1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 4. THE SUBCONVEXITY PROBLEM

L i/2

«e: L lcil

2

+ (qL)"'-( L lcil)

2

q

l~L l~L

( l,q)=l (l,q)=l

If we take now ( c1) to be the coefficients given in the third lecture

Ct=

{

x^6 (p) , for l = p^12 :::;; L , (p, q) = 1

• x4(p), for z = Ps:::;; Ls/12, (p,q) = 1


• x(p), for l = P2 :::;; L2/ 1 2' (p, q) = 1
  0 else,.

we obtain from (2.19), choosing N = q^2 /(1+^2 a) (for a:= 1/ 12) that

fSP(q, x)f «e: q^5 /7+e:.

249

D

Remark 4.5. This proof is an application of the amplification method where we

have amplified an entire set (here sfxotic(q, x, 0)) inside a family (an orthogonal

basis of .C 1 ( q , x )). One might be a bit surprised by the fact that this improvement

comes from the embedding of our original family inside a much larger one; in

fact, the embedding into a spectrally complete family makes Kuznetsov's formula

available and the improvement arises because of the lacunarity of the amplifier: in

our case the individual bounds provided by Kuznetsov's formula are stronger than

the averaged bound provided by the large sieve.

4.2.3. Further Miscellaneous Applications

In fact, the amplification methods can be used in quite a variety of contexts to give

non-trivial upper bounds for many arithmetic sums.

Another example concerns the L= -norm of Lr normalized weight zero Maass

forms with large Laplace eigenvalues on a modular curve. From its Fourier expan-

sion, one can show that for g E S 0 (q, i t 9 ) that is Lr normalized, one has

f f g (z)f f= «e:,q (1/ 4 + t; )^1 /4+e:.

In fact, this is a special case of a generic bound of Seeger and Sogge on the L=

norm of eigenfunctions of the Laplacian on a Riemann surface [SS]. When g

is a Hecke-eigenform, Iwaniec/ Sarnak [ISl] gave the first improvement over the

generic bound using the amplification method:

Theorem 4.6. Let g E S'b(q, i t 9 ) be a primitive weight zero Maass form on X 0 (q)
(with L r norm 1). Then for any c > 0, one has
f fg(z)f f= «e:,q (1/4 + ftg[2)1/4- 1 /24+e:.
Finally, we would like to mention an unusual application of amplification to
the rather technical question of bounding the following arithmetic quadratic form

B(M,N) := LL ambne(am)

n

m~M,n~N

(m,n)=l

(for some fixed a =f. 0); this sum is bounded trivially by

(MN)l/2(L [am[2 L [bn [2)1/2.

m n