48 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS
Note that E(z, ~+it) is not an £^2 -function on r\H; neither should it be since it
occurs as a continuous summand in ( 1.1). For the precise meaning of the continuous
term in (1.1) see [29].
The decomposition (1.1) gives the eigenvalues of~ in A:
{
0 on the constant summand
(1.2) A = ~ + t^2 on E(-, ~ +it)
An> 0 on fnNow replace r by a congruence subgroup, for instance ro(N) = { ( ~ ~) E
81(2, Z) : c = O[N]} (N ~ 1). Then we have a similar spectral decomposition,
the only important difference with (1.1) being that there will be several continuous
terms, associated to the several cusps of the quotient r\H. We can then state
SELBERG'S CONJECTURE. -For all n, An ~ ~.
By comparing with (1.2) we see that this says the discrete spectrum of~ should
be (except for A = 0) embedded in the continuous spectrum.
1.3. We now introduce the Hecke operators. For p a prime, define(1.3) (Tpf)(z) = f(pz) + L f C + b) (z EH).
bmodp pIf f is a function on 1i invariant by r, so is Tp(f). For a congruence subgroup we
will consider only the Tp for pf N. The Tp commute, commute with ~' and are
self-adjoint. They are bounded; in fact we have the easy
Lemma. -
(i) llTpfll2:::; (p + l)llfll2 (f EA)
(ii) If f is a non - constant eigenf orm:
Tp(f) = Apf (f E A)
then IApl < p + 1.Note that p+ 1 is the degree of Tp, i.e. the number of terms occurring in (1.3).
The proof is left to the reader (or see [18, p. 210]).
We can now refine our decomposition (1.1) by taking a basis fn of cusp forms
that are eigenforms both for ~ and for the Tp (pf N). Note that
(1.4) Ap = {p +^1.. on the constants
p^112 (p•t + p-•t) on E(·, ~+it)If fn is a cusp form and Ap = Ap,n the corresponding eigenvalue we may write
Ap = p1/2(cxp + {Jp), exp, {Jp E ex ' cxp{Jp = 1.Since Ap is real, and IApl < p + 1, an easy computation shows that
(1.5) {
either lapl = lfJPI = 1
or exp, {Jp E JR. and p-1;2 < lapl, lfJPI < p1/2