Class of mathematical functions
"℘" redirects here; the symbol can also be used to denote a
power set .
In mathematics , the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass . This class of functions are also referred to as ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy script p . They play an important role in the theory of elliptic functions, i.e., meromorphic functions that are doubly periodic . A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice.
Symbol for Weierstrass
℘
{\displaystyle \wp }
-function
Model of Weierstrass
℘
{\displaystyle \wp }
-function
Motivation [ edit ]
A cubic of the form
C
g
2
,
g
3
C
=
{
(
x
,
y
)
∈
C
2
:
y
2
=
4
x
3
−
g
2
x
−
g
3
}
{\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }=\{(x,y)\in \mathbb {C} ^{2}:y^{2}=4x^{3}-g_{2}x-g_{3}\}}
, where
g
2
,
g
3
∈
C
{\displaystyle g_{2},g_{3}\in \mathbb {C} }
are complex numbers with
g
2
3
−
27
g
3
2
≠
0
{\displaystyle g_{2}^{3}-27g_{3}^{2}\neq 0}
, cannot be rationally parameterized .[1] Yet one still wants to find a way to parameterize it.
For the quadric
K
=
{
(
x
,
y
)
∈
R
2
:
x
2
+
y
2
=
1
}
{\displaystyle K=\left\{(x,y)\in \mathbb {R} ^{2}:x^{2}+y^{2}=1\right\}}
; the unit circle , there exists a (non-rational) parameterization using the sine function and its derivative the cosine function:
ψ
:
R
/
2
π
Z
→
K
,
t
↦
(
sin
t
,
cos
t
)
.
{\displaystyle \psi :\mathbb {R} /2\pi \mathbb {Z} \to K,\quad t\mapsto (\sin t,\cos t).}
Because of the periodicity of the sine and cosine
R
/
2
π
Z
{\displaystyle \mathbb {R} /2\pi \mathbb {Z} }
is chosen to be the domain, so the function is bijective.
In a similar way one can get a parameterization of
C
g
2
,
g
3
C
{\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }}
by means of the doubly periodic
℘
{\displaystyle \wp }
-function (see in the section "Relation to elliptic curves"). This parameterization has the domain
C
/
Λ
{\displaystyle \mathbb {C} /\Lambda }
, which is topologically equivalent to a torus .[2]
There is another analogy to the trigonometric functions. Consider the integral function
a
(
x
)
=
∫
0
x
d
y
1
−
y
2
.
{\displaystyle a(x)=\int _{0}^{x}{\frac {dy}{\sqrt {1-y^{2}}}}.}
It can be simplified by substituting
y
=
sin
t
{\displaystyle y=\sin t}
and
s
=
arcsin
x
{\displaystyle s=\arcsin x}
:
a
(
x
)
=
∫
0
s
d
t
=
s
=
arcsin
x
.
{\displaystyle a(x)=\int _{0}^{s}dt=s=\arcsin x.}
That means
a
−
1
(
x
)
=
sin
x
{\displaystyle a^{-1}(x)=\sin x}
. So the sine function is an inverse function of an integral function.
[3]
Elliptic functions are the inverse functions of elliptic integrals . In particular, let:
u
(
z
)
=
−
∫
z
∞
d
s
4
s
3
−
g
2
s
−
g
3
.
{\displaystyle u(z)=-\int _{z}^{\infty }{\frac {ds}{\sqrt {4s^{3}-g_{2}s-g_{3}}}}.}
Then the extension of
u
−
1
{\displaystyle u^{-1}}
to the complex plane equals the
℘
{\displaystyle \wp }
-function.
[4] This invertibility is used in
complex analysis to provide a solution to certain
nonlinear differential equations satisfying the
Painlevé property , i.e., those equations that admit
poles as their only
movable singularities .
[5]
Definition [ edit ]
Visualization of the
℘
{\displaystyle \wp }
-function with invariants
g
2
=
1
+
i
{\displaystyle g_{2}=1+i}
and
g
3
=
2
−
3
i
{\displaystyle g_{3}=2-3i}
in which white corresponds to a pole, black to a zero.
