1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

It turns out that the best choice for U is U = min(X, Y) = X (say) and consequently
one has R = U^112 = X^112 , which implies that

(4.10)

() i () i

~Er(x,y,h) «ij

u'xuJy '

. j
1 £1£2 R i+ ·

(r R + l£1x - £2y - hi) X i+i (-;:) J ·

We apply the Voronoi summation formula of the second lecture to both vari-
ables m and n. For simplicity, we assume that g has level^4 1: by Lemma 2.3.1, the
sum is transformed into
(4.11)
l:.w(g,£1,£2,h) = ~ L (£1£2, r) ~- ( I I ) ( )
r^2 L>..^9 (m)>..^9 (n)S -l^1 m+l^2 n ,-h;r Ir m , n , h ,
r~R m,n

with

(4.12)

here we have set l~ = lif (r , li), i = 1, 2. By integrating the Bessel function by parts
several times (see (2.22) (2.23)), one shows using ( 4.10) that I,. ( m , n , h) is very
small, unless

m < l' 1 __!!___ R2-c = l' 1 X^0 l^2 ' n < l' 2 2-R2-c = l' 1 x^0!^2 X y.

In the remaining range, the bound ( 4.10) and a trivial estimate show that

X(£1£2,r)

Ir(m, n , h) « £

1

£

2

logU.

(In fact, ¢ was introduced precisely to ensure this last estimate.)

The above considerations together with Weil's bound for Kloosterman sums

IS(a, b; r)I ~ T(r)(a, b, r )^112 r^112

yield
l:.w(g,£1,£2,h) «c,g (XY)"'X^1 l^4 y^1 l^2.

This improves on the trivial bound as soon as £ 1 + £ 2 ~ X^314 / Y^112. O

Remark 4.8. This method applies equally to Maass forms and to Eisenstein series.
For instance, it applies to the Eisenstein series g(z ) = E'(z, 1/ 2) of the second
lecture, the main difference in that case being that S w(g, ... ) has a main term
coming from the constant term of the Eisenstein series (see [DFI3]).

Remark 4.9. Note that any non-trivial bound for Kloosterman sums (i.e.

IS(a, b; r)I « (a, b, r)^112 r^8

for some B < 1 and any c: > O) is sufficient to give a non-trivial (although weaker)
bound for the SCP (for example, Kloosterman's original bound with e = 3/4). It
is interesting to note that Kloosterman derived his bound by elementary methods
that used families and moments (see [IS] Chap. 3).
On the other hand, it is possible to improve the given bound for the SCP beyond
Weil's bound with more advanced techniques: starting with (4.11), it is possible
to use Kuznetzov's trace formula backwards so as to transform ( 4 .11) into sums

(^4) For non-trivial level, some more (purely technical) work is necessary; for this we refer to the papers
[KMV2, Mi]; we also mention the paper of Harcos [Ha2] for a somewhat different treatment based on
a variant of the a-symbol due to Jutila.