LECTURE 4. THE SUBCONVEXITY PROBLEM 257Finally by the unfolding method one has(u (. s ) u·)= 2s-1;oj(h)r(s-~+itj)r(s-~-itj)
h ' ' J lhls- 1 2 2
so that
~ 2s-lp-(h) S - l +it S - l - it -
(4.16) l= L.., lhls~l f (^2
2(^1) )f ( 2
2
(^1) )(uj,V)+EisensteinContr.
j~l
with
Eisenst ein Con tr. =
1 (^00 2s-^1 7Jt(h) s - l +it s - l - it -
4n ~Jo lhls
0
~ 1 f( ~ )r( ~ )(Ea(.,~+ it) , V )dt.
From this decomposition one sees that D(g, s; .e 1 , .e 2 , h) has analytic continuation up
to Wes> We(s - ~ ± itj ) ~ ~ + e.
The bound for D (g, s; f 1, f2, h) follows now by bounding each term of the spec-
tral decomposition (4.16).
4.4.2.1. Bounding the Fourier coefficients. We start with Pj ( h): if Uj = fj / (fj, f j)^112
for some primitive Hecke-eigenform fj, one has lhl^112 pj(h) = ±AJ; (lhl)/ (fj, fj)^112 ;
hence H 2 (B) and the estimate (2.16) give
(4.17) Pj (h) «,, (lhlN(~ltjl)))" ch(~j )lhllJ-l/^2
A similar bound for all Uj would follow from producing an explicit orthonormal
basis, constructed from old and new forms as in [ILS] (for squarefree N ). However,
for general levels, the corresponding computations can be quite messy. In fact, we
need this bound only on average over an orthonormal Hecke eigenbasis B o ( N) =
{uj}j~o, (4.17): one has (see [Mi]), for any c > 0,
(4.18) ~ lhllPj(h) l
2
«,, (lh lN T)"T^2 lh l^29.
L.., ch ( ntj )
u;EBo(N)
lt;l:s:;T
4.4.2.2. Bounding the triple products (uj, V). It remains to bound the integral of the
triple product ( Uj, V) and, more precisely, one needs a bound with an exponential
decay like exp (-nltjl/2) (as ltjl --+ +oo) in order to compensate for the factor
ch(~) from ( 4.17). This is a key point and it remained, until recently, the main
obstacle to a general solution of the SCP along the lines proposed by Selberg; the
exponential decay had been obtained in several specific cases (see [Go, TV, Ju2,
Ju3]) but the first truly general treatment was given in [Sa3] and made explicit
further in [Sa4]. For any Uj, one has
(4.19) (uj, V) «k l lYk/^2 gll~VlV(l + ltjl)k+le-~lt;I,
and the same bound holds for (E 0 (., ~+it), V) with tj replaced by t. Using (4.19)
together with Weyl's law,
(4.20) L l = vol(~~(N))T^2 +0,,((l+T)l+"),
it;l:s:;T