LECTURE 4. HOLOMORPHY AND BOUNDEDNESS; APPLICATIONS 325Corollary [K4]. The representation Sym^4 (n) is automorphic, where
Sym^4 (n) = ®vSym^4 (nv)
and 7f = ®v1fv is a cusp form on GL2(Ap).Proof. Apply A^2 to Sym^3 n;
A^2 (Sym^3 n) = Sym^4 n ® w11" EB w;
The reservations at vl2 and vl3 can be removed. DProposition 4. 7 (Kim-Shahidi [KS3]). Sym^4 ( n) is cuspidal unless n is of di-
hedral, tetrahedral or octahedral type.
Theorem 4.6 is proved by using
G = Spin 2 k+8> MD = SLk+1 x SL4, k = 0, 1, 2, 3.
We get L(s,n®CJ, A^2 p 4 ®Pk+I) for each cusp form CJ on GLk+i(Ap), k = 0, 1 ,2,3.
Applications:
Theorem 4.8 (Kim-Shahidi [KS3]). Let F be an arbitrary number field. Let
7f = ®vnv be a cusp form on GL2(Ap). Assume 1fv is parametrized bytv = ( ~v iJ E GL2(C).
Then
q:;;^1!^9 < lavl and l.Bvl < qy^9.
Similar inequality holds at archimedean places (.A> 0.23765432 ... ).This is proved using the techniques of [Sh2] (which led to 1/5 using Sym^2 n and
groups of either type F 4 or E6, by applying a general theorem from [Sh2] which
implies L(s, 1fv, Sym^5 (p 2 )) is holomorphic for Re(s) 2: 1 for all such v). When this
general theorem is applied to a simply connected group of type Es with a Levi M for
which MD = SL 5 x SL 4 , together with a representation related to Sym^4 n® Sym^3 n
leading to L(s,nv,Sym^9 (p 2 )), one gets 1/9. We refer to [CPSS] for an important
application of Theorem 4.8. Next, we have
Theorem 4.9 (Kim-Sarnak [KSa]). Suppose F = Q, then
p-7/64::; lapl and I.BPI ::; p7/64.
At the archimedean places we get the estimate A 2: J; 956 = 0.2380371 for the first
positive eigenvalue of 6. , the hyperbolic Laplacian.
Proof. This is proved by means of analytic methods of Duke-Iwaniec [DI] applied
to L(s, Sym^4 n, Sym^2 ), (cf. [BG]) along the lines of Bump-Duke-Hoffstein-Iwaniec
[BDHI] which led to 5/28 +cover Q, when applied to L(s, Sym^2 n, Sym^2 ). D
Quite a bit more work than what appeared in [BDHI] is carried out in [KSa],
both from the point of view of analytic number theory and L-functions. In fact,
using the twisting by a highly ramified character 'r/ which provides us with the nec-
essary analtyic properties of L(s, Sym^4 (n®ry), Sym^2 ) by means of our method, one
no longer needs to deal with the difficulties arising from the lack of our knowledge
of local L-functions at ramified and archimedean places that one has to take into
account if one uses the method of Rankin- Selberg as in [BG].