MODULAR FORMS-page56

MODULAR FORMS-page56 - 52 LECTURE 6. MODULAR FORMS...

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Unformatted text preview: 52 LECTURE 6. MODULAR FORMS represented by a rational number x or the stabilizer group x is conjugate to a subgroup of SL(2 , Z ) . In fact, if g x = for some g SL(2 , Z ), then g x g- 1 = . Since + + = = 0 , we have g x g- 1 { 1 1 , Z } Let h be the smallest positive occured in this way. Then it is immediately seen that g x g- 1 is generated by the matrices T h = 1 h 1 ,- I = - 1- 1 . The number h is also equal to the index of the subgroup g x g- 1 in SL(2 , Z ) = ( T,- I ) . In particular, all x from the same cusp of define the same number h . We shall call it the index of the cusp. Let f ( ) be a holomorphic function satisfying (6.1). For each x Q {} consider the function ( ) = f ( ) | k g- 1 , where g x = for some g SL(2 , Z ). We have ( ) | k g x g- 1 = f ( ) | k g- 1 g x g- 1 = f (...
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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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