MODULAR FORMS-page88

# MODULAR FORMS-page88 - 84 LECTURE 8. THE MODULAR CURVE This...

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84 LECTURE 8. THE MODULAR CURVE This is called the Hurwitz formula . The number n here is called the degree of the meromorphic function f . Formula (8.21) says that this number is equal to # f - 1 ( y ) for almost all y P 1 ( C ). We shall deﬁne the triangulation of X as follows. Take a triangulation T of P 1 ( C ) in which each point y j is a vertex. Consider the pre-image T 0 of this triangulation in X . Since, the restriction of f to P 1 ( C ) \ S is a covering map, the open cell of our triangulation are equal to connected components of the pre-images of open cells of the triangulation of the sphere. Let d 0 ,d 1 ,d 2 be the number of 0-,1-, and 2-cells T . Then we have nd 1 1- and nd 2 2-cells in T 0 . We also have P y S # f - 1 ( y ) 0-cells in T 0 . By the Euler formula we have e ( X ) = X y S # f - 1 ( y ) - nd 1 + nd 2 = X y S # f - 1 ( y ) + n ( e ( P 1 ( C )) - n # S = 2 n - X y S ( n - # f - 1 ( y )) . Comparing this with (8.22) we obtain the assertion of the Theorem.
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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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