262 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £ -FUNCTIONSwork of Duke/ Friedlander/ Iwaniec [DFI3] for f a holomorphic form with trivial
nebentypus and g(z ), the Eisenstein series E' (z, 1/2); in view of the identity
L(f 0 E ' (z, 1/2), s ) = L(f, s)^2 ,
this is equivalent to solving the SCP for L(f, s) in the level aspect. Some general-
izations to the case of g a general cusp form have been given in [KMV2, Mi], and
we describe here the proof of the following special but interesting case:
Theorem 4.13. [Mi] Let g E sr, ( q', x') be a holomorphic^6 cusp form and f E
S~ ( q, x, it) be a general primitive cusp form. Then one has
L(f ® g, s ) «c:,s ,g q1/2- 1;1os4.
Proof. (Sketch) The conductor of L(f 0 g, s), q10 9 =: Q^2 say, satisfies
q2 «g (q/q')2 ~ Q 2 ~ (qq')2 «g q2,
and the convexity bound is then L(f 0 g , s) «g,c:,J= Q^1 /2+c:. Thus our objective is
to bound non-trivially the linear form of length Q = q10 9,'£v(f x g,Q) = L >-1(~(n)V(~),
n
for V some function with rapid decay; to simplify presentation we suppose that V
is compactly supported in [1, 2]. The next step is to choose an appropriate family F
as follows- When f E S k ( q, x) is holomorphic of weight k ~ 2, one can take for f an
orthonormal basis of Sk(q, x) containing f /(f,!)^112 , and use Petersson's
formula (2.1). - If f is a Maass form of weight k = 0 or 1, it is natural to take for F
an orthonormal basis of Ck(q, x) containing f(z )/ (!, !)^112 and use the
Kuznetsov/ Petersson formula (2.25), or its smoothed version (2.28). - Obviously, the first possibility is not available when f is holomorphic of
weight one; instead one considers F(z ) := y^112 f(z ) as a Maass form of
weight 1, and makes the above choice for F. - In fact, for technical purposes, it is useful to enlarge the family fur-
ther by taking an orthonormal basis of forms of level [q, q'] containing
f / (!, f) [q,q' ] as an old form.
Remark 4.12. Actually, even for holomorphic forms of weight k ~ 2, there are
some technical advantages to considering F (z ) = ykf^2 f(z ) inside an orthogonal
basis of non-holomorphic forms of weight k : for small weights, Petersson's formula
is only slowly convergent in the c variable making its use delicate. (The intrinsic
reason is that the corresponding eigenvalue ~( 1 - ~), for small k, is not enough
separated from the continuous spectrum of .C 2 (q, x ).)
By the amplification method, it is sufficient to give a bound for the amplified
second moment of
'£v(f x g, Q) x L c1.A1(l ) = L 1x 9 (C),
l~L
(l ,qq')=l(^6) When g is a Maass form, the corresponding bound h as now been proven in [HM]