MODULAR FORMS-page98

MODULAR FORMS-page98 - 94 LECTURE 9. ABSOLUTE INVARIANT AND...

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Unformatted text preview: 94 LECTURE 9. ABSOLUTE INVARIANT AND CROSS-RATIO Theorem 9.1. The cross-ratio R defines a bijective map R : X/ GL(2 , C ) → C \ { , 1 } . Proof. Let ( x 1 ,x 2 ,x 3 ,x 4 ) ∈ X . Solving a system of three linear equations with 4 unknowns a,b,c,d we find a transformation g : ( x,y ) → ( ax + by,cx + dy ) such that g · ( a 2 ,b 2 ) = (1 , 0) , g · ( a 3 ,b 3 ) = (0 , 1) , g · ( a 4 ,b 4 ) = (1 , 1) , g · ( a 1 ,b 1 ) = (1 ,λ ) , for some λ 6 = 0 , 1. We recall that two proportional vectors define the same point. This allows us to choose a representative of each orbit in the form ( λ, , ∞ , 1), where we now identify points in P 1 ( C ) \{∞} with complex numbers. Since the cross-ratio does not depend on the representative of an orbit, we obtain from (9.1) R ( x 1 ,x 2 ,x 3 ,x 4 ) = λ. Since λ takes any value except 0 and 1, we obtain that the image of R is equal to C \{ , 1 } . Also it is immediate to see that λ and hence the orbit is uniquely determined by the value of R...
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This note was uploaded on 01/08/2012 for the course MATH 300 taught by Professor Ontonkong during the Fall '09 term at SUNY Stony Brook.

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