Cryptography Reference
In-Depth Information
Define the divisor of a function f to be
div( f )=
w
(ord w f )[ w ] .
F
THEOREM 9.1
Let f be a doubly periodicfunction for the lattice L and let F be a fundam ental
parallelogram for L .
1. If f has no poles, then f isconstant.
2. w∈F Res w f =0 .
3. If f isnotidentically0,
deg(div( f )) =
w
ord w f =0 .
F
4. If f isnotidentically0,
w · ord w f ∈ L.
w
F
5. If f isnotconstant,then f : C C ∪∞ issurjective. If n isthe sum
of the orders of the poles of f in F and z 0 C ,then f ( z )= z 0 has n
solutions (counting m ultiplicities).
6. If f has onlyonepo ein F ,then thispolecannotbeasimp epo e.
Allofthe above sum s over w
F have only finitely m any nonzero term s.
PROOF Because f is a meromorphic function, it can have only finitely
many zeros and poles in any compact set, for example, the closure of F .
Therefore, the above sums have only finitely many nonzero terms.
If f has no poles, then it is bounded in the closure of F , which is a compact
set. Therefore, f is bounded in all of C . Liouville's theorem says that a
bounded entire function is constant. This proves (1).
Recall Cauchy's theorem, which says that
f ( z ) dz =2 πi
w
Res w f,
∂F
F
where ∂F is the boundary of F andthelineintegralistakeninthecoun-
terclockwise direction. Write (assuming ω 1 2 are oriented as in Figure 9.1;
otherwise, switch them in the following)
f ( z ) dz =
∂F
ω 2
f ( z ) dz + ω 2 + ω 1
ω 2
f ( z ) dz + ω 1
ω 1 + ω 2
f ( z ) dz + 0
ω 1
f ( z ) dz.
0
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