Let
ω
1
,
ω
2
∈
C
{\displaystyle \omega _{1},\omega _{2}\in \mathbb {C} }
be two complex numbers that are linearly independent over
R
{\displaystyle \mathbb {R} }
and let
Λ
:=
Z
ω
1
+
Z
ω
2
:=
{
m
ω
1
+
n
ω
2
:
m
,
n
∈
Z
}
{\displaystyle \Lambda :=\mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}:=\{m\omega _{1}+n\omega _{2}:m,n\in \mathbb {Z} \}}
be the period lattice generated by those numbers. Then the
℘
{\displaystyle \wp }
-function is defined as follows:
℘
(
z
,
ω
1
,
ω
2
)
:=
℘
(
z
)
=
1
z
2
+
∑
λ
∈
Λ
∖
{
0
}
(
1
(
z
−
λ
)
2
−
1
λ
2
)
.
{\displaystyle \wp (z,\omega _{1},\omega _{2}):=\wp (z)={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right).}
This series converges locally uniformly absolutely in the complex torus
C
∖
Λ
{\displaystyle \mathbb {C} \setminus \Lambda }
.
It is common to use
1
{\displaystyle 1}
and
τ
{\displaystyle \tau }
in the upper half-plane
H
:=
{
z
∈
C
:
Im
(
z
)
>
0
}
{\displaystyle \mathbb {H} :=\{z\in \mathbb {C} :\operatorname {Im} (z)>0\}}
as generators of the lattice . Dividing by
ω
1
{\textstyle \omega _{1}}
maps the lattice
Z
ω
1
+
Z
ω
2
{\displaystyle \mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}}
isomorphically onto the lattice
Z
+
Z
τ
{\displaystyle \mathbb {Z} +\mathbb {Z} \tau }
with
τ
=
ω
2
ω
1
{\textstyle \tau ={\tfrac {\omega _{2}}{\omega _{1}}}}
. Because
−
τ
{\displaystyle -\tau }
can be substituted for
τ
{\displaystyle \tau }
, without loss of generality we can assume
τ
∈
H
{\displaystyle \tau \in \mathbb {H} }
, and then define
℘
(
z
,
τ
)
:=
℘
(
z
,
1
,
τ
)
{\displaystyle \wp (z,\tau ):=\wp (z,1,\tau )}
.
Properties [ edit ]
℘
{\displaystyle \wp }
is a meromorphic function with a pole of order 2 at each period
λ
{\displaystyle \lambda }
in
Λ
{\displaystyle \Lambda }
.
℘
{\displaystyle \wp }
is an even function. That means
℘
(
z
)
=
℘
(
−
z
)
{\displaystyle \wp (z)=\wp (-z)}
for all
z
∈
C
∖
Λ
{\displaystyle z\in \mathbb {C} \setminus \Lambda }
, which can be seen in the following way:
℘
(
−
z
)
=
1
(
−
z
)
2
+
∑
λ
∈
Λ
∖
{
0
}
(
1
(
−
z
−
λ
)
2
−
1
λ
2
)
=
1
z
2
+
∑
λ
∈
Λ
∖
{
0
}
(
1
(
z
+
λ
)
2
−
1
λ
2
)
=
1
z
2
+
∑
λ
∈
Λ
∖
{
0
}
(
1
(
z
−
λ
)
2
−
1
λ
2
)
=
℘
(
z
)
.
{\displaystyle {\begin{aligned}\wp (-z)&={\frac {1}{(-z)^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(-z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)\\&={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z+\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)\\&={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)=\wp (z).\end{aligned}}}
The second last equality holds because
{
−
λ
:
λ
∈
Λ
}
=
Λ
{\displaystyle \{-\lambda :\lambda \in \Lambda \}=\Lambda }
. Since the sum converges absolutely this rearrangement does not change the limit.
The derivative of
℘
{\displaystyle \wp }
is given by:[6]
℘
′
(
z
)
=
−
2
∑
λ
∈
Λ
1
(
z
−
λ
)
3
.
{\displaystyle \wp '(z)=-2\sum _{\lambda \in \Lambda }{\frac {1}{(z-\lambda )^{3}}}.}
℘
{\displaystyle \wp }
and
℘
′
{\displaystyle \wp '}
are doubly periodic with the periods
ω
1
{\displaystyle \omega _{1}}
and
ω
2
{\displaystyle \omega _{2}}
.[6] This means:
℘
(
z
+
ω
1
)
=
℘
(
z
)
=
℘
(
z
+
ω
2
)
,
and
℘
′
(
z
+
ω
1
)
=
℘
′
(
z
)
=
℘
′
(
z
+
ω
2
)
.
