Cryptography Reference
In-Depth Information
If (6) were false for a function
f
,then
f
would give a bijective (by (5)) map
from the torus to the Riemann sphere (=
C
). This is impossible for many
topological reasons (the torus has a hole but the sphere doesn't).
∪∞
So far, we do not have any examples of nonconstant doubly periodic func-
tions. This situation is remedied by the
Weierstrass
℘
-function
.
THEOREM 9.3
Given a lattice
L
,define the W eierstra ss
℘
-function by
ω
2
.
z
2
+
ω∈L
ω
℘
(
z
)=
℘
(
z
;
L
)=
1
1
(
z − ω
)
2
−
1
(9.1)
=0
Then
1. T he sum defining
℘
(
z
)
converges absolutelyanduniform ly on com pact
setsnotcontaining elem entsof
L
.
2.
℘
(
z
)
is m erom orphicin
C
and has a doublepo eateach
ω ∈ L
.
3.
℘
(
−z
)=
℘
(
z
)
for all
z ∈
C
.
4.
℘
(
z
+
ω
)=
℘
(
z
)
for all
ω ∈ L
.
5. T he set of doubly periodicfunctions for
L
is
C
(
℘, ℘
)
.Inother w ords,
every doubly periodicfunction isarationalfunction of
℘
and itsderiva-
tive
℘
.
PROOF
Let
C
be a compact set, and let
M
=Max
{|z||z ∈ C}
.If
z ∈ C
and
|ω|≥
2
M
,then
|z − ω|≥|ω|/
2and
|
2
ω − z|≤
5
|ω|/
2, so
=
1
(
z − ω
)
2
−
1
ω
2
z
(2
ω − z
)
(
z − ω
)
2
ω
2
(9.2)
M
(5
|
ω
|
/
2)
|ω|
=
10
M
|ω|
≤
3
.
4
/
4
The preceding calculation explains why the terms 1
/ω
2
are included. With-
out them, the terms in the sum would be comparable to 1
/ω
2
. Subtracting
this 1
/ω
2
makes the terms comparable to 1
/ω
3
. This causes the sum to con-
verge, as the following lemma shows.
LEMMA 9.4
If
k>
2
then
1
|ω|
k
ω∈L
ω
=0
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