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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 preimage
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 preimages of open cells of the
triangulation of the sphere. Let
d
0
,d
1
,d
2
be the number of 0,1, and 2cells
T
. Then
we have
nd
1
1 and
nd
2
2cells in
T
0
. We also have
P
y
∈
S
#
f

1
(
y
) 0cells 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.
 Fall '09
 ONTONKONG

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