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

By Hervé Jacquet (auth.)

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15: ~(~l,Vl) We assume that ~I is absolutely cus~idal or ~i = ~ and all the p~oducts ~iV2 , ~I~2 , Vl~ 2 , VlV 2 , are ramified. For ~0i in ~(~i,~) (or characters of j n R x) in Z and all quasi-characters X of we set ~i(n,x) = ~RX ~i(~n)x(e)d~ where de is the normalized Haar measure of transform of ~0i The operator ~i(w) RE . The formal Mellin is the family of formal series is defined by the formula A -I -I ~i(w)~i~,x) = ci(×,x)~i( x -I -I w i ,x ~i (w)), where ci(x,x) =I ci(n'x)x= is a suitable family of formal series.

W-l(d) = 0 if k E K' , otherwise. Hence we find (S,Wl,W2,~) = (meas R×) 2 . The lemma follows. As usual, we shall need more precise results. 8: There are Euler factors L(s,~) with the an__~d e(s,~) following properties. (I) Then (2) Set T(S,Wl,W2,~) = L(s,n)E(s,WI,W2,~) , ~(s,WI,W2,~) = L(s,~)E(s,WI,W2,~) • E(s,WI,W2,~) and ~(s,WI,W2,#) One can choose families i i i ~(s,WI,W2,~ ) = I W Ii , W 2i (resp. ~C are polynomi@is in _and _ #i q -s and q s so that i i i I) =(s,WI,W2,~ ) = • l (3) form There is a function cq -si e(s,n,$) which, as a function of s , has the so that (1-s ,W 1 ,W2 ,~) = w 2 (-1) e (s, ~, ~)E (s ,W1 ,W 2 ,~ ) .

As for ~I its support is contained and it is therefore invariant under CI ~i 0 ~ where x E R -39So we find m(-I)~(s,WI,W2,~ ) = ~l (w)~t (a)~2 (-a) I~ is-1 dXa . 10o2) we see that the integrals in t are extended, in fact, to t E ~-URX and x ~ ~UR . and x Hence w(-1)~(1-S,Wl,W2,~) = ¢(2s-1,~) x ~r /i x Again we use the fact that ~ 9°2 is we observe that the support of ~'UR - {0} . ~I 1 K ~w]~ 2 (-~) LaI"s w "I (a)dX~ invariant. 12: We assume that ~i = ~ I ~ i ) and ~2 = ~ 2 " ~ 2 ) with -40u I = 0(w) = 0(~1~2) = 0(~1) > u 2 = 0(~2 ) , o(~1) = o(~2) = o .

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