1549380232-Automorphic_Forms_and_Applications__Sarnak_

12. EISENSTEIN SERIES 37

we may as well assumes to be real (i.e. in X(A)). If x varies in a compact set D

then, by Section 5.7, there exists constants c, c' > 0 such that

(99) ca(r)s+p :S a(rx)s+p :S c'a(x)^5 +Pa(r)5+P (x ED,'}' Er)

therefore the uniform convergence on D and the convergence at one point are equiv-

alent (and the moderate growth will follow); these conditions are also equivalent

to

(100) in L a(rx)^5 +Pdx < oo

,,Er p 0 \r

Assume D small enough so that nn-^1 n r = {l}. Then:

L r a(rx)^5 +Pdx = L 1 a(x)^5 +Pdx:::; 1 a(y)5+Pdy = J

,,Erp 0 \ r Jn " ,,n <rPo \r)O.

By Proposition 11.5, we can write a(r) E a 0 - Ap, for some a 0 E Ap. Therefore,

since the 1D are disjoint:

J:::; r a(x)s+p dx

lrp\NpMp(aoA?)K

The Haar measure can be written as a-^2 Pdmdndkda. Since rP \MpNpK has

finite volume, we are reduced to showing that

r a^5 -pda = a~-p J a^5 -Pda < 00

Jao -Ap -A

Take as coordinates on A the ta = a/3"', notation of Section 11. l. Write

P = l:a r af3a, s = l:a saf3a. Then the last integral is

II {l t~"'-r"' dt a

a Jo ta

which converges if Sa - r a > 0 for all a E b...

12.6. Bernstein's reduction of the general case to Section 12 .5 relies on the following
crucial observation:
(*) Let f be an automorphic form on r\G. There exists t E X(A), in the
domain of convergence of Ec,p 0 (l, s), such that

(101) lf(x)I-< Ec,P 0 (1,t)(x) (x E r\G)

Proof. First note that for s real, the series defining Ec,p 0 (l, s) consists of positive
terms, the first of which is a(x)^5 +P. Since f has moderate growth on r \^0 G, we can
find ti, in the domain of convergence of Ee ,Po ( 1, s), such that

(102) lf(x)I-< Ea,p 0 (l,t1)(x) (x Er\^0 G).

Combined with Proposition 5.8, this shows the existence of finitely many μi E
X(AG) (i EI) such that

(103) lf(x)I-< Ea,P 0 (l, t1)(x)(L a(x)μ'), (x E r \ G).

But ifμ E X(AG), it is left-invariant under r (or even °G), so that it follows from
the definition that

(104) Ec,P 0 (l, t1)(x)a(x)μ = Ec,P 0 (1, t1 + μ)(x).