MODULAR FORMS-page85

MODULAR FORMS-page85 - 81 zero (resp. the order of pole )...

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Unformatted text preview: 81 zero (resp. the order of pole ) of f at x . We have the following easily verified properties of x ( f ): Lemma 8.3. Let x X and f,g be two meromorphic functions on X . Then (i) x ( fg ) = x ( f ) + x ( g ) ; (ii) x ( f + g ) = min { x ( f ) , x ( g ) } if f + g 6 = 0 . A meromorphic function on X is called a local parameter at x if x ( f ) = 1. Lemma 8.3 (i) allows us to give an equivalent definition of x ( f ). It is an integer such that for any local parameter t at x , there exists an open neighborhood U in which f = t x ( f ) for some invertible function O ( U ). Let Div( X ) be the free abelian group generated by the set X . Its elements are called divisors . One may view a divisor as a function D : X Z with finite support. It can be written as formal finite linear combinations D = P a x x , where a x = D ( x ) Z ,x X . For any D = P a x x Div( X ) we define its degree by the formula: deg( D ) = X a x . (8.10) There is an obvious order in Div(...
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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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