LECTURE 5. CONVERSE THEOREMS 151I(s; U€,ip') = ( W€ (~ IO ) W~,(h)I det(h)ls-(n-ml/^2 dh
}Nm (A}\ GLm(A} n-m=II r w€v (hv
0
) w~, (hv)I det(hv)l~-(n-m)/^2 dhv
v }Nm(kv}\ GLm(kv} Q Jn-m v=II Wv(s; w€v> w~,J.
v
Note that unless 1f^1 is generic, this integral vanishes.
Assume first that 1f^1 is cuspidal. Then from the local theory of L-functions from
Lecture 3, for almost all finite places we have Wv(s;W€v> W~,J = L(s,ITv x H'v)
and for the other places Wv(s;W€v>W~,J = ev(s;W€v>W~,JL(s,ITv x 1f^1 v ) with
the ev(s; w€v' w~, J entire. So in this case I(s; U€, ip') = e(s)L(s, IT x 1f^1 ) with e(s)
entire. Since L(s; IT x 1f^1 ) is assumed nice we may conclude that I(s; U€, ip') has
an analytic continuation to an entire function. When 1f^1 is not cuspidal, it is a
subrepresentation of a representation that is induced from (possibly non-unitary)
cuspidal representations ai of GLr, (A) for Ti < m with 2:.: Ti = m and is in fact,
if our integral doesn't vanish, the unique generic constituent of this induced rep-
resentation. Then we can make a similar argument using this induced representa-
tion and the fact that the L(s, IT x ai) are nice to again conclude that for all H',
I(s; U€, ip') = e(s)L(s, IT x H^1 ) = e'(s) TI L(s, IT x ai) is entire. (See [9] for more
details on this point.)
Similarly, consider I(s; V€, ip') for ip' E V7r' with H^1 an irreducible subrepresen-
tation of the space of automorphic forms on GLm(A), still with
I(s; v€, <p^1 ) = 1 IP'~~ (h 1) ip'(h)I det(h)ls-^1 /^2 dh.
GLm(k)\ GLm (A)Now this integral converges for Re(s) << 0. However, when we unfold, we find
I(s;V€,<p^1 ) =II Wv(l - s;p(wn,m)W€v> w~,J = e(l - s)L(l - s,fi x ;')
as above. Thus I(s; ~' ip') also has an analytic continuation to an entire function
of s.
Now, utilizing the assumed global functional equation for L(s, IT x 1f^1 ) in the
case where 1f^1 is cuspidal, or for the L(s, IT x ai) in the case 1f^1 is not cuspidal, as
well as the local functional equations at v E S 00 U Sn U S7r' US.,μ as in Lecture 3 one
finds
I(s; u€, ip') = e(s)L(s, IT x 1f^1 ) = e(l -s)L(l - s, fix;')= I(s; v€, <p^1 )
for all ip' in all irreducible subrepresentations 1f^1 of GLm(A), in the sense of analytic
continuation. Then an application of the Phragmen-LindelOf principle implies that
these functions are bounded in vertical strips of finite width. This concludes our
use of the L-function.
We now rewrite our integrals I(s; U€, ip') and J(s; V€, ip') as follows. We first
stratify GLm(A). For each a E Ax let GL~(A) = {g E GLm(A) I det(g) =a}. We
can, and will, always take GL~(A) = SLm(A) · (a Im_J. Let
(IP'~U€, ip')a = ( IP'~U€ (h
1
) ip'(h) dh
JsLm(k}\ GL~(A}