G1 . As in the previous
section, the group G1 acts by the right regular representation R on the space L2 ( \G1 )
and for f 2 Cc1 (G1 ), the operator R(f ) is an integral operator with kernel
X
K(x, y) = f (x 1 y)
2

with x, y 2 \G1 . Unlike the cocompact case, the kernel K is not square-integrable on
\G1 ⇥ \G1 . It is not even integrable on the diagonal. Let 2 . Let G1 be the centralizer
of in G1 . It is easy to see that the coefficient

a ( ) = vol( \G1 )

is infinite for diagonal and non central and that the orbital integral
Z
J (f ) = f (g 1 g) dḡ
G1 \G1

4
is generally infinite for a non-trivial unipotent element. So the geometric expansion (2.1)
does not have any sense. What is worse is that the discrete spectral decomposition (2.2)
is no longer true and R(f ) is not a trace class operator. At first glance, there seems to
be no hope to have a trace. But, as we shall see, what can be computed is the trace of a
truncated avatar of R(f ) which still has a rich spectral decomposition.

4 Arthur’s truncation operator

4.1. Constant term. — Let B be the subgroup of upper triangular matrices in G. Let
N be the subgroup of unipotent matrices in B. The quotient N \ \N is isomorphic to
Q\A and therefore is compact.
For any 2 L2 ( \G1 ). By Fubini’s theorem, the function

n 2 N \ \N 7! (nx)

belongs to L2 (N \ \N ) for almost all x 2 G1 and thus also to L1 (N \ \N ).

The constant term of (“along” B) is the function B defined for almost all g 2 G1 by
Z
B (x) = (nx) dn.
N \ \N

We shall say that is cuspidal if B = 0 (almost everywhere). Let

L2cusp ( \G1 ) = { 2 L2 ( \G1 ) | B = 0 (a.e.)}

be the closed subspace of L2 ( \G1 ) generated by cuspidal functions. Note that this sub-
space is stable by R(f ).
4.2. The H-function. — Let T be the subgroup of diagonal matrices in B. Let
Y
⌦ = O2 (R) GL(2, Zp ).
p prime

This is a maximal compact subgroup. Then we have the Iwasawa decomposition

G = BK = N T ⌦.

We define a function H from T to R by

✓ ◆
a 0 1 a
(4.1) H( ) = log(| |A ),
0 b 2 b

for all a, b 2 A⇥ . For any g 2 G, the Iwasawa decomposition enables us to write g = ntk
with n 2 N , t 2 T and k 2 ⌦. We extend the function H to all G by the formula

(4.2) H(g) = H(t).

5
We thus obtain a function on G which is invariant on the left by N (T \ ) and on the right
by ⌦.
Let ✓ ◆
0 1
w=
1 0
be (a representative) of the non-trivial element of the Weyl group of the diagonal torus.
We have the following “positivity lemma”.
Lemma 4.1. — For any g 2 G, the inequality

H(g) H(wg) > 0

is true.
Proof. — It is a simple (local) computation which is left to the reader. (cf. e.g. the
computation of the function H in lemma 6.6 of [21]). ⇤

4.3. Reduction theory. — For any X 2 R, let ⌧X be the characteristic function of the
set

(4.3) {g 2 G | H(g) > X}.

Let T1 be the group of diagonal matrices (a, b) with a, b 2 R⇥ 1

+ . Let T be the subgroup
of T of diagonal matrices (a, b) with a and b in A . In particular, T = T 1 T1 .
1

Let ! ⇢ N T 1 be a compact subset. Let X1 2 R. We define the Siegel set

S = S(X1 , !) = {x = ptk 2 G | p 2 !, t 2 T1 , k 2 ⌦, , ⌧X1 (x) = 1}.

The relevance of Siegel sets comes in part from the following lemma. (cf. e.g. proposition
7.10 of [21])
Lemma 4.2. — If S is sufficiently large then G = S.
Let ! and X1 < 0 be such that the conclusion of the previous lemma is true. Let X 2 R
and
S(X, X1 , !) = {x 2 S(X1 , !) |⌧X (x) = 0}.
It is clear that S(X, X1 , !) \ G1 is compact. Let F G (., X) be the characteristic function of
the projection of S(X, X1 , !) on \G.
Lemma 4.3. — For any g 2 G, the sum
X
⌧X ( g)
2B\ \

is finite. If X > |X1 | then for all g 2 G we have

X
F G (g, X) + ⌧X ( g) = 1.
2B\ \

6
Proof. — We will not prove the first assertion (however cf. e.g. lemma 7.7 in [21]).
The Bruhat decomposition gives the disjoint union

= (B \ ) [ (B \ )w(B \ ).

