1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3. LARGE SIEVE INEQUALITIES

3.2.1. The large sieve inequality for forms of weight one

Theorem 3.4. For any c: > 0, N;:::: 1 and any (an)n~N E c N,

(3 .10) L IL AJ(n)an l2 «c qc(q + N) L lanl2·

fES'{(q,x) n~N n~N

229

Proof. We provide here an elementary proof of (a slightly stronger form of) (3.9):

L IL ann

112

P1(n)l

2

« (1 +;) L lanl

2

.

fEBi(q,x) n~N n~N

By duality, it is sufficient to prove that

Set g = LJEBi(q,x) bjf; then L fEBi(q,x) P1(n)fab1 = n^112 p 9 (n) and the inequal-

ity will follows from the inequality

N N

L ln^112 p 9 (n)l2 ~ (1 + -)(g,g) = (1 + -) L lb11^2 -

n~N q q f

We use the following argument of Iwaniec:

L ln^1!^2 p 9 (n)l^2 exp(-4nny) = f

1

lg(x + iy)l^2 dx

n Jo

and hence

r+= dy r dxdy

L ln1/2pg(n)l2 « L ln1/2Pg(n)l2 J1 exp(-4nny)y2 = }F lg(z)l2y-2-

n~N n l/N Y P(l/N) Y

where P(l/N) = {z E H , 0 ~ ~ez < 1, <:smz > 1 /N}. The number of fundamental
domains for r 0 (q) that are needed to cover P(l/N) is bounded by (which we leave
as an exercise)

Hence

N

Max lb E ro(q),'Yz E P(l/N)}I ~ (1+10-).

zEP(l/N) q

N

L ln1/2pg(n)l2 « (1 + -)(g,g).

n~N q

0

Remark 3.4. By the same method and Weyl's law, one could prove (3.8), without
the use of the Petersson-trace formula.

3.2.2. Large sieve inequalities for automorphic forms of higher rank

For quite general families of automorphic £-functions, large sieve inequalities simi-
lar to (3.5) are almost a formal consequence of the above duality principle and the
existence of Rankin-Selberg theory.