Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 265

Therefore, there is a constant C such that

u ) e u + 2 u 2

3

| (1 − − 1 |≤ C | u |

for u near 0. In particular, this inequality holds when u = z/ω for |ω| su-

ciently large and z in a compact set. Recall that if a sum |a n | converges,

then the product (1+ a n ) converges. Moreover, if (1+ a n ) =0forall n ,then

the product is nonzero. Since |z/ω|

3 converges by Lemma 9.4 with k =3,
the product defining σ ( z ) converges uniformly on compact sets. Therefore,
σ ( z ) is analytic. This proves (1). Part (2) follows since the product of the
factors, omitting one ω , is nonzero at z = ω .
To prove (3), differentiate the logarithm of the product for σ ( z )toobtain

1
z − ω + ω 2 . z +

ω∈L
ω =0

dz log σ ( z )= 1

d 1

ω +

z

Taking one more derivative yields the sum for − ℘ ( z ). This proves (3).

Let ω ∈ L .Since

d 2

dz 2 log σ ( z + ω )

=0 ,

σ ( z )

there are constants a = a ω and b = b ω such that

log σ ( z + ω )
σ ( z )

= az + b.

Exponentiating yields (4). We can restrict z in the above to lie in a small re-
gion in order to avoid potential complications with branches of the logarithm.
Then (4) holds in this small region, and therefore for all z ∈ C , by uniqueness

of analytic continuation.

We can now state exactly when a divisor is a divisor of a function. The
following is a special case of what is known as the Abel-Jacobi theorem ,
which states when a divisor on a Riemann surface, or on an algebraic curve,
is the divisor of a function.

THEOREM 9.6

Let D = n i [ w i ] be a divisor. T hen D isthe divisor of a function ifand

onlyif deg( D )=0 and n i w i ∈ L .

PROOF Parts (3) and (4) of Theorem 9.1 are precisely the statements

that if D is the divisor of a function then deg( D )=0and n i w i ∈

L .

Conversely, suppose deg( D )=0and n i w i =

∈ L .Let

σ ( z − )
i

σ ( z )

w i ) n i . f ( z )= σ ( z −