1549380232-Automorphic_Forms_and_Applications__Sarnak_

214 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS

defines an isomorphism (which is in fact an isometry up to some explicit scalar )

between the spaces S,,,(q, x, it) and Sk(q, x, it). This shows that the study of general

Maass forms of weight k can be reduced either to that of Maass forms of weight

K = 0, 1 with positive 6.k-eigenvalue, or to that of holomorphic forms of weight;;:: 1;

however, for several technical purposes it may be useful to consider holomorphic

forms of weight k in terms of Maass forms of weight k.

Selberg's conjecture (which is the Ramanujan/ Petersson conjecture for GL 2 ,q

at the infinite place) is the statement that whenever .A > 0, then .A ;;:: 1 /4 (i.e.

t E R). Note that fork = 1 this holds trivially by (2.4), while fork = 0 the best

result toward this conjecture so far is l'Smtl ~ 7 / 64 [KiSa].

2.1.2.2. Fourier expansion. By periodicity f(z + 1) = f(z), and by separation of

variables, one shows that a Maass cusp form f with 6. k-eigenvalue .A = (1/ 2 +

it)(l/2 - it) has a Fourier expansion at infinity of the form

+oo

(2.7) f(z) = °""" L...,, PJ(n)W _I!._!i_ lnl 2' it(47rlnly)e(nx),

n=-oo

n#O

where Wa,.e(Y) denotes the Whittaker function; the {PJ(n)}nEZ-{O} are called the

Fourier coefficients of f. The Eisenstein series have a similar Fourier expansion,

Ea(z, 1/ 2 + it)= Oay^112 +it + 'Pa(l/2 + it)y^112 -it

(2.8)

+oo

+ L-t °""" Pa(n, t)W _I!._!i_ ln l 2 ' it (47rlnly)e(n x ),

n=-oo

n#O

where Oa = 0 unless a = oo, 000 = 1, and 'Pa(l/2 +it) is the entry (oo, a) of

the scattering matrix. Recall that the Whittaker function W a,it(2y) is the unique

solution of the differential equation

W

11

(y) + (.Ay-^2 + 2ay-^1 - l)W(y) = 0

that decays exponentially as y-+ +oo; more precisely, we have Wa,it(Y) ,..., y°'e-Yl^2.

In the particular case where it= a - 1/2, we have an exact formula

(2.9) W a,a-1/2 ( ) y -- y °' e -y/2.

In particular, for F(z ) E Sk(q, x), if we denote f(z) = ykf^2 F(z) E Sk(q, x, k2l ),

one has the following relation between the Fourier coefficients:

pp(n) = (47rll^2 Pt(n).

2.1.2.3. The reflection operator. Next we introduce the reflection operator X, which

acts on functions by

(Xf)(z) = f(-z).

The operator X sends forms of weight - k isometrically to forms of weight k and
satisfy X^2 = 1; moreover, X commutes with the Laplacian, i.e. 6._kX = X 6.k , so
that X Sk(q, x, it) = S_k(q, x, it). In fact, if we denote by Qit, k the operator

Qit,k := o(it, k)X II L1,

-k<l~k

l=k(2)