1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 2. A REVIEW OF CLASSICAL AUTOMORPHIC FORMS 217

In particular, under H 2 (B) one has

1 1

(2.11) AJ = 4 + t^2 ~ 4 -e^2 , and l>-1(n)I:::;; T(n)ne.

Recall that for holomorphic forms, RPC was proved by Deligne and by Deligne/

Serre [De, De2, DS]: hence f E sr ( q , x) one has

(2.12)

In general, by the work of Kim/Shahidi, Kim and Kim/Sarnak [KiSh, KiSh2, Ki,

KiSa], one knows that H 2 (7 /64) holds.

From the above identification, it follows that the classical Hecke £-function

matches the automorphic one: L(7r 1 , s) = L(f, s). Following the general theory of

automorphic forms, one can also form the automorphic Rankin/Selberg £-function

L( 7r / © 7r 9 , s), which is an Euler product of degree 4 and whose local factors match

the local factors of L(f x g, s) at every prime not dividing the conductors off and

g. Also, by a result of Gelbart/Jacquet [GeJl], there exists an automorphic rep-

resentation of G L 3 , the symmetric square of 7r 1 , noted sym^2 7r 1 , whose £-function,

L(sym^2 7r 1 , s) has the same local factors as L(sym^2 f, s) at every prime not dividing

q 1. These automorphic £-functions have a priori a less explicit definition (in partic-

ular they do not match their classical counterpart in general), but they are in many

aspects the most natural objects to study. In particular, their functional equation

takes a more natural form and is proved in full generality. As long as the analytic

techniques have not been completely translated to the automorphic setting, it will

be useful to take advantages of both aspects; the switch between the automorphic

and the classical aspects is formalized via the following factorizations:

(2.13)

(2.14)

(2.15)

L

>-1(n)x(n)

L(7rJ·X, s) = F(7rJX, s)L(f xx, s) , with L(f x x , s) = n s ,

n

L(7rJ © 'ffg, s) = F(7rJ © 'ffg, s)L(f x g , s),

L

>- 1 (n)>- (n)

with L(f x g, s) = L(x1x 9 , 2s) n s^9 ,

n

L(sym^2 7rf, s) = F(sym^2 7rf, s)L(sym^2 f , s) ,

-1(n2)
with L(sym^2 f, s ) = L(x}, 2s) L ~-
n
Here the fudge factors F are finite Euler products supported at the ramified
primes and are well controlled by the known bounds for the local parameters of 7r f
and Jr 9 • In particular, it follows from H 2 (B) (for any fixed e < 1/ 4) that

(qN 9 )-^0 « 0 F(7rJX, s), F(7rJ © 7r 9 , s) , F(sym^2 7rj, s) «c (q1q 9 )°

for any c: > 0, uniformly for ~es ~ 1/ 2 - o, where o is some positive absolute
constant.
We can then utilize the analytic results on automorphic £-functions of the first
lecture to derive upper and lower bounds for several classically defined quantities,
such as the inner product (f, f). From the method of the proof of (1.19), the Siegel