1549380232-Automorphic_Forms_and_Applications__Sarnak_

156 J.W. COGDELL, £-FUNCTIONS FOR GLn

5.4.2. The case of non-empty S

Let S be our non-empty set of finite places of k. Since we have restricted our

ramification at the places in S , we no longer know that L(s, II x 7r^1 ) is nice for

all 7r^1 E T(n - 2) and so Proposition 5.1 a bove is not immediately applicable. In

this case, for each place v E S we fix a vector ~~ E Vrrv as in the above Lemma.

(So we must assume we have chosen 'ljJ so it is unramified at the places in S.) Let

~s = I1vES ~~ E IIs. Consider now only vectors ~ of the form e ® ~s with ~s

arbitrary in Vrrs and ~s fixed. For these vectors, the functions lP'~_ 2 U~ (h 1 ) and

lP'~_ 2 ~ (h

1

) are unramified at the places v ES, so that the integrals I(s; U~, cp')

and I(s;~,cp') vanish unless cp'(h) is also unramified at those places in S. In

particular, if 7r^1 E T ( n -2) but 7r^1 rf-ys ( n -2) these integrals will vanish for all cp' E

V7r'· So now, for this fixed class of~ we actually have I(s; U~, cp') = I(s; V~, cp') for

all cp' E V7r' for all 7r^1 E T(n - 2). Hence, as before, lP'~_ 2 U~(In_i) = lP'~_ 2 ~Un-1)

for all such ~.

Now we proceed as before. Our Fourier expansion argument is a bit more

subtle since we have to work around our local conditions, which now have been

imposed before this step, but we do obtain that U~(g) = V~(g) for all g E G' =

(I1vES Koo,v(P~v)) Gs. The generation of congruence subgroups goes as before. We

then use weak approximation as above, but then take for II' any constituent of

the extension of II^8 to an automorphic representation of GLn(A).There no use of

strong multiplicity one nor any further use of the £-function in this case. More

details can be found in [12].

5.5. Remarks on the proof of Theorem 5.3

Let us now sketch the proof of Theorem 5.3. Details can be found in [9].

We fix a non-empty finite set of places S, containing all archimedean places,

such that the ring os of S-integer has class number one. Recall that we are now

twisting by all cuspidal representations 7r^1 E Ts(n - 1), that is, 7r^1 which are un-

ramified at all places v rf-S. Since we have not twisted by all of T(n - 1) we a re

not in a position to apply Proposition 5.1. To be able to apply that, we will now

have to place local conditions at all v rf-S.

We begin by recalling the definition of the conductor of a representation IIv

of GLn(kv) and the conductor (or level) of II itself. Let Kv = GLn(ov) be the

standard maximal compact subgroup of GLn(kv)· Let Pv C Ov be the unique prime

ideal of Ov and for each integer mv ;:::: 0 recall that

and Ki,v(P;;-'v ) = {g E Ko,v(P;;-'v ) I 9n,n = 1 (mod p;;-'v))}. Note that for mv = 0
we have Ki,v(P~) = Ko,v(P~) = Kv. Then for each local component IIv of II there
is a unique integer mv ;:::: 0 such that the space of K 1 ,v (p;;-'v )- fixed vectors in IIv
is exactly one. For almost all v , mv = 0. We take the ideal p;;-'v = f(IIv) as the