Thus the union of {1} and w(N \ ) is a set of representatives of the quotient B \ \ .
Since the second assertion only depends on the class g, we have to consider only two
cases : either ⌧X (g) = 1 or ⌧X ( g) = 0 for all 2 . In the first case, by the positivity
lemma 4.1, we have for any n 2 N

H(wng) 6 H(ng) = H(g) 6 X 6 X.

Thus ⌧X ( g) = 0 if 2/ B \ . Moreover it is clear that F G (g, X) = 0.

In the second case, by lemma 4.2, there exists 2 such that g 2 S. Since ⌧X ( g) = 0
we have F G (g, X) = 1.
⇤

4.4. Arthur’s truncation operator — It is denoted by ⇤X and it depends on a real

parameter X. It is defined for any 2 L2 ( \G1 ) or 2 C 1 ( \G1 ) by the formula
X
⇤X (x) = (x) ⌧X ( x) B ( x)
2B\ \

for x 2 G1 .
Remark. — Thanks to lemma 4.3, we see that the sum over can be taken over a fi-
nite set and that the constant function 1 satisfies ⇤X (1) = F G (., X) if X is sufficiently large.

Here are the main properties of the truncation operator. (for some details see e.g. [21]
§7, for more complete proofs see [2] and perhaps also [32]).
Proposition 4.4. — For any 2 L2 ( \G1 ), we have ⇤X 2 L2 ( \G1 ). Moreover, the
following assertions are true :
1. ⇤X = for all 2 L2cusp ( \G1 ) ;
2. ⇤X = ⇤X ⇤X ;
3. ⇤X transforms smooth functions of “uniform moderate growth” into “rapidly decrea-
sing” ones ;
4. h⇤X , i = h , ⇤X i for and 2 Cc1 (G1 ) ;
5. ⇤X extends to a bounded operator on L2 ( \G1 ) which a orthogonal projector.

Let us denote ⇤X K, resp. K⇤X , the action of ⇤X on the first variable of the kernel
K, resp. the second variable. It is easy to check that the operator ⇤X R(f ) acting on
L2 ( \G1 ) is integrable with kernel ⇤X K. But thanks to the property 3 of the proposition

7
above, one can show that the kernel ⇤X K is square-integrable on \G1 ⇥ \G1 (see for
example the upper bounds for ⇤X K in the proof of proposition 3.5 of [19]). We deduce the
following theorem.
Theorem 4.5. — For any f 2 Cc1 (G1 ), the operator ⇤X R(f ), acting on L2 ( \G1 ) is a
Hilbert-Schmidt operator.
That theorem has the same consequences as in the cocompact case. Before stating them,
let us introduce the discrete part of L2 ( \G1 ) that is the closed subspace L2disc ( \G1 )
generated by the irreducible subrepresentations of G1 in L2 ( \G1 ). This is the part of
L2 ( \G1 ) whose spectral decomposition looks like the decomposition (2.2).
Corollary 4.6. — We have the inclusion
L2cusp ( \G1 ) ⇢ L2disc ( \G1 ).
More precisely, the space L2cusp ( \G1 ) is the Hilbert sum of unitary irreducible representa-
tions of G1 with finite multiplicities.
Proof. — (Sketch) Let Rcusp be the regular representation of G1 on L2cusp ( \G1 ). By
property 1 of the proposition 4.4, the operator Rcusp is the restriction of the Hilbert-
Schmidt operator ⇤X R(f ) to the closed subspace L2cusp ( \G1 ). It is also a Hilbert-Schmidt
operator. Then the proof is the same as in the cocompact case. (cf. the proof of proposition
2.1 and for more details, see e.g. the proof of theorem 1.5 of [21]). ⇤
Remark. — The irreducible components (and their multiplicities) of L2cusp ( \G1 ) are
in the heart of the Langlands’ program. In our case and more generally for GL(n) the
multiplicities are 0 or 1 (cf. [34]).
Corollary 4.7. — The operator ⇤X R(f )⇤X is an integral operator with kernel ⇤X K⇤X .
Moreover it is of trace class and its trace is the integral over the diagonal over its kernel
Z
X X
(4.4) trace(⇤ R(f )⇤ ) = ⇤X K⇤X (x, x) dx.
\G1

