6 TRACES OF HECKE OPERATORS

(2-7) " E £ f(g)x(det(g))dg.

The terms (2.1)-(2.4) constitute the geometric side of the trace formula,

and consist of orbital integrals or weighted orbital integrals over the con-

jugacy classes in G(Q). The remaining terms form the spectral side of the

trace formula. We now give a brief elaboration of each geometric term:

(2.1) This is the identity term coming from the conjugacy class {f

n 1

)}.

(2.2) Elliptic terms: 7 G G(Q) is elliptic when it is not conjugate to an

upper triangular matrix (over Q), or equivalently, when the eigenvalues of

7 lie outside Q. For any element 7 G G(Q), [7] is the G(Q)-conjugacy class

of 7, and G7(Q) is the centralizer of 7 in G(Q). The sum is taken over all

elliptic conjugacy classes in G(Q).

(2.3) Hyperbolic terms: An element of G(Q) is hyperbolic if it is con-

jugate to a nonscalar diagonal matrix in G(Q). The sum is taken over all

hyperbolic conjugacy classes in G(Q). Note that G7(A) = M(A) when 7

is diagonal. The weight function v is defined by v(g) = H(g) + H(wg),

where w = (

1 n

J, and H is the height function defined in Section 7.

(2.4) Unipotent term: Here

and f.p.Zp(s) denotes the "finite part" at s = 1 (i.e. the constant term of

s=l

the Laurent expansion about s — 1) of the meromorphic zeta-function

ZF(s)= [

F(a)\a\sd*a

JA*

defined by Tate.

The remaining terms (2.5)-(2.7) are the noncuspidal spectral terms.

These terms do not contribute to the traces of Hecke operators on holo-

morphic cusp forms.

(2.5) Continuous terms: This is the contribution of the continuous ker-

nel. We follow [G2] for the notation. The summation is over pairs of Hecke

characters (xi, X2) such that X1X2 = ^ and p(x, s) denotes the induced rep-

resentation space IndB/Au^4^y |§|^), where we write b — (

n

, 1 G B(A).

(For the definition of this induced representation, see page 390 of [Knl],

or [GJ] §4A.) Letting

\w

= (X2Xi) M(s) = M(s,x) is the intertwining