{\displaystyle {\begin{aligned}\wp (z+\omega _{1})&=\wp (z)=\wp (z+\omega _{2}),\ {\textrm {and}}\\[3mu]\wp '(z+\omega _{1})&=\wp '(z)=\wp '(z+\omega _{2}).\end{aligned}}}
It follows that
℘
(
z
+
λ
)
=
℘
(
z
)
{\displaystyle \wp (z+\lambda )=\wp (z)}
and
℘
′
(
z
+
λ
)
=
℘
′
(
z
)
{\displaystyle \wp '(z+\lambda )=\wp '(z)}
for all
λ
∈
Λ
{\displaystyle \lambda \in \Lambda }
.
Laurent expansion [ edit ]
Let
r
:=
min
{
|
λ
|
:
0
≠
λ
∈
Λ
}
{\displaystyle r:=\min\{{|\lambda }|:0\neq \lambda \in \Lambda \}}
. Then for
0
<
|
z
|
<
r
{\displaystyle 0<|z|<r}
the
℘
{\displaystyle \wp }
-function has the following Laurent expansion
℘
(
z
)
=
1
z
2
+
∑
n
=
1
∞
(
2
n
+
1
)
G
2
n
+
2
z
2
n
{\displaystyle \wp (z)={\frac {1}{z^{2}}}+\sum _{n=1}^{\infty }(2n+1)G_{2n+2}z^{2n}}
where
G
n
=
∑
0
≠
λ
∈
Λ
λ
−
n
{\displaystyle G_{n}=\sum _{0\neq \lambda \in \Lambda }\lambda ^{-n}}
for
n
≥
3
{\displaystyle n\geq 3}
are so called
Eisenstein series .
[6]
Differential equation [ edit ]
Set
g
2
=
60
G
4
{\displaystyle g_{2}=60G_{4}}
and
g
3
=
140
G
6
{\displaystyle g_{3}=140G_{6}}
. Then the
℘
{\displaystyle \wp }
-function satisfies the differential equation[6]
℘
′
2
(
z
)
=
4
℘
3
(
z
)
−
g
2
℘
(
z
)
−
g
3
.
{\displaystyle \wp '^{2}(z)=4\wp ^{3}(z)-g_{2}\wp (z)-g_{3}.}
This relation can be verified by forming a linear combination of powers of
℘
{\displaystyle \wp }
and
℘
′
{\displaystyle \wp '}
to eliminate the pole at
z
=
0
{\displaystyle z=0}
. This yields an entire elliptic function that has to be constant by
Liouville's theorem .
[6]
Invariants [ edit ]
The real part of the invariant g 3 as a function of the square of the nome q on the unit disk.
The imaginary part of the invariant g 3 as a function of the square of the nome q on the unit disk.
The coefficients of the above differential equation g 2 and g 3 are known as the invariants . Because they depend on the lattice
Λ
{\displaystyle \Lambda }
they can be viewed as functions in
ω
1
{\displaystyle \omega _{1}}
and
ω
2
{\displaystyle \omega _{2}}
.
The series expansion suggests that g 2 and g 3 are homogeneous functions of degree −4 and −6. That is[7]
g
2
(
λ
ω
1
,
λ
ω
2
)
=
λ
−
4
g
2
(
ω
1
,
ω
2
)
{\displaystyle g_{2}(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-4}g_{2}(\omega _{1},\omega _{2})}
g
3
(
λ
ω
1
,
λ
ω
2
)
=
λ
−
6
g
3
(
ω
1
,
ω
2
)
{\displaystyle g_{3}(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-6}g_{3}(\omega _{1},\omega _{2})}
for
λ
≠
0
{\displaystyle \lambda \neq 0}
.
If
ω
1
{\displaystyle \omega _{1}}
and
ω
2
{\displaystyle \omega _{2}}
are chosen in such a way that
Im
(
ω
2
ω
1
)
>
0
{\displaystyle \operatorname {Im} \left({\tfrac {\omega _{2}}{\omega _{1}}}\right)>0}
, g 2 and g 3 can be interpreted as functions on the upper half-plane
H
:=
{
z
∈
C
:
Im
(
z
)
>
0
}
{\displaystyle \mathbb {H} :=\{z\in \mathbb {C} :\operatorname {Im} (z)>0\}}
.