Proof. — The first assertion is easy. For the second, the Dixmier-Malliavin’s factorization
theorem (cf. section 1) enables us to write ⇤X R(f )⇤X as a sum of products of the type
⇤X R(g)R(h)⇤X with g and h in Cc1 (G1 ). We have just to remark that the adjoint operator
of R(h)⇤X is ⇤X R(h_ ) with h_ (x) = h(x 1 ) for x 2 G1 . Thus ⇤X R(f )⇤X is a sum of
products of Hilbert-Schmidt operators. Hence it is of trace class and its trace is the sum of
the Hilbert-Schmidt scalar products of the operators R(h)⇤X and (⇤X R(g))⇤ which gives
the last statement by an easy computation. ⇤
We have dramatically improve the situation : we have now a trace class operator
⇤X R(f )⇤X . Its trace is the integral of its kernel ⇤X K⇤X over the diagonal. Moreover,
this trace “includes” the trace of the operator Rcusp (f ) which carries interesting arithme-
tic informations. As in the cocompact case, one can hope to get eventually a geometric
decomposition of the integral of ⇤X K⇤X . But we still need a full spectral decomposition
of L2 ( \G1 ) that we shall now review.

8
5 Langlands’ spectral decomposition of L2( \G1)
Let A1 be the group of diagonal matrices (a, a 1 ) with a 2 R⇥ + . Let us recall that T
1

is the subgroup of T of diagonal matrices (a, b) with a and b in A .

1

The space H of measurable functions

: N (T \ )A1 \G1 ! C

such that for any x 2 G1 the function

t 2 T 1 7! (tx)

belongs to L2 ((T \ )\T 1 ) and such that

Z Z
2
k k = | (tk)|2 dtdk < 1
K (T \ )\T 1

is a Hilbert space.
For any s 2 C, we define an induced action Rs of G1 on H by the formula

Rs (y) (x) = exp( (s + 1)H(x)) s (xy)

where
s (x) = (x) exp((s + 1)(H(x))
for any x, y 2 G1 . The function H has been defined in the previous section. This is a
unitary representation of G1 if s 2 iR.
For any 2 H, x 2 G1 and s in a suitable subset of C, we define the Eisenstein series
X
E(x, , s) = s ( x)
2B\ \

and the intertwining operator

Z
(M (s) )(x) = exp((s 1)H(x)) s (wnx) dn.
N

Note that formally

E(xy, , s) = E(x, Rs (y) , s)
and
M (s)Rs (y) = R s (y)M (s).

9
Theorem 5.1. — (Langlands’ decomposition)
1. The series and the integral that define respectively E(x, , s) and M (s) are convergent
and analytic for Re(s) > 1. Moreover, they admit a meromorphic continuation to the
whole complex plane, which is analytic on iR. They satisfy the functional equations
E(x, M (s) , s) = E(x, , s)
and
M ( s)M (s) = 1.
2. Let F the Hilbert space of functions
F : iR ! H
which satisfy
F ( s) = M (s)F (s)
and Z
1
2
kF k = kF (s)k2 ds < 1
2 iR
The map defined over a dense subspace of F by
Z
1
F 7! E(x, F (s), s) ds
2 iR
extends to an isometry from F onto the orthogonal complement denoted L2cont ( \G1 )
of L2disc ( \G1 ) in L2 ( \G1 ).
Recall that L2cusp ( \G1 ) ⇢ L2disc ( \G1 ). Let L2res ( \G1 ) be the orthogonal complement
of L2cusp ( \G1 ) in L2disc ( \G1 ). This is the so-called residual spectrum (because it is even-
tually obtained from residues of Eisenstein series). In fact, in our situation, the residual
spectrum admits a very explicit description
M
L2res ( \G1 ) = det

where belongs to the group of continuous characters of Q\A1 .

We have an orthogonal decomposition
L2 ( \G1 ) = L2cusp ( \G1 ) L2res ( \G1 ) L2cont ( \G1 ).
Let f 2 Cc1 (G1 ). Each term in the above decomposition is stable by the operator R(f ).
By restriction, we obtain integrable operators Rcusp (f ), Rres (f ), and Rcont (f ) with kernels
respectively denoted by Kcusp , Kres , and Kcont .
Let B be an orthonormal basis of H. The Langlands’ decomposition gives the following
expression Z X
1
Kcont (x, y) = E(x, Rs (f ) , s)E(y, , s) ds
2 iR 2B

10
for any x, y 2 G1 .
We have already shown that ⇤X R(f )⇤X is of trace class (cf. corollary 4.7). Its kernel
⇤X K⇤X satisfies

⇤X K⇤X (x, y) = Kcusp (x, y) + ⇤X Kres ⇤X (x, y)

