1549380232-Automorphic_Forms_and_Applications__Sarnak_

266 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS

bound of Theorem 4.8 replaced by Burgess's bound for Dirichlet £-functions ( 4.2).

With these bounds at hand, one can repeat the arguments of section 4.4.2 to find

that (4.31) is bounded by

«c,g Qjq- 8 L 3+2A L lc1l2,

l::;:,.L

which gives (4.30) as soon as L::::; q^8 /(3+^2 A), and is more than enough to solve the

s~. o

Remark 4.14. As we have seen, the case where x has large conductor is deduced

from (known) cases of subconvexity for £-functions of lower ranks (1 and 2); in

fact this phenomenon already occurred in the work of Duke/Friedlander/Iwaniec

[DFI8]. That a "reduction of the rank" principle exists for the Subconvexity Problem

is very encouraging for its resolution in higher degrees. A posteriori, the possibil-

ity of this reduction could have been anticipated, firstly because of the inductive

structure of the automorphic spectrum of G Ld ([MW2]) and secondly because this

principle is already present in Deligne's proof of the Weil conjectures [De2, De3].

4.5.3. Questions of Uniformity II

So far we have the ScP for Rankin-Selberg £-functions L(j © g, s) when g is fixed

and the level of f grows. One may wonder what happens when the level of g

grows too. The discussion of Section 4.4.4 enables one to solve the SCP in the q

and q' aspects simultaneously, as long as q' is smaller than a (small but explicit)

power of q. For example, suppose that f and g have coprime levels with trivial

nebentypus and that g is holomorphic. By keeping track of the dependency on q'

in our estimates and by using respectively (4.23), (4.24), or (4.28), one can show

that for ates = 1/ 2 one has

L(j © g , s) « (qq')1f2- 8

for some positive o (the implied constant depending on sand on the parameters at

infinity off and g), as long as q' ::::; qrJ for some fixed 'TJ, where

'r/ < (1/ 2 - e)/(3/ 2 + e), 'r/ < (1/ 2 - e)/(1/ 2 + e), 'r/ < (1/ 2 - e)/(1/ 4 + e),
respectively. In particular, since e = 1/ 9 is admissible and (1/2-1/9)/(1/4+ 1/9) >
1, the bound (4.28) would solve the Subconvexity Problem for L(j © g, s) in the
qq' -aspect when q and q' are coprimes with no further restrictions on the relative
sizes of q and q'.