MODULAR FORMS-page98

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Ask a homework question - tutors are online