1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 4. THE SUBCONVEXITY PROBLEM 251

of Dirichlet characters of modulus q and form the averaged mean square

LIL:v(g, x)l^2 I L x(Z)c1l^2 :=LI L x(m)bml^2 := Q((bm)m~qL),

x(q) l~L x(q) m~Lq

say. We want to bound this sum essentially by the contribution coming from the

diagonal terms i.e. to obtain the bound «e,g (qL )eq l:i~L lc1 12. Opening the square,

one has

Q((bm)) = cp(q) L bmbn = cp(q) L I L bml

2

~ cp(q) LI L bml^2 ;

m:=n(q) a(q) m:=a(q) a(q) m:=a(q)

(mn,q)=l (a,q)=l

opening the square again and averaging over a(modq), one obtains

Q((bm)) ~ cp(q) L L bmbn = cp(q) L L Ce 1 ce 2 L:v(g; h , t\,£2)

h:=O(q) m-n=h h:=O(q) e1,e2~L

where

This sum is much like a partial sum of Rankin-Selberg type but with an extra ad-
ditive shift given by h. When h = 0, this is a true Rankin-Selberg type partial sum
and one easily sees by (2.17) that

cp(q) L ce1ce2L:v(g;0,£1,e2) «e,k ql+eL lctl^2.

Remark 4.6. In fact, one can get an asymptotic formula for this sum and see that
this bound is nearly optimal (because of the lower bound in (2 .16) for L( sym^2 g, 1)).

When h =f. 0, the additive shift is non-trivial, and one expects some cancellation

in the averaging of .Ag(m).Ag(n) (the trivial bound being «e qe). We will see in the

next section that this is indeed the case; the outcome is

L:v(g; h, £1,£2) «e,g (qL)e L3/4q-l/4.

It follows that

IL:v(g, Xo)l^2 I L Xo(l)ctl^2 ~ LIL:v(g, q)i21 L x(Z)ctl^2

l~L x(q) l~L

«e,g ql+e L lctl2 + q3/4+e L5/2+1/4 L ictl2;

choosing c 1 = x 0 (l) (in that case, a = 1) and optimizing L, one obtains

L:v (g, Xo) «e,k ql/2-l/22+e.

Remark 4. 7. The original exponent 1 /2 -1/22 = 0.4545 ... of [DFil] is not the best
possible; in fact we will see in the next section that its size is directly dependent
on H 2 (B). In particular, the truth of H 2 (7 /64) due to Kim/ Sarnak yields the slightly
better subconvexity exponent 1 /2 - 25/448 < 0.442.