Let
τ
=
ω
2
ω
1
{\displaystyle \tau ={\tfrac {\omega _{2}}{\omega _{1}}}}
. One has:[8]
g
2
(
1
,
τ
)
=
ω
1
4
g
2
(
ω
1
,
ω
2
)
,
{\displaystyle g_{2}(1,\tau )=\omega _{1}^{4}g_{2}(\omega _{1},\omega _{2}),}
g
3
(
1
,
τ
)
=
ω
1
6
g
3
(
ω
1
,
ω
2
)
.
{\displaystyle g_{3}(1,\tau )=\omega _{1}^{6}g_{3}(\omega _{1},\omega _{2}).}
That means
g 2 and
g 3 are only scaled by doing this. Set
g
2
(
τ
)
:=
g
2
(
1
,
τ
)
{\displaystyle g_{2}(\tau ):=g_{2}(1,\tau )}
and
g
3
(
τ
)
:=
g
3
(
1
,
τ
)
.
{\displaystyle g_{3}(\tau ):=g_{3}(1,\tau ).}
As functions of
τ
∈
H
{\displaystyle \tau \in \mathbb {H} }
g
2
,
g
3
{\displaystyle g_{2},g_{3}}
are so called
modular forms.
The Fourier series for
g
2
{\displaystyle g_{2}}
and
g
3
{\displaystyle g_{3}}
are given as follows:[9]
g
2
(
τ
)
=
4
3
π
4
[
1
+
240
∑
k
=
1
∞
σ
3
(
k
)
q
2
k
]
{\displaystyle g_{2}(\tau )={\frac {4}{3}}\pi ^{4}\left[1+240\sum _{k=1}^{\infty }\sigma _{3}(k)q^{2k}\right]}
g
3
(
τ
)
=
8
27
π
6
[
1
−
504
∑
k
=
1
∞
σ
5
(
k
)
q
2
k
]
{\displaystyle g_{3}(\tau )={\frac {8}{27}}\pi ^{6}\left[1-504\sum _{k=1}^{\infty }\sigma _{5}(k)q^{2k}\right]}
where
σ
a
(
k
)
:=
∑
d
∣
k
d
α
{\displaystyle \sigma _{a}(k):=\sum _{d\mid {k}}d^{\alpha }}
is the
divisor function and
q
=
e
π
i
τ
{\displaystyle q=e^{\pi i\tau }}
is the
nome .
Modular discriminant [ edit ]
The real part of the discriminant as a function of the square of the nome q on the unit disk.
The modular discriminant Δ is defined as the discriminant of the polynomial on the right-hand side of the above differential equation:
Δ
=
g
2
3
−
27
g
3
2
.
{\displaystyle \Delta =g_{2}^{3}-27g_{3}^{2}.}
The discriminant is a modular form of weight 12. That is, under the action of the
modular group , it transforms as
Δ
(
a
τ
+
b
c
τ
+
d
)
=
(
c
τ
+
d
)
12
Δ
(
τ
)
{\displaystyle \Delta \left({\frac {a\tau +b}{c\tau +d}}\right)=\left(c\tau +d\right)^{12}\Delta (\tau )}
where
a
,
b
,
d
,
c
∈
Z
{\displaystyle a,b,d,c\in \mathbb {Z} }
with
ad −
bc = 1.
[10]
Note that
Δ
=
(
2
π
)
12
η
24
{\displaystyle \Delta =(2\pi )^{12}\eta ^{24}}
where
η
{\displaystyle \eta }
is the Dedekind eta function .[11]
For the Fourier coefficients of
Δ
{\displaystyle \Delta }
, see Ramanujan tau function .
The constants e 1 , e 2 and e 3 [ edit ]
e
1
{\displaystyle e_{1}}
,
e
2
{\displaystyle e_{2}}
and
e
3
{\displaystyle e_{3}}
are usually used to denote the values of the
℘
{\displaystyle \wp }
-function at the half-periods.
e
1
≡
℘
(
ω
1
2
)
{\displaystyle e_{1}\equiv \wp \left({\frac {\omega _{1}}{2}}\right)}
e
2
≡
℘
(
ω
2
2
)
{\displaystyle e_{2}\equiv \wp \left({\frac {\omega _{2}}{2}}\right)}
e
3
≡
℘
(
ω
1
+
ω
2
2
)
{\displaystyle e_{3}\equiv \wp \left({\frac {\omega _{1}+\omega _{2}}{2}}\right)}
They are pairwise distinct and only depend on the lattice
Λ
{\displaystyle \Lambda }
and not on its generators.
