2

TRACES OF HECKE OPERATORS

level, and Hijikata [H], who gave the trace of Tn, with no restriction on the

level N, for (n, N) — 1. Hijikata's computation builds on work of Shimizu

([Sh], which applies Selberg's ideas to the Hilbert modular setting) and

Saito ([Sa], which generalizes Eichler's work). The most general formula

for the trace of Tn on Sk(N,u), valid for all n and iV, was given in 1977

by Oesterle in his thesis ([Oe]; see [Coh] for a description). This explicit

formula is known as the Eichler-Selberg trace formula. A statement of

the formula is given on page 370.

The first goal of these notes is to provide a reference with a comprehen-

sive self-contained proof of this fundamental formula, using the more modern

methods provided by the Arthur-Selberg trace formula for the adelic group

GL»2(A). We evaluate the trace formula using a function / : GL2(A) — C

which is constructed from double cosets at the finite places in the same way

as the classical Hecke operator Tn, and whose infinite component / ^ is a

matrix coefficient for the weight k discrete series representation of GLi2(R).

Because this matrix coefficient is not integrable when k = 2, we need to

require k 2. We also assume (n, N) — 1.

This technique is basic in the theory of automorphic forms. For example,

it is used in Langlands' general strategy for computing the Hasse-Weil zeta

function of a Shimura variety in terms of automorphic L-functions. Roughly,

an analytic expression coming from the trace formula for a function like our

/ (which can be evaluated in terms of automorphic L-factors) is compared

with a geometric expression involving the traces of Frobenius elements acting

on the cohomology of the variety (in terms of which the zeta function can

be evaluated). See [LI], [L2] and [Ro2].

In Sections 3 through 11 we have attempted to assemble the necessary

background from representation theory and number theory in one place for

anyone who wishes to understand the whole story without having to jump

between too many sources. This includes detailed treatments of modular

forms and Hecke operators, adeles and ideles, structure theory and strong

approximation for GL(2), integration theory, Poisson summation for func-

tions on the adeles, adelic zeta functions, representation theory for locally

compact groups, and the unitary representations of GL2(R).

The heart of the text begins in Section 12 where we give a thorough

account of the passage from the classical setting to the adelic one. In the

sections that follow, we essentially reprove the convergence of the truncated

terms on the geometric side of the trace formula for GL(2). This discussion

is quite general and overlaps significantly with the articles [G2] and [GJ],

however we have tried to include more detail than these sources, particularly

on convergence issues. Some extra care is required since our test function is

not compactly supported.

Lastly, we hope that the explicit computations of orbital integrals for

GL(2) over R and p-adic fields in Sections 24-26 will be interesting for any-

one studying the trace formula or local harmonic analysis. We will not

discuss zeta functions further (and indeed the most natural application in