158 J.W. COGDELL, £-FUNCTIONS FOR GLnZ(ks) of Gs. The requisite growth properties are satisfied and hence the map
~s t--7 <I?€s defines an embedding of IIs into the space A(fo(ns)\ Gs;ws,xs) of
classical automorphic forms on Gs relative to the congruence subgroup fo(ns)
with Nebentypus xs and central character ws.
We now need to lift our classical automorphic representation back to an adelic
one and hopefully recover the rest of II. By strong approximation for GLn and
our class number assumption we have the isomorphism between the space of clas-
sical automorphic forms A(fo(ns)\ Gs; ws, xs) and the Kf (n) invariants in the
space A(GLn(k)\ GLn(A);w) where w is the central character of II. Hence IIs
will generate an automorphic subrepresentation of the space of automorphic forms
A(GLn(k)\ GLn(A);w). To compare this to our original II, we must check that, in
the space of classical forms, the <I?i; 50 i;o are Hecke eigenforms for a classical Hecke
algebra and that their Hecke eigenvalues agree with those from II. We do this only
for those v tJ_ S which are unramified, where it is a rather standard calculation. As
we have not talked about Hecke algebras, we refer the reader to [9] for the details.
Now if we let II' be any irreducible subrepresentation of the representation
generated by the image of IIs in A(GLn(k)\ GLn(A);w), then II' is automorphic
and we have II~ '.:::::'. IIv for all v E S by construction and II~ '.:::::'. IIv for all v tJ_ S' by
the Hecke algebra calculation. Thus we have proven Theorem 5.3.5.6. A useful variant
For the applications of any of these Converse Theorems to the problem of lifting
of automorphic representations to GLn, which we will take up in the next Lecture,
the following simple variant of these theorems is extremely useful [13]. If T is
one of the twisting sets from above and rJ is a fixed idele class character, we set
T ® T) = { 1f^1 I 7f^1 = 1fb ® T) with 7fb E T} where we view T) as a character of any
GLm by composition with the determinant.
Observation 5.1. Let II be as in Theorem 5.1, 5.2, or 5.3. Suppose that T) is a
fixed character of kx \Ax. Suppose that L(s, II x 7r^1 ) is nice for all 7f' E T ®T), where
T is any of the twisting sets of those theorems. Then II is cuspidal automorphic or
quasi-automorphic as in those theorems.
The only thing to observe, say by looking at the local or global integrals, is that
if 7fb ET then L(s, II x (7rb ®TJ)) = L(s, (II®TJ) x 7rb) so that applying the Converse
Theorem for II with twisting set T ® T) is equivalent to applying the Converse
Theorem for II® T) with the twisting set T. So, by either Theorem 5.1, 5.2, or 5.3,
whichever is appropriate, II® T) is cuspidal automorphic or quasi-automorphic and
hence II is as well.
- Global fields of characteristic p -f. 0
When the global field k is of characteristic 0, that is, is a number field, the state-
ments of the Converse Theorems we have given in terms of the analytic properties of
the £-functions are the most appropriate and applicable. However when the global
field k is of characteristic p # 0, that is, the field of functions of a curve over a finite
field, while the statements we have presented are still true, it is most appropriate
and useful to have the Converse Theorems stated in terms of the global £-functions
as rational functions, as was done in Piatetski-Shapiro's original paper [65]. While