1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 4. THE SUBCONVEXIlY PROBLEM 265

shows that the h =I= 0 terms of ( 4.31) contribute (essentially)

« Qc:LL1~Llc1l

2

r:;;-q* Q3/ 2+i!L3+211« Qc:(q~x')1;2q11L4+211'"'lcl2

c:,g f [q, q']2 y 'ixx' c:,g f q &;;., l ,

where q~x' is the conductor of the character xx'. Thus the above bound solves the

Subconvexity Problem for L(f x g, s) (and g fixed) only as long as q~x' :::;:; qTJ for

some fixed 'r/ < 11 ~ 211 ; in particular, even RPC is not sufficient to solve our problem

when x is primitive.

The problem comes from the size of the Gauss sums Gxx' (h; c), which get larger

with q~x'; hence it is clear that the oscillations of Gxx' ( h; c) must be exploited in

the averaging over h: this is reasonable since h varies over rather long intervals (of

length"' £ 1 f 2 Q when xis primitive). This effect is best seen when one considers the

extreme (but most crucial) case of xx' being primitive (q~x' = [q, q']): for simplicity,

we examine only the contribution coming from c = [q, q'] (the other terms are

treated similarly). Under these conditions, one has Gxx'(h;c) = Gxx'(l;c)xx'(h)

and the corresponding term becomes (see the notations of Section 4.4)

Gxx^1 (l; c) L xx'(h)~w(g, f1f2, 1, h).
hfO
By (4.13), (4.15) and (4.16), the above sum has the following spectral decomposi-
tion

1 J (2n)s+k-l2s-l

Gxx'(l; c)-

2

. 1 x
ni r(s + k - l)(f1f2)2
(1/2+ll+c:)

(

s-l+it· s-l-it· _ - W(hs) )

I:r(^2

2

(^1) )r( 2
2
(^1) )(uj,V)2:xx'(h)pj(h) lhls~i + ... ds,
j~l h#O
where... denotes the contribution from the continuous spectrum. We want to
bound the sum LhfO xx'(h)pj(h)W(h, s) : for simplicity we assume that Uj is of
the form Uj = fj / (fj, fj)^1 /^2 where fj is primitive. The above sum then equals
~ '""' x(±l ) ( Pj ±1) ~x'""' -( x' h )-( Aj h ) w hs-1/2 ( ±h, s) ·
± h>O
By averaging trivially over h, one has, using (2 .17),
~h>DXX'(h)>..j(h) ~~~~/ 2
8
) «c:,k' (N(l + 1tjl))c:(QL^2 )^3 /2+c:
which is as good as having RPC for the individual Aj(h). However, as we have seen
previously, this is -just barely!-not sufficient.
One can do better by considering the h-sum in terms of the twisted £-function
L(xx' x fj, z) and by using the subconvexity bound (in the q~[aspect) proven
before. Indeed, Theorem 4.8, leads to an upper bound of the form

L:xx'(h)>..j(h) ~~~~/ 2

8

) «c:,k' (N(l + ltjl))A(q~x')^1 l^2 -^0 (QL^2 )i+c:

h>O

for some A ;;,: 0, some 6 = 1/22 > 0, and any E: > 0. The analysis of the continuous
spectrum contribution follows essentially the same lines, with the subconvexity