[12]
e
1
{\displaystyle e_{1}}
,
e
2
{\displaystyle e_{2}}
and
e
3
{\displaystyle e_{3}}
are the roots of the cubic polynomial
4
℘
(
z
)
3
−
g
2
℘
(
z
)
−
g
3
{\displaystyle 4\wp (z)^{3}-g_{2}\wp (z)-g_{3}}
and are related by the equation:
e
1
+
e
2
+
e
3
=
0.
{\displaystyle e_{1}+e_{2}+e_{3}=0.}
Because those roots are distinct the discriminant
Δ
{\displaystyle \Delta }
does not vanish on the upper half plane.
[13] Now we can rewrite the differential equation:
℘
′
2
(
z
)
=
4
(
℘
(
z
)
−
e
1
)
(
℘
(
z
)
−
e
2
)
(
℘
(
z
)
−
e
3
)
.
{\displaystyle \wp '^{2}(z)=4(\wp (z)-e_{1})(\wp (z)-e_{2})(\wp (z)-e_{3}).}
That means the half-periods are zeros of
℘
′
{\displaystyle \wp '}
.
The invariants
g
2
{\displaystyle g_{2}}
and
g
3
{\displaystyle g_{3}}
can be expressed in terms of these constants in the following way:[14]
g
2
=
−
4
(
e
1
e
2
+
e
1
e
3
+
e
2
e
3
)
{\displaystyle g_{2}=-4(e_{1}e_{2}+e_{1}e_{3}+e_{2}e_{3})}
g
3
=
4
e
1
e
2
e
3
{\displaystyle g_{3}=4e_{1}e_{2}e_{3}}
e
1
{\displaystyle e_{1}}
,
e
2
{\displaystyle e_{2}}
and
e
3
{\displaystyle e_{3}}
are related to the
modular lambda function :
λ
(
τ
)
=
e
3
−
e
2
e
1
−
e
2
,
τ
=
ω
2
ω
1
.
{\displaystyle \lambda (\tau )={\frac {e_{3}-e_{2}}{e_{1}-e_{2}}},\quad \tau ={\frac {\omega _{2}}{\omega _{1}}}.}
Relation to Jacobi's elliptic functions [ edit ]
For numerical work, it is often convenient to calculate the Weierstrass elliptic function in terms of Jacobi's elliptic functions .
The basic relations are:[15]
℘
(
z
)
=
e
3
+
e
1
−
e
3
sn
2
w
=
e
2
+
(
e
1
−
e
3
)
dn
2
w
sn
2
w
=
e
1
+
(
e
1
−
e
3
)
cn
2
w
sn
2
w
{\displaystyle \wp (z)=e_{3}+{\frac {e_{1}-e_{3}}{\operatorname {sn} ^{2}w}}=e_{2}+(e_{1}-e_{3}){\frac {\operatorname {dn} ^{2}w}{\operatorname {sn} ^{2}w}}=e_{1}+(e_{1}-e_{3}){\frac {\operatorname {cn} ^{2}w}{\operatorname {sn} ^{2}w}}}
where
e
1
,
e
2
{\displaystyle e_{1},e_{2}}
and
e
3
{\displaystyle e_{3}}
are the three roots described above and where the modulus
k of the Jacobi functions equals
k
=
e
2
−
e
3
e
1
−
e
3
{\displaystyle k={\sqrt {\frac {e_{2}-e_{3}}{e_{1}-e_{3}}}}}
and their argument
w equals
w
=
z
e
1
−
e
3
.
