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

3

this direction would be to compute the zeta functions of modular curves,

which would require the k = 2 case), but we include some more modest

applications and examples in the last chapter. These include the dimension

formula for

S^(N1ujf)1

the integrality of Hecke eigenvalues, and the asymp-

totic equidistribution of eigenvalues of Tp as k + N — oo.

Other references for the traces of Hecke operators include Duflo and

Labesse [DL], who used the trace formula for GL2(A) to sketch a derivation

of the formula for the traces of Hecke operators. Miyake's book [Mi] contains

a proof (for k 2) using the trace formula for SL2(R). Miyake's exposition

is based on [Sh] and [H], and includes the case of cusp forms attached to

the unit groups of indefinite quaternion algebras due to Hijikata. Zagier

gave a proof for level 1 and weight k 3 (also using the classical language)

in [Zl] and [Z2]. In [Oe], Oesterle removed the condition (n, N) = 1, and

allowed for half-integer weights by building on work of Shimura. We adopt

Oesterle's notation for the final form of the trace formula.

Acknowledgements . We would like to thank J. Rogawski for his en-

couragement and helpful advice, without which these notes could not have

been written. We also thank the anonymous referees for their detailed com-

ments. This work was supported in part by NSA grants H98230-05-1-0028

and H98230-06-1-0039. The second author thanks the UCLA math depart-

ment for support when most of this work was done, as well as the Taiwan

National Center for Theoretical Sciences which provided travel support for

a visit to Maine.

2. Th e Arthur-Selberg trace formula for GL(2)

We begin with a review of the trace formula for GL(2) for a compactly

supported function. Although we will not use it explicitly, this formula

provides the framework for the trace formula derived in these notes. Nearly

all of the definitions and concepts which are mentioned briefly in this section

will be discussed in greater detail later on. A good introduction to the trace

formula is given in Lecture 1 of Gelbart's book [G2].

Let A be the adele ring of Q, and let A* be the idele group (see Section

5.2 below for definitions and topology).

Let G be the group GL2. Thus for any ring R (we always assume rings

are commutative with 1), G(R) is the group of 2 x 2 invertible matrices with

entries in R. We use this notation for any linear group. For example let

B C G denote the Borel subgroup of invertible upper triangular matrices.

Then B(R) = M(R)N(R) where M(R) is the group of diagonal matrices

with invertible entries in i?, and N(R) is the group of unipotent matrices

(

n 1

I for t G R. The Iwasawa decompositio n of Gp = G(QP) (or

Goo - G(R)) is

Gp = BpKp (o r G Q O = ^ 0 0 ^ 0 0 ) 5