Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 257

Ω 3

Ω 1

Ω 2

0

Figure 9.1
The Fundamental Parallelogram

functions, so we define a doubly periodic function to be a meromorphic
function

f : C → C ∪∞

such that

f ( z + ω )= f ( z )

for all z ∈ C and all ω ∈ L . Equivalently,

f ( z + ω i )= f ( z ) , i =1 , 2 .

The numbers ω L are called the periods of f .

If f is a (not identically 0) meromorphic function and w

∈

∈ C ,thenwecan

write

w ) r + a r +1 ( z w ) r +1 + f ( z )= a r ( z − − ··· ,

with a r = 0. The integer r can be either positive, negative, or zero. Define
the order and the residue of f at w to be

r =ord w f
a - 1 =Res w f.

Therefore, ord w f is the order of vanishing of f at w , or negative the order of
a pole. The order is 0 if and only if the function is finite and nonvanishing at
w . It is not hard to see that if f is doubly periodic, then ord w + ω f =ord w f
and Res w + ω f =Res w f for all ω ∈ L .
A divisor D is a formal sum of points:

D = n 1 [ w 1 ]+ n 2 [ w 2 ]+ ··· + n k [ w k ] ,

where n i ∈ Z and w i ∈ F . In other words, we have a symbol [ w ]foreach

w F , and the divisors are linear combinations with integer coe cients of

these symbols. The degree of a divisor is
deg( D )= n i .

∈