Z X
1
+ ⇤X E(x, Rs (f ) , s)⇤X E(y, , s) ds.
2 iR 2B
The trace of ⇤X R(f )⇤X is the integral of its kernel over the diagonal hence
trace(⇤X R(f )⇤X ) = trace(Rcusp (f )) + trace(⇤X Rres (f )⇤X )
Z XZ
1
+ ⇤X E(x, Rs (f ) , s)⇤X E(x, , s) dx ds.
2 iR 2B \G1
Note that the inner integral is nothing else than the scalar product of two trunca-
ted Eisenstein series, which is not difficult to compute (cf. [21] proposition 7.13). After
integration we obtain :
Z XZ
1
⇤X E(x, Rs (f ) , s)⇤X E(x, , s) dx ds
2 iR 2B G(Q)\G(A)1
Z Z
1 1 0
= trace(M (s) M (s)Rs (f )) ds + X trace(Rs (f ))) ds
2 iR iR
1
+ trace(M (0)R0 (f )) + r(X)
4
where the last term r(X) vanishes at infinity
lim r(X) = 0.
X!+1

Finally, we also have

lim trace(⇤X Rres (f )⇤X ) = trace(Rres (f ))

X!+1
In conclusion, we obtain the theorem.
Theorem 5.2. —For any f 2 Cc1 (G1 ),
1
trace(⇤X R(f )⇤X ) = trace(Rdisc (f )) + trace(M (0)R0 (f ))
4
Z Z
1
+ trace(M (s) 1 M 0 (s)Rs (f )) ds + X trace(Rs (f ))) ds + "(X)
2 iR iR
with
lim "(X) = 0.
X!+1

Remarks. —

11
 – Besides the trace of R(f ) in the discrete spectrum denoted by trace(Rdisc (f )), we have
an other discrete contribution namely trace(M (0)R0 (f )) which comes from Eisenstein
series. R
– The continuous contribution iR trace(M (s) 1 M 0 (s)Rs (f )) ds is built upon the so-
called weighted characters trace(M (s) 1 M 0 (s)Rs (f )) which are not invariant by conju-
gacy by G.
– the coefficient of X is an invariant distribution : it is the integral of the trace of
representations induced from T .

6 The geometric side : the cut-o↵ kernel

As in the cocompact case, we can expect the trace of the operator ⇤X R(f )⇤X
Z
X X
trace(⇤ R⇤ (f )) = ⇤X K⇤X (x, x) dx
\G1

to have a geometric decomposition according to conjugacy classes in . In this section, it

is our task to provide such a decomposition. Unfortunately, the fact that the truncation
operator ⇤X does not act diagonally on the kernel K makes our task harder.
According to Arthur, it is more convenient to introduce before another truncated kernel
whose geometric decomposition is more tractable.
Let f 2 Cc1 (G1 ). Let RB the right regular action of G1 on
L2 (N (T \ )\G1 )
As before, one defines a convolution operator RB (f ) which is an integral operator which a
kernel denoted KB . We have the following explicit expression for this kernel
Z X
KB (x, y) = f (x 1 ny) dn.
N 2T \

for x and y 2 G1 . Then we can introduce what we call the “cut-o↵” kernel (to distinguish
it from the truncated kernel ⇤X K)
X
k X (x, f ) = K(x, x) ⌧X ( x)KB ( x, x).
2B\ \

The function ⌧ was defined in (4.3). Again, by lemma 4.3, the sum over can be taken
over a finite set.

7 Characteristic polynomials
Instead of conjugacy classes, we are going to expand the cut-o↵ kernel according to
characteristic polynomials. Let X be the set of characteristic polynomials of elements of .
For any 2 , we denote by the characteristic polynomial of .

12
 For any 2 X, x, y 2 G1 , we set
X
1
K (x, y) = f (x y)
{ 2 | = }

and Z X
1
KB, (x, y) = f (x ny) dn.
N { 2T \ | = }

Thus we have the expansion

X
k X (x, f ) = k X (x, f ).
2X

We can now state the first theorem.

