MODULAR FORMS-page88

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

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

View Full Document Right Arrow Icon
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.
Background image of page 1
This is the end of the preview. Sign up to access the rest of the 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