By Wu X.
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Extra resources for 8-ranks of Class Groups of Some Imaginary Quadratic Number Fields
Rn, c i , d i , i = 1,. . 2 implies that an estimate of the desired form is satisfied for some fixed 6 > 0 and all v E a*,(6) with llvll 2 T for some T > 0. Since the set (v E azl llRe vII I6, llIm vIJ I TI is compact and since p ( u , v ) is holomorphic at all v with Re v = 0, the result now follows. Note. Since Ip,,, is generically irreducible, Vogan’s minimal K-type theorem (Vogan 111) implies that there exists y E I? with multiplicity one in I,. 6. 1. The literature on the intertwining integrals is vast and no doubt we will do a disservice to some researchers in the subject.
U is surjective and V is injective. Furthermore, U is a (g, K )-module homomorphism of IP,a,,+h 8 F* to If,,,, and V is a (g, K)-module homomorphism of IF,,, ,to IF,~,, 8 F*. 4)we see that U is surjective. If V ( f )= 0, then C S(f 8 ui) Q u? = 0. Thus, S(f 8 u ) = 0 for all u E F . 5 implies that f = 0. The intertwining assertions follow from the corresponding intertwining assertions for T and S . Let P,(v) be the projection of IP,o,,+h Q F* onto (IP,a,u+h 8 F * ) x ~ +and y let Q 2 ( v )be the projection of IF,,,,+^ 8 F* onto IF,,,,+^ @F*)*A+Y.
U(ffTxffv) = Horn,( I " ( Y ) , J " ( 7 ) ) * It is now clear that if f,,,(v> r u ( 2 )=fI,. 4. # 0 for all y , T E k, and if f E Z, then Lemma. There exists a meromorphic function qf,, on a: such that JfIP( 4Jqf ( 4 = (Pf,Av)l, wherever the left hand side is defined. We use the notation of the previous number. Let y E k,. Let v be such # 0. Thus, Schur's lemma implies that E , J P , P ( v ) J F , f ( v ) E ,= (PyWE, * Since dim( I,( 7 1)P,< v ) = tr( E, Jf IF( v ) JPl f ( v ) E , ) for such v, Q, extends to a meromorphic function in v.
8-ranks of Class Groups of Some Imaginary Quadratic Number Fields by Wu X.