Theorem 7.1. — For large X, we have
XZ
|k X (x, f )| dx < 1.
2X \G1

(For a sketch of the proof see [19] lecture II, theorem 4.3)
Thanks to the previous theorem, we can set, for all 2 X, and for large X
Z
X
J (f ) = k X (x, f ) dx.
\G1

For applications, it is important to find more explicit expressions for the distributions
X
J (f ) and to understand their behaviour in X. Before considering this problem, we need a
coarse classification of characteristic polynomials. We introduce the subsets Xell of elliptic
characteristic polynomials, Xpar of parabolic ones and Xsing of singular ones. By definition,
we have
1. Xell is the subset of irreducible polynomials (over Q) in X ;
2. Xpar is the subset of polynomials in X which have two distincts roots in Q ;
3. Xsing is the subset of polynomials in X which have one double root in Q.
So we have a disjoint union
X = Xell [ Xpar [ Xsing .
Let , 0 be elements of . Assume that belongs to Xell [ Xpar . Then is semi-simple
and regular. Here regular semi-simple just means diagonalizable over an algebraic closure
of Q with two distinct eigenvalues. Moreover = 0 if and only if and 0 are conjugate
in . In that sense, we can identify the set Xell [ Xpar with the set of conjugacy classes of
semi-simple regular elements in .
On the other hand, belongs to Xsing if and only the semi-simple part of in the
Jordan decomposition is central (in other terms a scalar matrix). Moreover = 0 if and
only if and 0 have the same semi-simple part.

13
 We will now describe the distributions

J X (f )

according to those subsets of X.

8 Weighted orbital integrals

8.1. Elliptic polynomials — Let 2 such that = belongs to Xell . In that case,
we have obviously 62 T \ . Thus the -part of the cut-o↵ kernel reduces to the -part
of the kernel :
k X (x, f ) = K (x, x)
does not depend on X. As a consequence, for the elliptic part of the cut-o↵ kernel, we can
perform the same calculations as in the cocompact case. Let us state the result.
Proposition 8.1. — For any 2 such that = belongs to Xell , we have
Z
J X (f ) = vol(G1 \ \G1 ) f (g 1
g) dḡ
G \G

where G , resp. G1 , is the centralizer of in G, resp. G1 .

Remark. — The centralizer of in GL(2) is a torus. The quotient G1 \ \G1 is in fact

compact. So its volume is indeed finite.
8.2.
R Parabolic polynomials — For elliptic polynomials, we have got orbital integrals
G \G
f (g 1 g) dḡ as in the cocompact case. For parabolic polynomials, new distributions
appear : the so-called weighted orbital integrals. First, we need to define the weight v. For
any g 2 G, we set
v(g) = H(g) H(wg),
where H is the function defined in (4.1) and (4.2). It is an amusing exercise to check that
this defines a non-negative function on G which is invariant on the left by T and on the
right by K.
We can now state the result.
Proposition 8.2. —For any non-scalar 2 T \ and = , we have
Z Z
X 1 1 1
J (f ) = vol(T \ \T ) ⇥ 2X f (g g) dḡ + f (g g)v(g) dḡ
T \G T \G

where T 1 is the group of diagonal matrices (a, b) with a, b 2 A1 .

This proposition is quoted in [19] (cf. lecture IV proposition 1.1) where the reader will
find some references.

14
Remarks. —
– The quotient T \ \T 1 is compact hence of finite volume.

– For any 2 Xpar , one can find 2 T \ such that = . Of course, is not
central. Thus we have a description of J X (f ) for any 2 Xpar . R
– The group T is really the centralizer of in G. So the integral T \G f (g 1 g) dḡ is
nothing else than the orbital integral associated to .
R
Because of the weight v, the so-called weighted orbital integral T (A)\G(A) f (g 1 g)v(g) dḡ
is a distribution that is not invariant by conjugacy by G.
8.3. Singular polynomials — There exists a similar but more complicated expression
for singular characteristic polynomials. We will not give it (however cf. [19] lecture IV
proposition 1.2). The reasons for this complication are easy to track : first a singular
characteristic polynomial cannot be identified to a single conjugacy class. Second, the global
unipotent integrals do not converge (we already mentionned this fact when we introduced
the truncation).
Let us emphasize that J X (f ) for singular is still a polynomial in X of degree at most
1. Moreover, the distributions J X for singular are in some sense in the closure of the
distributions J X for elliptic or parabolic.

9 The trace formula

So far, we have introduced the cut-o↵ kernel and we have got a geometric decomposition
of its integral in terms of (weighted) orbital integral. The following theorem will enable us
to compare the geometric decomposition of the cut-o↵ kernel with the former truncated
kernel.
Theorem 9.1. — Let f 2 Cc1 (G1 ). There exists Xf such that for any X > Xf we have
Z Z
X
⇤ K(x, x) = k X (x, f ) dx.
\G1 \G1

(for a sketch of the proof, see for example [19] p.26)

In theorem 5.2, we saw that the left-hand side is a polynomial in X up to a negligible
error term. In section 7 and 8, we saw that the right-hand side is exactly a polynomial in
X. Equaling the constant terms (at X = 0), we obtain the following corollary.
Corollary 9.2. —(Arthur’s trace formula for GL(2)) Let f 2 Cc1 (G1 ). We have the
following equality X
Jdisc (f ) + Jcont (f ) = J (f )
2X

where

15
 – the discrete part of the spectral side of the trace formula is
1
Jdisc (f ) = trace(Rdisc (f )) + trace(M (0)R0 (f )) ;
4
– the continuous part of the spectral side of the trace formula is
Z
1
X
Jcont (f ) = trace(M (s) 1 M 0 (s)Rs (f ))ds ;
2 iR
– for 2 X, the geometric distribution J (f ) is the value at X = 0 of J X (f ). Thus,
J (f ) is an orbital integral for 2 Xell and a weighted orbital integral for 2 Xpar .

