Automorphic Forms on GL(2) Part II by Hervé Jacquet (auth.)

By Hervé Jacquet (auth.)

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17). Finally there is no harm in replacing the pair pair (~I ® X'n2 ® X-l) X where (~13~2) is a quasi-character of particular, by taking the order of X by the ~ . In to be large enough~ we may as- sL~ne furthermore that: L(S,~l) = L(S,~l) = L(s,~2) = L(s,~2) = I and ¢(S,~i) = q'2SU¢(0~i) where u , 0~2) = 0(~2) = u is an integer strictly larger than c,d~z and v. (of. 8). 2) will be the following lemma. measure on ~ 2 is convergent for value is ~iven by Res e ~ Rx , = e(1-s, i) be the Haar measure of dx for large enouKh.

Or P0 = 0 and are relatively P0,Q0 P0 = I . 5) follows that Then L(S,~) = Qo(q-S) -I is the unique Euler factor satisfying the conditions (11 and (2). is, similarly, a unique Euler factor conditions. L (s,~) Then there is a function of equation (31. s satisfying the two first which satisfy the functional By (11 and (2) it is a polynomial in there is a similar factor ¢(s,~,\$) . There q -s and q Exchanging the roles of s ~ But and we find ~(s,~,~) ¢(l-s,~,~) Hence ¢(s,~,\$) must be a monomial. 8. 5) ¢(s,~,~) = ~ (bx) (b E FX) is replaced by ~(s,~,~t)= w2(b)Ib l4(s-~) s(s,~,~) .

Which, on the left. 2). k s a is a billnear form satisfying the assumptions of Lemma So there is a function c(s) of s , defined for Res large enough, so that ks = c(S)~s Coming back to the definition of 5 s and Vs we see that~ for large enough. ~)Idet gl ~ . 2) we find that Res -19- ~(l-S,Wl,W2,#) = for Res c(s)T(s,Wi,W2,~) large enough. 3) it is enough to show that is a rational function of q-S c(s) To do that it will suffice to apply the following lemma. 5: for all One can find W i , i = 1,2 and ~ i__nn g(F 2) so that, s , T(s,WI,W2,~) = 1 .