264 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS
Here
x'(R2)Bxx'U\ m - fz n, O; c) = x(f 2)Gxx'(f 1f 2m - n ; c)
is the Gauss sum of character x x' and modulus c, and J(m, n ; c) is some appro-
priate Bessel transform of I. Thus we have considerably simplified the picture,
by replacing the Kloosterman sums by the much simpler Gauss sums. By [Mi]
Lemma 4.1, one sees that J(m, n ; c) is essentially bounded and is very small unless
m"' Q, n "'f 1 f 2 Q. Changing the variables by setting h = f 1 f2m - n, one is led to
the following instance of the SCP:
hwith W(x,y) = J(e 1 xe 2 , y ;c) and (with the notations of section 4.4) X "'Y"'
f1f2Q.
Remark 4.13. Assume for simplicity that xx' is trivial. An easy estimate shows that
the global contribution of the terms above is bounded by Oe:(qe: L^2 ('£e~L lcel)^2 );
this is not quite sufficient, but at least Voronoi's formula has placed us back in an
acceptable position.
4.5.1. The contribution of the term h = 0
The contribution of the degenerate frequency h = 0 is void unless xx' is the triv-
ial character. When xx' is trivial, the h = 0 term brings another main contribu-
tion that can be computed quite explicitly; this contribution is called the first off-
diagonal main term: its existence is the first manifestation that the variables m and
n have reached a critical range, since for smaller ranges, the main terms only arise
from diagonal contributions. We will not describe the calculation any further (see
[DFI3, KMV2, DFIS] for instance), but instead merely mention that if k =f=. k' (2), or
if g is holomorphic of weight k' ;:::: 2, this term is as small as the diagonal contribu-
tion; this fact is a consequence of an orthogonality property of the Bessel functions
arising in the computation of this term. When k = k' (2) and g has weight 0 or 1
the Bessel functions are no longer orthogonal and the first off-diagonal main term
becomes much larger than the diagonal. At this point, it is possible for the ampli-
fication method to break down... Eventually, Duke/Friedlander/Iwaniec resolved
this problem and identified the true origin of the first off-diagonal main term in this
case: this term is nicely compensated by the contribution from the Eisenstein spec-
trum (noted +... in the left hand side of ( 4.30)), up to an admissible error term.
The verification of the matching of both terms is carried out in [DFIS, Sect. 12
and 13], and uses Burgess's bound together with a delicate identification of rather
different complex integrals.
4.5.2. The contribution of the term h f= 0
The remaining contribution (from the "non degenerate" frequencies h f= 0) is called
the off-off-diagonal term and is estimated by solving the SCP with the methods of
Section 4.4. The off-off-diagonal term turns out to be an error term when g is
cuspidal, but brings another third main contribution when g is an Eisenstein series
(see[KMVl]).
An application of Theorem 4.14 and the trivial bound for Gauss sums
IGxx'(h;c)[ ~ (q~x')1f2(h,c),