Remark. — As we stated it, the trace formula we obtained does not compute the trace
of anything. Nonetheless, it gives a formula for the trace of R(f ) in the discrete spectrum
in terms of both geometric and spectral data.

10 A simple trace formula

For some more restricted classes of functions, the trace formula is simpler (cf. [19]
Lecture V, §2). Here is an useful example. Let V be the set of all places of Q. For v 2 V ,
let Qv be the completion of Q at v. Let Gv = GL(2, Qv ) and Tv the group of diagonal
matrices in Gv . By identifying Gv with a subgroup of G in the usual manner, we define the
weight function v on Gv . We have the following splitting formula due to Arthur. (see e.g.
proposition 1.1 p.46 of [19]). (of course, we tacitly assume some compatibility conditions
among the various Haar measures).
Proposition 10.1. — Let 2 T \ and f = ⌦v2V fv . The the global weighted orbital
integral Z
1
f (g g)v(g) dḡ
T \G

is equal to the sum over v 2 V of the product

Z YZ
1
fv (g xg)v(g) dḡ ⇥ fw (g 1 xg) dḡ.
Tv \Gv w6=v Tw \Gw

Here almost all the factors of the product are in fact 1. Moreover for almost all v, the local
weighted orbital integral Z
fv (g 1 xg)v(g) dḡ
Tv \Gv

is 0 and the product above is 0.

Let us fix two distinct places v1 and v2 and let us consider functions f = ⌦v2V fv such
that for v 2 {v1 , v2 } the local orbital integrals of fv vanish for all x 2 Tv :
Z
(10.1) fv (g 1 xg) dḡ = 0.
Tv \Gv

16
The proposition 10.1 implies the parabolic part of the geometric side of the trace formula
vanishes for such functions f . Besides one can show that the singular part reduces to a sum
over the set Z of central elements in (cf. [19] lecture V, §2). More precisely, we obtain
for the geometric side the expression
X X
J (f ) + vol( \G1 ) f (z).
2Xell z2Z

The vanishing asumption (10.1) implies that the trace of fv in the induced representations
is 0 for v 2 {v1 , v2 }. But we have also splitting formulae for the spectral terms (see e.g.
proposition 1.3 p.48 of [19]). As consequence we can also simplify the spectral side of the
trace formula. Finally, the trace formula reduces for such functions f to the equality
X X
(10.2) trace(Rdisc (f )) = J (f ) + vol( \G1 ) f (z).
2Xell z2Z

11 Application : the Jacquet-Langlands correspondence

One of the most striking applications of the trace formula is to obtain some special cases
of what is known as Langlands’ functoriality. Such applications are based on a comparison
of di↵erent trace formulae for di↵erent groups. Let us discuss a typical case : the Jacquet-
Langlands correspondence. Let G0 = D⇥ be the group of invertible elements of a quaternion
algebra D over Q. We will use a 0 to denote objects relative to G0 . Let G0 (A)1 be the
subgroup of G0 (A) of elements g 2 G0 (A) such that |Nrd(g)|A = 1 where Nrd is the reduced
norm. Then the quotient G0 (Q)\G0 (A)1 is compact. Therefore using the same method as
in the section 2, we can prove that for any test function f 0 on G0 (A)1
X
(11.1) trace(R0 (f 0 )) = J 0 (f 0 )

where as usual R0 is the regular representation of G0 (A)1 on L2 (G0 (Q)\G0 (A)1 ). On the
right-hand side, there is a sum indexed by conjugacy classes in G0 (Q) of J 0 (f 0 ) which is
a global orbital integral times a volume. We can still classify the elements of G0 (Q) accor-
ding to their characteristic polynomials. Moreover since there are no non-trivial unipotent
element in G0 (Q), two elements in G0 (Q) which have the same characteristic polynomial are
in fact conjugate. Moreover there are no element in G0 (Q) which has a parabolic characte-
ristic polynomial. The characteristic polynomial map induces a bijection between the set
of conjugacy classes in G0 (Q) and the set of elliptic or singular characteristic polynomials.
In this way, the formula (11.1) can be written
X X
(11.2) trace(R0 (f 0 )) = J 0 (f ) + vol(G0 (Q)\G0 (A)1 ) f 0 (z),
2Xell z2Z 0 (Q)

where Z 0 is the center of G0 wich we identified to the center Z of GL(2, Q). Of course,
Z 0 ' Z. This formula looks like the simple trace formula (10.2) we obtained for GL(2).

