260 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £ -FUNCTIONSholomorphic of weight k and level q', the bound (4.21) gives
(4.23) L l(uj, V)l^2 e71'1ti 1 +Eisenstein contr. «k,E (NT)Eq'^4 (f 1 f 2 )^2 T^2 k+4,
ltil~Tin view of the trivial bound I lyk/^2 g(z)l [ 00 «k q^1112 log q' of [AU]. There is no doubt
that this method can be refined further.
We pass to the methods of Bernstein/ Reznikov (and its improvements by
Krotz/ Stanton): the dependency with respect to the level of g was computed by
Kowalski and provides stronger result ([Kow], [KrS] Thm. 8.5); namely, one has
for T ~ 1
(4.24) L l(uj, V)l^2 e71'1ti 1 +Eisenstein contr. «k (f 1 f2q')^2 T^2 k,
ltil~T
which is stronger than ( 4.23) by a factor ~ q'^2 T^4.
Although such results are quite good and solve new instances of the convexity
problem, the given uniformity is not quite sufficient for the most advanced pur-
poses. We will now discuss a third (tentative) approach to this question. This ap-
proach is far less general (and somewhat more technical), but makes full use of the
arithmetic context (it applies only to arithmetic lattices). It is suggested by triple
product identities of Harris/Kudla, Gross/Kudla and Bocherer/ Schulze-Pillot and
Watson[HK, GK, BS, Wat], which relate periods of products of triples of automor-
phic forms to central values of Rankin triple product L-functions L(n 1 Q9 n 2 Q9n 3 , s ).
To fix ideas, we consider these identities in the special case n 1 = n 9 , n 2 =
ii" 9 , n 3 = n f, where f and g are primitive forms of squarefree level q' with trivial
nebentypus, f a Maass form of weight zero and g either a holomorphic form of
weight k or a Maass form (of weight k = O); elaborating on the above cited works,
Watson gives a completely explicit relation between the period and the central
critical value of the triple L-function L( n 9 Q9 n 9 Q9 n f, 1/2). More precisely, one has
I fxo(q') yklg(z)[2 f(z)dxdy/y2l2 Tiplq' Qp A(ng Q91l'g Q91l'J, 1/2)
(4.25)(g, g)2(J, !) q^12 A(sym2n 9 , 1)^2 A(sym^2 n f, 1)'
where A(s) = L 00 (s)L(s) is the completed L-function and Qp is an explicit, uni-
formly bounded, local factor depending on n f,p, n g,p· From the value of the factors
at infinity (see [Wat]) and(2.19) and (2.16), one deduces that[(yk[g(z)[2, (!, f) 112 )[
2
«e,k (QJQ 9 nl + [tJ[)^2 k-^2 exp(-nltt l)L(n 9 Q9n 9 Q9nf,1/2)
for any c > 0.
This formula shows that GLH provides a very sharp (probably best possible)
bound in all aspects but, of course, we are looking for unconditional results. In
fact, the convexity bound
L(ng Q91l'g Q91l'J,1/2) «e,k q'5/4(1+1tJl)8/4is known unconditionally: it follows from the factorization
L(n 9 Q9n 9 Q91l'J, s) = L(sym^2 n 9 Q91l'J, s)L(nJ, s)
and the fact that the convexity bound is known for each factor. For the first factor
this is a consequence of work of Molteni [Mol] and the truth of Hypothesis H 2 (1/9),
due to Kim/ Shahidi [KiSh]. This yields an individual bound stronger than ( 4.19)