1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. MOSTLY SL(2) 53

There is an obvious notion of upper density, and Q(S) ;::: 1 - J(P - S); in fact

Q(S) + J(P - S) = 1; finite sets of primes have density 0.

Theorem 1.3 (Ramakrishnan). -Q(S1) 2: 190.

This has been improved by Kim and Shahidi who obtain Q 2: ~:. See [48b,

34b]. Since the basic argument is quite simple, we give a proof that Q 2: ~-

Recall that the (incomplete) £-function off is

LN(s j) =IT 1

' pfN (l - app-s)(l - /Jpp-s).

It is meromorphic in the s-plane, and holomorphic at s = 1. (See Cogdell's
lecture for the complete £-function). It is known by the work of Rankin and others
that

LN (s, f x f) =IT - -s -s

1

• -s -s
  pfN (1 - apapp )(1-ap/JpP )(1 - ap/JpP )(1 - /3p/3pP )

is also meromorphic. Moreover, it has a simple pole at s = 1, with > 0 residue.
Consider, in the usual fashion, and for s real , 1 +, the function A ( s)
logLN(s). By standard arguments, A(s) grows for s , 1+ like its term "of degree
l" involving the first power of p-^8 :

A(s) = 2:>-~°X~p-s +(convergent near 1).

p

Here >.~ = ap + /Jp = p-^1 /^2 Ap· On the other hand, by the meromorphic
behaviour of LN,

A(s) = -log(s - 1) +(bounded near 1).

Let T = P-S. For p ET, i>-~I > 2. Thus, for s ____, 1+:

• log(s - 1) "'A(s) 2: L l.A~l^2 p-s ( + Cst)
  pET
  (+ Cst).

This implies J(T) :::; t, q.e. d.

Note that for any c > 0 the same argument yields

(1.13)

This is of interest in view of the Sato-Tate conjecture, according to which the
ap should by equidistributed on the circle z = eilJ for the measure ~ sin 282 - except
for very particular forms. Amongst these are the forms f with A (eigenvalue of the

Laplacian) = t, conjectured to correspond to Artin representations of Gal(Q/Q)

with even determinant. This is again compatible with (1.13), by the following

(^2) Since only {ap,<ip} = {ap,,Bp} is determined we may take() E [0,7r].