Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 278

If z ∈ C , recall that ord z f is the order of f at z .Thatis,

z ) ord z ( f ) g ( τ ) , f ( τ )=( τ −

with g ( z ) =0 ,∞ . We can also define the order of f at i∞ . Suppose

f ( τ )= a n q n + a n +1 q n +1 + ··· , (9.15)

with n ∈ Z and a n = 0, and assume that this series converges for all q close

to 0 (with q =0when n< 0). Then

ord i∞ ( f )= n.

Note that q → 0as τ → i∞ ,soord i∞ ( f ) expresses whether f vanishes ( n> 0)
or blows up ( n< 0) as τ → i∞ .

PROPOSITION 9.16
Let f be a function m erom orphicin H su ch that f isnotidenticallyzeroand
su ch that

f aτ + b
cτ + d

= f ( τ )

for all ab

∈ SL 2 ( Z ) .

cd

Then

2 ord i ( f )+
z

ord i∞ ( f )+ 1

3 ord ρ ( f )+ 1

ord z ( f )=0 .

= i,ρ,i ∞

REMARK 9.17 The function f can be regarded as a function on the
surface obtained as follows. Identify the left and right sides on F to get a
tube, then fold the part with to

a point. This gives a surface that is topologically a sphere. The proposition
expresses the fact that the number of poles of f equals the number of zeros on
such a surface, just as occurred for doubly periodic functions in Theorem 9.1.
The point i is special since a small neighborhood around i contains only half
of a disc inside F . Similarly, a small neighborhood around ρ includes only
1/3 of a disc from F (namely, 1/6 near ρ and 1/6 near 1 + ρ , which is folded
over to meet ρ ). This explains the factors 1/2 and 1/3 in the proposition. For
a related phenomenon, see Exercise 9.3.

| z | =1at i . Thenpinchtheopenendat i ∞

PROOF Let C be the path shown in Figure 9.4. Essentially, C goes around
theedgeof F . However, it consists of a small circular arc past each of ρ ,1+ ρ ,
and i . If there is a pole or zero of f at a point on the path, we make a small
detour around it and a corresponding detour at the corresponding point on
the other side of F . The arcs near ρ ,1+ ρ ,and i have radius ,where is
chosen small enough that there are no zeros or poles of f inside the circles,
except possibly at ρ ,1+ ρ ,or i . Similarly, the top part of C is chosen to have