MODULAR FORMS-page97

MODULAR FORMS-page97 - P 1 ( C )) 4 \ of ordered fourtuples...

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Lecture 9 Absolute Invariant and Cross-Ratio 9.1 Let x 1 = ( a 1 ,b 1 ) , x 2 = ( a 2 ,b 2 ) , x 3 = ( a 3 ,b 3 ) , x 4 = ( a 4 ,b 4 ) be four distinct points on P 1 ( C ). The expression R = ˛ ˛ ˛ ˛ a 1 b 1 a 2 b 2 ˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ a 3 b 3 a 4 b 4 ˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ a 1 b 1 a 3 b 3 ˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ a 2 b 2 a 4 b 4 ˛ ˛ ˛ ˛ (9.1) is called the cross-ratio of the four points. As is easy to see it does not depend on the choice of projective coordinates of the points. Also it is unchanged under the projective linear transformation of P 1 ( C ): ( x,y ) ( ax + by,cx + dy ) . If none of the points is equal to the infinity point = (0 , 1) we can write each x i as (1 ,z i ) and rewrite R in the form R = ( z 2 - z 1 )( z 4 - z 3 ) ( z 3 - z 1 )( z 4 - z 2 ) . (9.2) One can view the cross-ratio function as a function on the space X = (
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Unformatted text preview: P 1 ( C )) 4 \ of ordered fourtuples of distinct points in P 1 ( C ). Here denotes the diagonal, the set of 4-tuples with at least two coordinates equal. The group GL(2 , C ) acts naturally on X by transforming each ( x 1 ,x 2 ,x 3 ,x 4 ) in ( g x 1 ,g x 2 ,g x 3 ,g x 4 ) and R is an invariant function with respect to this action. In other words, R descends to a function on the orbit space R : X/ GL(2 , C ) C . The following is a classical result from the theory of invariants: 93...
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