LECTURE 5. CONVERSE THEOREMS 157conductor of IIv. Then the ideal n = f(II) = ITv P~" Co is called the conductor of
II. For each place v we fix a non-zero vector~~ E IIv which is fixed by K 1 ,v(P~"),
which at the unramified places is taken to be the vector with respect to which the
restricted tensor product II= @'IIv is taken. Note that for g E Ko,v(P~") we have
IIv (g )~~ = wn,, (gn,n)~~ ·
Now fix a non-empty finite set of places S, containing the archimedean places
if there are any. Then the compact subring ns = ITv~S P~" C ks, or the ideal
it determines ns =kn ksns c os, is called the S-conductor of II. Let Kf (n) =
f1v~sK1,v(P~") and similarly for Kg(n). Let C = ®v~s~~ E IIs. Then this vector
is fixed by Kf (n) and transforms by a character under Kg(n). In particular, since
ITv~sGLn-1(ov) embeds in Kf(n) via hr---; (h 1 ) we see that when we restrict
IIs to GLn-l the vector C is unramified.
Now let us return to the proof of Theorem 5.3 and in particular the version
of Proposition 5.1 we can salvage. For every vector ~s E IIs consider the func-
tions UE.s®E.o and l/(s®f,o. When we restrict these functions to GLn-l they become
unramified for all places v tt S. Hence we see that the integrals I(s;UE.s®f.o,<p^1 )
and I(s; vf,s®f,O) cp') vanish identically if the function <p^1 E v11', is not unramified for
v tt S, and in particular if cp' E V11'' for 7r^1 E T(n - 1) but 7r^1 tt Ts(n - 1). Hence,
for vectors of the form ~ = ~s 0 ~^0 we do indeed have that I(s; Uf.s®f,o, cp') =
I(s;Vf.s®f.o,<p') for all cp' E V11'' and all 7r^1 E T(n -1). Hence, as in Proposition
5.1 we may conclude that Uf,s®E,o(In) = VE.s®E,o(In) for all ~s E Vn 5. Moreover,
since ~s was arbitrary in Vn 5 and the fixed vector ~^0 transforms by a character
of Kg ( n) we may conclude that UE.s®E.o (g) = VE.s®E.o (g) for all ~s E Vn 5 and all
g E GsKg(n).
What invariance properties of the function uf.s®f,O have we gained from our
equality with vf,s®f,O. Let us let ri(ns) = GLn(k) n Gs Kf (n) which we may view
naturally as congruence subgroups of GLn(os). Now, as a function on Gs Kg (n),
UE.s®E,o(g) is naturally left invariant under fo,P(ns) = Z(k) P(k) n Gs Kg(n) while
'\l(s®f.o (g) is naturally left invariant under fo,q(ns) = Z(k) Q(k) n Gs Kg (n) where
Q = Qn-l · Similarly we set f1,dns) = Z(k) P(k) n Gs Kf (n) and f1,q(ns) =
Z(k) Q(k) n Gs Kf (n). The crucial observation for this Theorem is the following
result. ·
Proposition 5.2. The congruence subgroup ri(ns) is generated by the subgroups
fi,P(ns) and fi,Q(ns) for i = 0, l.
This proposition is a consequence of results in the stable algebra of GLn due
to Bass which were crucial to the solution of the congruence subgroup problem for
SLn by Bass, Milnor, and Serre. This is reason for the restriction to n 2: 3 in the
statement of Theorem 5.3.
lF'rom this we get not an embedding of II into a space of automorphic forms
on GLn(A), but rather an embedding of IIs into a space of classical automorphic
forms on Gs. To this end, for each ~s E Vn 5 let us set
q.f.S (gs)= UE,[email protected] ((gs, 1 s)) = vf,s@f,O ((gs, 1 s))
for gs E Gs. Then f.s will be left invariant under f1(ns) and transform by a
Nebentypus character xs under fo(ns) determined by the central character wns
of IIS. Furthermore, it will transform by a character WS = WIIs under the center