2. TH E ARTHUR-SELBERG TRAC E FORMULA FO R GL(2) 5

orthogonal Hilbert space direct sum of countably many irreducible repre-

sentations ([GGPS], Chapter 3 §4.6):

* ) =

©*•

(Here 0 denotes an orthogonal Hilbert space direct sum, i.e. the closure

of the usual algebraic direct sum.) See also Chapter 3 of [Bu], or [Knl].

An irreducible representation of G(A) with central character u is cuspidal

if it occurs in this decomposition. Jacquet and Langlands proved that the

multiplicity of each cuspidal representation in the above sum is one (cf. [JL],

[PS], or Chapter 3 of [Bu]).

Let

Cc(G(A),u~1)

be the space of continuous functions / : G(A) — C

with compact support modulo Z(A) satisfying f(zg) —

ou(z)~1f(g).

Then

we define an operator R(f) on

L2(u)

by

R(f)p(9) = L f(x)(p(gx)dx,

JG(A)

for / G

CC{G(A),UJ~1)

and tp G

L2(u).

In general, R(f) is an infinite

rank operator and not of trace class. However its restriction Ro(f) to the

subspace of cuspidal functions is of trace class, so tri?o(/) is meaningful

([GJ] Corollary 2.4 and [Knl] Corollary 6.3).

The proof of the theorem below can be found in [G2] and [GJ], which

specialize Arthur's formula for a general reductive group ([Arl],[Ar4]) to

the case GL(2). The statement is from [GJ], Theorem 6.33. See also Theo-

rem 7.14 in the survey [Knl]. Haar measures will be fixed in Section 7.

THEOREM

2.1 (Arthur-Selberg Trace Formula for GL2). For

feCdGiA),*-1),

tr/Jo(/) =

(2.1) meas(G(Q)\G(A))/(l)

(2.2) + E £ -

f(9-l79)dg

[7]CS(Q) elliptic ^ W ) \ ^ ( A )

(2.3) -^meas(Q*\A

1

) E £- - f^lQ^Wg

2

1 / 7 G

M ( Q ) MA)\G(A )

(2.4) +f.p;ZF(s)

s=l

1 f°°

(2'5)

+ i ^ E / tr(M(-it)M'(it)p(X,it)(f))dt

XlX2

= u ;

(2-6) - ^ t r ( M ( 0 ) p (

X

, 0 ) ( / ) )

4