Download PDF by Murakami M.: A bound for the orders of the torsion groups of surfaces

By Murakami M.

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We have We isolate a part of the last expression. If v is as before and if T E Hom,$V,, H J , then set Notice that this integration is actually taking place in a finite dimensional space. 5. 4 (2) implies that if (2) , T = T, CP2lPl(LJ,Y , v ) = det(JP2(PI(v)IIv(Y))' then = det(Ap2,pI(v,7 ,v ) ) d ( Y ) , with d ( y ) = dim V,, as usual. 13. 4. A Generalization of L Cohn’s Determinant Formula in that number and set K , the map = K nP, K , =K 35 n P , (= K n P,). 2. 13 ( 1 ) translates into A P ~ ~ P I ( ~ , T , v )8 ~ (ST ) = ~ ( ( A * ~ , ~ * ~ I ( o , ~ , ~8VS) ) T ) for T E Hom,$W,,(H,),), S E Hom,l(V,, W,).

The product formula for minimal parabolic subgroups was given in the form of the material of Section 1 by Schiffman [l]. His work involves a generalization of the method of Gindikin-Karpelevic [ 11for the Harish-Chandra c-function. In this method, the calculation is reduced to reductive groups of R-rank 1 rather than to parabolic subgroups with one 48 10. IntertwiningOperators dimensional split components. In essentially the same generality it can be found in Knapp-Stein [2] and Harish-Chandra [16].

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.

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A bound for the orders of the torsion groups of surfaces with c2 1 = 2x - 1 by Murakami M.


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