MODULAR FORMS-page86

MODULAR FORMS-page86 - 82 LECTURE 8 THE MODULAR CURVE when restricted to U ∩ U Here dt U dt U is the derivative of the function g U,U = t U ◦ t

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Unformatted text preview: 82 LECTURE 8. THE MODULAR CURVE when restricted to U ∩ U . Here dt U dt U is the derivative of the function g U,U = t U ◦ t- 1 U : t U ( U ∩ U ) → t U ( U ∩ U ). Two meromorphic differentials are said be equal if they coincide when restricted to the subcover formed by intersections of their defining covers. Let ω = { f ( t U ) dt U } be a meromorphic function on X . Define ν x ( ω ) = ν x ( f U ) . (8.13) Since the function dt U dt U is invertible at x , we see that this definition is independent of the choice of an open neighborhood U of x . The divisor div( ω ) = X x ν x ( ω ) x. (8.14) is called the divisor of the meromorphic differential ω Since X is compact and hence can be covered by a finite set of locally compact subsets, we see that div( ω ) is well-defined. Lemma 8.4. Let ω and ω be two meromorphic differentials on X . Then their divisors div( ω ) and div( ω ) are linearly equivalent....
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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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