17
 We would like to compare the geometric parts of these formulae of G and G0 for enough
pairs of test functions (f 0 , f ). We normalize the Haar measures on G and G0 (A) by taking
the Tamagawa measures (cf. [19] p.53). These measures decompose locally in a product
of Haar measures. Let f 0 be a smooth compactly supported function on G0 (A)1 such that
f 0 = ⌦v2V fv0 . For v 2 V , we denote G0v the group G0 (Qv ). Let S be the finite set of places
v where Gv and G0v are not isomorphic (note that |S| > 2). Thus for the places v 2 / S,
we may and we will take fv = fv0 . For the places v 2 S, we will use a function fv which
satisfies the following proposition (“existence of the transfer”).
Proposition 11.1. — Let v 2 V . For any test function fv0 on G0v there exists a test function
fv such that
1. for any 2 Tv the orbital integral of fv vanishes
Z
fv (g 1 g) dḡ = 0 ;
Tv \Gv

0
2. for any 2 Gv and 2 G0v with matching characteristic polynomials we have
Z Z
fv (g 1
g)dḡ = fv0 (g 1 0 g)dḡ
G ,v \Gv G0 0 ,v \G0v

(when and 0 match, their centralizers G ,v and G0 0 ,v are isomorphic which enables
us to take compatible Haar measures on G ,v and G0 0 ,v .)

The function f = ⌦v2V fv is smooth and compactly supported. Using the simple form
(10.2) of the trace formula for G and f and the formula (11.2) above, we write
X X
trace(Rdisc (f )) trace(R0 (f 0 )) = vol( \G1 ) f (z) vol(G0 (Q)\G0 (A)1 )) f 0 (z).
z2Z z2Z 0 (Q)

With the identification Z ' Z 0 (Q), it is possible to show that we have for the pair (f, f 0 ) the
equality : f (z) = f 0 (z) for all z 2 Z (this equality is true locally only up to a Kottwitz sign
which disappears globally). Thanks to Weyl’s integration formula, it is easy to show that the
contributions of the 1-dimensional subrepresentations to trace(Rdisc (f )) and trace(R0 (f 0 ))
match. Let us denote R00 the regular representation of G0 (A)1 on L20 (G0 (Q)\G0 (A)1 ) the
subspace of L2 (G0 (Q)\G0 (A)1 ) orthogonal to all the invariant 1-dimensional subspaces. We
obtain
X
trace(Rcusp (f )) trace(R00 (f 0 )) = (vol( \G1 ) vol(G0 (Q)\G0 (A)1 )) f (z).
z2Z

But this equality is possible only if both sides are zero. Let us explain the rough idea
which comes from functional analysis : take v 2/ S an “unramified” place. Then the formula
above is an equality of distributions for fv in the Hecke algebra. But the left-hand side
is a discrete sum of unramified unitary representations whereas the right-hand side is a

18
continuous integral of tempered representations against the Plancherel measure (thanks to
the Plancherel formula). By Riesz uniqueness theorem, both distributions are zero. As a
consequence, we have

(11.3) vol( \G1 ) = vol(G0 (Q)\G0 (A)1 )

and

(11.4) trace(Rcusp (f )) = trace(R00 (f 0 )).

From the last equality, we can eventually prove that there is a natural injective map
from the irreducible constituents of L20 (G0 (Q)\G0 (A)1 ) to the irreducible constituents of
L2cusp (G0 (Q)\G0 (A)1 ) such that for v 2
/ S the local components at v match. Moreover it
is possible to characterize the image of the map (cf. [19] lecture VI). This is the global
Jacquet-Langlands correspondence.

