256 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS
We present this approach for g E Sk ( q', x) holomorphic. It is then easy to see
that a non-trivial bound for ~w(g, £ 1 , £ 2 , h) follows from the following: for ~es> 1,
we set
~ ~ -( Ag m ) ( Ag n )( e v£1£2mn g )k-1( £1m + f2n )-s.
E1m-£2n=h 1m+ 2nThis series is absolutely convergent for ~es > 1 and analytic there; the next theo-
rem show that it can be analytically continued beyond this natural barrier.
Theorem 4.11. Assume H 2 ( e) holds. For any c > 0, D (g, s; £ 1 , £ 2 , h) extends holo-
morphically to ~es ~ 1/2 + e + c and in this region, it satisfies the upper bound
D(g, s;£1,f2, h) «e,g,f)i (lhl£1£2)E(£1£2)^1 /^2 lhl^1 /^2 +ll-a(l + jtl)^3 ,
where s = a-+ it and the implied constant depends only on c and g.
To pass from this bound to one for ~w (g, £ 1 , £ 2 , h), we note that(4.13) ~w(g,£1,£2,h) = 11 -
2
ni D(g,s;£ 1 ,£2,h)W(h,s)ds
(2)with
W(h )=1+=W(u+h u-h)( 4u
2
)k'21 5 du
,s 0 2 ) 2 u (^2) - h2 u u.
Shifting the contour to ~es = 1 /2 + e + c, we obtain, after several integrations by
parts and using (2.22) and (2 .23), the bound
(4.14)
Remark 4.10. Note that H 2 (B) for any e < 1 /2 would be sufficient to solve the
SCP; compare with Remark 4.9. However, improvements on H 2 (B) directly yield
improved bounds for the SCP, hence for cases of the ScP. In particular H 2 (7/64)
yields the subconvex exponent ~ - 1 ~ 0 in place of ~ - 212 in Theorem 4.8.
Proof. (of Thm 4.11) The proof follows from an appropriate integral representa-
tion of D(g, s; £1, £2, h). Setting N = q' £ 1 £ 2 , one expresses D(g, s ; £ 1 , £ 2 , h) in terms
of the integral of the I' 0 (N) invariant function
k k
V(z) = 'Sm(£1z)29(£1z)'Sm(£2z)2 g(f2z)
against an appropriate Poincare series
y(rz )^8 e(hx(rz));
1'Ero(N) 00 \r 0 (N)
more precisely, one has by the unfolding method
(4.15)
- 1 dxdy I'(s + k - 1)(£ 1 £ 2 )~
I= (Uh, V) = Uh(z, s)V(z)- 2 - = ( ) +k-l D(g, s; h).
r 0 (N)\H Y 2n^8On the other hand Uh and V can be decomposed spectrally (at least formally) over
a orthonormal basis for r 0 ( N) and by Parseval we have