Download e-book for kindle: Automorphic Forms on GL(2) Part II by Hervé Jacquet (auth.)

n = -a-b , C2(-a-b,X ) = (l-q'l)21~2(~-a)v2(~-b)~(7~2,m-a)~(Xv2,m -b) .

If n2 ~2 = 92 we may exchange the roles of ~i and ~s(%01,%02) is a and reduce ourselves to the previous case. Finally, if ~I = ~i and ~2 = 92 we see that sum of terms of the form f(a)~l~ 2(a) lal s vi(a)dXa where i = 0,1,2 nomial in q-S and and f E S(F) . qs If ~i~2 If ~i~2 is ramified this is a poly- is unramified this has the form L (S,~l~2)3 P(s) where P belongs to C[q'S,q s] . (Cf. 1). Hence the first -27assertion of the lemma follows. Assume now Chat prime. There are L(s,~ I ® ~2 )'I P and Q in and C[q-S] L(s,~ I ® v2)'l are relatively so that L(s,n I ® ~2)L(s,N I ® 92 ) = PL(s,~ I ® ~2 ) + QL(s,n I ® v 2) • There are also %OI and %o2 ; %o1(a)~2(a) tal s'~ i/ f q:)l(a)P'2 (a) and f in g(F) in ~(~i,~) d×a = lal s'~ dXa so that PL(s,rr 1 ® ~2) , _- QL(s,N I ® ~2) , so that If we set now I # ~2(a) = f(a)P,2(a)[al ~ , %o2(a) = f(a)v 2(a) lal~ we find t ,) + ~ s~%OI,%O2 .

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