By Hervé Jacquet (auth.)
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Extra resources for Automorphic Forms on GL(2) Part II
Where we set fl(a) = ~ v R ~I 0 = 0~11~2) = O(m) , 0(~;I ) = 0(92) = 0 . z corresponding to ~oi (y) " defined by the following conditions: ~0l(ae) = ~0l(a)~ I(¢) for ¢ in RX , [ ~iCw)~1(a)lalS-½dXa= LCS,~l)e(l-s,~I) ; ~02(a¢ ) = q02(a) for ¢ in R× , ~°2(a) lalS-~dXa = LCs,v2)¢Cl-s,~2 I) , In ~(s,WI~W2,~) we integrate for t in Rx and x in R X ° We find in that way ~0(-I)~ (s ,WI,W 2,~) = ~ ~I (w)~°l (a)~2 (w) f2 ('a)dXa where f2 is the element of f2(a ) = ~R x ~2~I ~(~2,~) defined by x~,,)V, "~ 2-I (x)dx ~P2(a) .
The integrals . 6). ~I W1 The above results, suitably modified, large enough, are rational functions of If Then are special, a similar remark can be made. -23§15. Explicit computations The purpose of §15 is to prove the following theorem. 1: rations of Let ~i ' i = 1,2 GL(2~F) . Let be two admissible irreducible represen- ~ be ~I X ~2 " Assume that neither ~I nor ~72 is one dimensional. Y2 = ~F L(s,n) = L(s,~ I ® ~2 ) then , L(s,~) = L(s,~ I ® 92 I) , ¢(s,~,~) = ¢(S,~l~2,~)¢(S,~l~2,~)L(S,nl~2)'IL(l-S,~l~2 I)- • The theorem implies the following result.
S,WI,W2,~) = L(s,~I~2)L(S,~I~2)L(s,~I~2)L(S,VlV2) (2) which is Then . We have L(s,~) = L(S,~l~2)L(s,~iv2)L(s,Vl~2)L(S,VlV ¢(s,~,~) (3) R2 If s = s(s,~j) = I 2) , . x , -33Then q0i(¢a) = %0i(a) for ¢ in RX . 'n n l_a aa1'btb" X2 = (~i-a'a"X) (l-a'b"X) (l-b'a"X) (l-b'b"X) We leave the proof to the reader as a refreshing exercise. (Hint: decompose the rational fractions into their simple elements). The quasi-character = WlW 2 = ~iui~2~2 is unramified. It has the form ~D = ols . 2) is true for that particular choice of all choices.
Automorphic Forms on GL(2) Part II by Hervé Jacquet (auth.)