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
,ω
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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