12 Final comments
Arthur has developped the trace formula for any reductive connected group (over a
number fields) for more than twenty-five years. Without being exhaustive, we can mention
several important steps. The reader can find the “coarse form” of the trace formula in the
papers [1] for the geometric side and [2] for the spectral side (which relies on the Langlands’
spectral decomposition cf. [29] or [32] for an adelic setting). The papers [9, 6, 5] are devoted
to express the geometric distribution in terms of (local) weighted orbital integral. The
papers [3, 4] do the same job for the spectral distributions and the weighted characters.
In the papers [7, 8], Arthur deduces from the non-invariant trace formula an invariant
one. As a byproduct, he establishes a very useful “simple” trace formula (in the spirit of
the section 10). In [14], Arthur and Clozel prove the general comparison between the trace
formula for GL(n) and one of its inner form [14]. Using their result, Badulescu recently
succeeded in proving a general global Jacquet-Langlands correspondence (cf. [15]).
When one tries to generalize the Jacquet-Langlands correspondence to other reductive
groups, one has to face a difficulty. For any reductive group one can still define a notion of
characteristic polynomial. Moreover, two groups which are inner forms of each other share
the same set of characteristic polynomials. But the fibers of the characteristic polynomial
map are usually not ordinary conjugacy classes but rather stable conjugacy classes (roughly
speaking the stable conjugacy is the conjugacy over an algebraic closure of the base field).
The two notions of conjugacy are the same for GL(n) but in general they do not coincide.
So, if we want to compare the geometric sides of the trace formulae for two groups which
are inner forms of each other, we have before to express these geometric sides in terms of
stable conjugacy classes. As we saw it for GL(2) in section 11, the comparison is based on a
transfer of orbital integrals. In the most general situation, the comparison should be based
on a transfer of stable orbital integrals. Therefore we have to express the trace formula on a
group in terms of stable distributions (those which are in the weak closure of stable orbital

19
integrals). At first glance, it is not possible. In fact, what should be possible to do is to write
the trace formula for a group G in terms of a stable part and an instable part which is a
sum of stable trace formulae for a family of groups called endoscopic groups : these groups
have the same rank as G but a smaller dimension. This is basically Langlands’ strategy
(cf. [30]). In return, the stabilization of the trace formula should give the fonctorialities of
endoscopic type.
The works of Langlands [30] and Kottwitz [22, 23] achieved this program for the elliptic
part of the geometric side of the trace formula but under two assumptions : the first one
is the existence of the transfer for stable orbital integrals (in the spirit of proposition 11.1
above). The second one is the so-called fundamental lemma : it is a precise version of the
tranfer at unramified places which says that the transfer of the unit of a Hecke algebra
is again the unit of a Hecke algebra. Kottwitz used the stabilization of the elliptic part
to prove that the Tamagawa numbers are the same for inner forms (cf. [24] : this is a
generalization of the equality (11.3) above).
In a subsequent work (cf. [10, 11, 12] and many other ancillary papers), Arthur achieved
the stabilization of the whole trace formula : his work is also based on two assumptions
namely the existence of the transfer and a generalized fundamental lemma extended to
weighted orbital integrals. The existence of the transfer at archimedean places is due to
Shelstad ([35, 36]). Waldspurger showed how one can deduce the transfer from the funda-
mental lemma ([38]). So the cornerstone of the theory is the fundamental lemma and its
weighted version. Some cases were known by works of Hales, Waldspurger (for SL(n)) and
Weissauer. A breakthrough was the introduction of a geometric approach due to Goresky-
Kottwitz-MacPherson and Laumon. In this way, Laumon-Ngô [31] proved the fundamental
lemma for unitary groups. Recently Ngô has found a proof for the general case. In gene-
ral, the weighted fundamental lemma remains a conjecture (however cf. [39] for GSp(4)).
We also emphasize that when one tries to compare the trace formula to a Lefschetz trace
formula one often has to use the stable form of the trace formula : this is the case when
one tries to express Hasse-Weil zeta functions of some Shimura varieties in terms of auto-
morphic L-functions (cf. for an example Clozel’s report [17]) on Kottwitz work).
To finish, we mention that there is a twisted version of the trace formula (unpublished
notes of Clozel-Labesse-Langlands). In full generality, a twisted stable trace formula was
not yet established (however see the works of Kottwitz-Shelstad [25], Labesse [28], Renard
[33] and Waldspurger [37]). Conjecturally, such a formula should give functorialities for the
classical groups to GL(n) (work in progress by Arthur, cf. §30 of [13]). Arthur and Clozel
have already used a twisted trace formula to prove the cyclic base change for GL(n) (cf.
[14]). To conclude, let us mention that Jacquet has introduced interesting variants of the
Arthur-Selberg trace formula (cf. e.g. [20]).

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CNRS et Université Paris-Sud

UMR 8628
Mathématique, Bâtiment 425
F-91405 Orsay Cedex
France

mél :
Pierre-Henri.Chaudouard@math.u-psud.fr

23