1549380232-Automorphic_Forms_and_Applications__Sarnak_

78 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS

Now return to the restriction (3.15). It follows from base change that 7rE,oo - a

representation of SL(2, q -is infinite-dimensional and so contains (under H(JR) =

SU(2)) representations 700 ';/!. C. By the restriction (3.15) and Lernrna 3.14, we

have:

Lemma 3.15. - If p occurs in (the support of the) restriction 7rE,plHp, p is tem-

pered.

However, this restriction is a tensor product: recall that 7rE,p = 1fp ® np, which

we restrict to the diagonal SL(2, Qp) - i.e., we take the "inner" tensor product. Its

support has been determined, as a function of 1fp, by Asmuth and Repka.

Assume 1fp is not tempered, so its Hecke matrix^8 is tp = (pa p-a) (a E

JR, 0 < (T < ~).

Theorem 3.16 (3).

(i) If a ::; ~ 1fp ® 1fp has tempered support.

(ii) If a > ~ it does not, in fact it contains (discretely) 7r( a') where a' =

2a - ~ > 0.

Of course 7r( a') is the representation with Hecke matrix (pa' p-a'). If a > ~,

we see that Theorem 3.16 implies a contradiction with Lemma 3.15, so we are done.

We now justify the title of this section.

We have used essentially:

• The Ramanujan conjecture for holomorphic forms (1973)


• The Jacquet-Langlands correspondance (1970)


• Base change for E /Q.
  Although we quoted Langlands's general theory for base change, the quadratic
  case is much easier and was proved by Jacquet in 1972 [30] by use of the converse
  theorem. The reader will easily understand his proof after reading part of Cogdell's
  lectures.

(^8) 7rp is an induced representation ind~L(^2 ,QP\x) where x(x) is a character of x E «J!i ~
p
(x x-l) E SL(2,Q!p)· We set tp = (X(P) X(P)_ 1 )= Hecke matrix of the original repre-
sentation G L2. Conversely t p defines 7r P.