4 A. KNIGHTLY AND C. LI

and the article [LiX] by X. Li, who also extended the formula to the setting of

Maass forms for SLn(Z), [Gld]. Related investigations have been carried out by

others, notably in the context of base change by Jacquet and Ye (cf. [Ja] and its

references).

Our primary purpose is to give a detailed account of this method over the adeles

of Q, for Maass cusp forms of arbitrary level N and nebentypus ω . We obtain a

variant of the Kuznetsov trace formula by using the kernel function attached to a

Hecke operator Tn. The final formula is given in Theorem 7.14 on page 79, and it

differs from the usual version by the inclusion of eigenvalues of Tn on the spectral

side. The cuspidal term thus has the form

(1.8)

uj ∈F(N)

λn(uj) am1 (uj)am2 (uj)

uj 2

h(tj)

cosh(πtj)

.

This is a complement to the article [KL1], which dealt with Petersson’s formula

from the same viewpoint. As we pointed out there, the above variant can alter-

natively be derived from the classical version (see Section 7.7 below). It is also

possible to invert the final formula to get a version with the test function appearing

on the geometric side rather than the spectral side, although we will not pursue

this. See Theorem 2 of [BKV] or [A], p. 135.

The incorporation of Hecke eigenvalues in (1.8) allows us to prove a result about

their distribution (Theorem 10.2). To state a special case, assume for simplicity

that the nebentypus is trivial, and that the basis F(N) is chosen so that a1(uj) = 1

for all j. Then for any prime p N, we prove that the multiset of Hecke eigenvalues

λp(uj), when weighted by

wj =

1

uj 2

h(tj)

cosh(πtj)

,

becomes equidistributed relative to the Sato-Tate measure in the limit as N → ∞.

This means that for any continuous function f on R,

lim

N→∞

∑

uj ∈F(N)

f(λp(uj))wj

∑

uj ∈F(N)

wj

=

1

π

2

−2

f(x) 1 −

x2

4

dx.

This can be viewed as evidence for the Ramanujan conjecture, which asserts that

λp(uj) ∈ [−2, 2] for all j. The above result holds independently of both p and the

choice of h from a large family of suitable functions. We discuss some of the history

of this problem and its relation to the Sato-Tate conjecture in Section 10.

The material in the first six sections can be used as a basis for any number

of investigations of Maass forms with the GL(2) trace formula. Sections 2-4 are

chiefly expository. We begin with the goal of explaining the connection between

the Laplace eigenvalue of a Maass form and the principal series representation of

GL2(R) determined by it. We then give a detailed account of the passage between

a Maass form on the upper half-plane and its adelic counterpart, which is a cuspidal

funcion on GL2(A). We also describe the adelic Hecke operators of weight k = 0

and level N corresponding to the classical ones Tn.

Although similar in spirit with the derivation of Petersson’s formula in [KL1],

the analytic diﬃculties in the present case are considerably more subtle. Whereas

in the holomorphic case the relevant Hecke operator is of finite rank, in the weight

zero case it is not even Hilbert-Schmidt. The setting for the adelic trace formula is