Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 266

If ω ∈ L ,then

f ( z + ω )
f ( z )

= e a ω z + b ω −a ω ( z− ) −b ω e n i ( a ω ( z−w i )+ b ω ) =1 ,

since n i =0and n i w i = . Therefore, f ( z ) is doubly periodic. The
divisor of f is easily seen to be D ,so D is the divisor of a function.

Doubly periodic functions can be regarded as functions on the torus C /L ,
and divisors can be regarded as divisors for C /L .Ifwelet C ( L ) × denote the
doubly periodic functions that do not vanish identically and let Div 0 ( C /L )
denote the divisors of degree 0, then much of the preceding discussion can be
expressed by the exactness of the sequence

0 −→ C × −→ C ( L ) × div −→ Div 0 ( C /L ) sum −→ C /L −→ 0 . (9.3)

The “sum” function adds up the complex numbers representing the points in
the divisor mod L . The exactness at C ( L ) × expresses the fact that a function
with no zeros and no poles, hence whose divisor is 0, is a constant. The
exactness at Div 0 ( C /L ) is Theorem 9.6. The surjectivity of the sum function
is easy. If w ∈ C , then sum([ w ] − [0]) = w mod L .

9.2 Tori are Elliptic Curves

The goal of this section is to show that a complex torus C /L is naturally
isomorphic to the complex points on an elliptic curve.
Let L be a lattice, as in the previous section. For integers k ≥ 3, define the
Eisenstein series

G k = G k ( L )=
ω∈L
ω

ω −k .

(9.4)

=0

By Lemma 9.4, the sum converges. When k is odd, the terms for ω and −ω
cancel, so G k =0.

PROPOSITION 9.7
For 0 < |z| < Min 0 = ω∈L ( |ω| ) ,

z 2 + ∞

℘ ( z )= 1 (2 j +1) G 2 j +2 z 2 j .

j =1