1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. ANALYTIC PROPERTIES OF INDMDUAL £-FUNCTIONS 207

1.3.3.1. What do we mean by "nice family"? In practice one may proceed slightly
differently: instead of averaging the 2k-th moment of the linear form L:v(7T, X)
over F, one starts by computing the k-th power L:v(7T 0 ,X)k. Since the arithmetic
function n ---) ,\71" ( n) is multiplicative, one finds that

L:v(7To, X)k = L Tk,:?nn) >-11" 0 (n) = L:;,,,71"0 (7ro, Xk)
n~Xk
(say); here Tk,11" 0 (n) is some arithmetical function that depends (mildly) on 7ro. In
fact, this function looks essentially like the standard k-th divisor function (hence
is bounded by «e: ne: for any c > 0)^3. Next one average over F the square of the
following linear form oflength Xk:

I;' V ,7ro ( 7T, Xk)= """""'TkL_; ,11"o(n),\ fa 7r (n) •

n«Xk

Hence, by inverting the summations, one obtains

E(IL:;,,,71"Q(7r,Xk)l^2 ,μF) = L Tk,11"o(~11"Q(n) r >.11"(m)>.11"(n)dμF(7T).

m,n«X k mn JF

Thus "nice family" means that form, n less than Xk, the expectations of the ran-
dom variables 7T ---) ,\71" ( m) ,\71" ( n) are well controlled. In practice, these expectations
are to be "close" to the Dirac symbol 6m=n, a property of the family that we name
approximate orthogonality. One of the purposes of the next lectures is to describe
several families of arithmetic objects satisfying the "approximate orthogonality''
property for m and n in appropriate ranges. Assuming that approximate orthogo-

nality holds for F and m, n « Xk (which is the hard step, especially when dealing

with the ScP), one can then derive a bound of the form

μF( {7ro} )JL:;,,(7ro, X)l^2 k « L

m,n«Xk

Tk,11"o(m)Tk,11"o(n) s: ,,.-,,.- (Q X)e:

y r,;;;;;;; ,,.,. Um=n "-"-o , k 11"0 •

1.4. Appendum: bounds for local parameters via families of £-func-

tions

In this section we complete the proof of Theorem 1.1, following Luo/ Rudnick/
Sarnak [LRS]. Hopefully this will provide the first concrete example in these lec-
tures of the usefulness of families in analytic number theory.
We present the bound for the local parameters at infinity assuming that 7T 00 is
unramified (i.e. spherical). By unitarity of 7T, it is sufficient to show that

1 1

Max~eμ71",i ~ Bd = 2° - d 2 + 1.

From (1.3), we see that L 00 ( 7T 0 7r, s) has a pole at s = f3o := 2 Max ~eμ11" ,i, hence
it is sufficient to show that L(7T 0 ir, s) does not vanish for s > 2Bd = 1 - d2~ 1 ,
since ( s - l)L 00 ( 7T 0 ir, s )L( 7T 0 ir, s) is entire there. Actually, instead of considering
this L-function alone, one considers a whole family of L-functions: namely the
L (x. 7T 0 7r, s), where x. 7T : = 7T 0 x ranges over the twists of 7T by caracters of AQ / Q x
trival at oo. These correspond to even primitive Dirichlet characters, which by an

(^3) This is pretty straightforward to check ford= 1, 2, but for higher d, a good control on Tk,.rro (n) seems
to require at least (1.23).