1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. ANALYTIC PROPERTIES OF INDMDUAL £-FUNCTIONS 203

where S is a (sufficiently large) product of primes containing the first few primes

and the ramified primes for 1r. Indeed, one has for ~es » 1,

L( w, s ) = ( '°' L..., μ 2( n ) >.'lr(n) )L s ( w, s ) IJ Lp(w, >. s) < ) ,

(n,S)=l n s pgS (1 + ; .P )

say; by Proposition 1.1 the two rightmost factors converge absolutely and are uni-

formly bounded for ~es ): 1 + J the implied constant depending only on J and d

(granted that ISi is sufficiently large with respect to d). By the trivial bound, and

the positivity of the Dirichlet coefficients >.7r 0 ir ( n), one has

μ^2 (nS)l>.7r(n)I :( 1 + μ^2 (nS)l>.7r(n)l^2 = 1 + μ^2 (nS)>.7r 0 ;r(n) :( 1 + >.7r 0 ;r(n).

Since L( w ® ii-, s) is uniformly bounded for ~es ): 3, the functional equation (1.4)

and the convexity principle imply that

L(w ®ii-, 1 + J) « 0 Q~ 0 ir

for some absolute A > 0. Moreover, one can prove (see [BH, RS2]) that Q: 0 ir :(
Q~ for some B depending on A and d (in fact B = 2dA is sufficient); it follows
that

(1.27) '°' L..., μ 2 ( ) n I >.'Ir nl+o ( n) I :( ( ( 1 + u s:) + L ( w ® w, - 1 + J ) «o,d Q'lr. B

(n,S}=l

This is the initial bound and we are going to improve it by bootstrapping. By mul-
tiplicativity of the arithmetic function >.'Ir ( n ), one has, for n 1 and n 2 two squarefree
integers, >.7r(n1)>.7r(n2) = >.7r(m)^2 >.7r(n 1 n 2 /m^2 ), where m = (n 1 ,n 2 ). Hence,

(

'°' 2( )l>.7r(n)l)

2

:o::::'°'μ

2

(m)l>.7r(m)l

2

)( '°' 2() ( )l>.7r(n)I)

L..., μ n n Ho "' L..., m 2+20 L..., μ n^7 n n 1+0

(n,S)=l m (n,S}=l

(

'°' 2 l>-7r(n)I) B

«o,d L..., μ (n) nl+o/2 «o,d Q ,

(n,S}=l

the latter inequality being deduced from Prop. 1.1 and (1.27). Hence, (1.27)
holds (up to changing the implied constant) with B replaced by B /2. Iterating the
process, we get (1.26), after O(l log(B/c)I) steps. 0

1.3.1. The Subconvexity Problem

In view of (1.22) one has the:

Subconvexity Problem (ScP). Find J > 0 (depending on d only) such that for any
automorphic cuspidal representation w of degreed, and ~es= 1 /2, one has

(1.28) L(w, s) «d Q7r(~ms)^114 -^0 ,

the implied constant depending on d.

Alternatively, one can also measure the size of L(w, s) with respect to three
quantities separately: the "height" of s, Jl/4 + 1tl^2 , the "arithmetic" conductor q'lr,

or the "conductor at the infinite place": q 00 := Tii=l...d(l + lμ7r,il). Thus one can

consider three weakened variants of the ScP and seek a subconvex exponent for
only one of these parameters, the others remaining fixed - eventually one can also
ask for polynomial control on the remaining parameters, if one is greedy. These