{\displaystyle w=z{\sqrt {e_{1}-e_{3}}}.}
Relation to Jacobi's theta functions [ edit ]
The function
℘
(
z
,
τ
)
=
℘
(
z
,
1
,
ω
2
/
ω
1
)
{\displaystyle \wp (z,\tau )=\wp (z,1,\omega _{2}/\omega _{1})}
can be represented by Jacobi's theta functions :
℘
(
z
,
τ
)
=
(
π
θ
2
(
0
,
q
)
θ
3
(
0
,
q
)
θ
4
(
π
z
,
q
)
θ
1
(
π
z
,
q
)
)
2
−
π
2
3
(
θ
2
4
(
0
,
q
)
+
θ
3
4
(
0
,
q
)
)
{\displaystyle \wp (z,\tau )=\left(\pi \theta _{2}(0,q)\theta _{3}(0,q){\frac {\theta _{4}(\pi z,q)}{\theta _{1}(\pi z,q)}}\right)^{2}-{\frac {\pi ^{2}}{3}}\left(\theta _{2}^{4}(0,q)+\theta _{3}^{4}(0,q)\right)}
where
q
=
e
π
i
τ
{\displaystyle q=e^{\pi i\tau }}
is the nome and
τ
{\displaystyle \tau }
is the period ratio
(
τ
∈
H
)
{\displaystyle (\tau \in \mathbb {H} )}
.
[16] This also provides a very rapid algorithm for computing
℘
(
z
,
τ
)
{\displaystyle \wp (z,\tau )}
.
Relation to elliptic curves [ edit ]
Consider the embedding of the cubic curve in the complex projective plane
C
¯
g
2
,
g
3
C
=
{
(
x
,
y
)
∈
C
2
:
y
2
=
4
x
3
−
g
2
x
−
g
3
}
∪
{
∞
}
⊂
C
2
∪
{
∞
}
=
P
2
(
C
)
.
{\displaystyle {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }=\{(x,y)\in \mathbb {C} ^{2}:y^{2}=4x^{3}-g_{2}x-g_{3}\}\cup \{\infty \}\subset \mathbb {C} ^{2}\cup \{\infty \}=\mathbb {P} _{2}(\mathbb {C} ).}
For this cubic there exists no rational parameterization, if
Δ
≠
0
{\displaystyle \Delta \neq 0}
.[1] In this case it is also called an elliptic curve. Nevertheless there is a parameterization in homogeneous coordinates that uses the
℘
{\displaystyle \wp }
-function and its derivative
℘
′
{\displaystyle \wp '}
:[17]
φ
(
℘
,
℘
′
)
:
C
/
Λ
→
C
¯
g
2
,
g
3
C
,
z
↦
{
[
℘
(
z
)
:
℘
′
(
z
)
:
1
]
z
∉
Λ
[
0
:
1
:
0
]
z
∈
Λ
{\displaystyle \varphi (\wp ,\wp '):\mathbb {C} /\Lambda \to {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} },\quad z\mapsto {\begin{cases}\left[\wp (z):\wp '(z):1\right]&z\notin \Lambda \\\left[0:1:0\right]\quad &z\in \Lambda \end{cases}}}
Now the map
φ
{\displaystyle \varphi }
is bijective and parameterizes the elliptic curve
C
¯
g
2
,
g
3
C
{\displaystyle {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}
.
C
/
Λ
{\displaystyle \mathbb {C} /\Lambda }
is an abelian group and a topological space , equipped with the quotient topology .
It can be shown that every Weierstrass cubic is given in such a way. That is to say that for every pair
g
2
,
g
3
∈
C
{\displaystyle g_{2},g_{3}\in \mathbb {C} }
with
Δ
=
g
2
3
−
27
g
3
2
≠
0
{\displaystyle \Delta =g_{2}^{3}-27g_{3}^{2}\neq 0}
there exists a lattice
Z
ω
1
+
Z
ω
2
{\displaystyle \mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}}
, such that
g
2
=
g
2
(
ω
1
,
ω
2
)
{\displaystyle g_{2}=g_{2}(\omega _{1},\omega _{2})}
and
g
3
=
g
3
(
ω
1
,
ω
2
)
{\displaystyle g_{3}=g_{3}(\omega _{1},\omega _{2})}
.[18]
The statement that elliptic curves over
Q
{\displaystyle \mathbb {Q} }
can be parameterized over
Q
{\displaystyle \mathbb {Q} }
, is known as the modularity theorem . This is an important theorem in number theory . It was part of Andrew Wiles' proof (1995) of Fermat's Last Theorem .
Addition theorems [ edit ]
Let
z
,
w
∈
C
{\displaystyle z,w\in \mathbb {C} }
, so that
z
,
w
,
z
+
w
,
z
−
w
∉
Λ
{\displaystyle z,w,z+w,z-w\notin \Lambda }
. Then one has:[19]
℘
(
z
+
w
)
=
1
4
[
℘
′
(
z
)
−
℘
′
(
w
)
℘
(
z
)
−
℘
(
w
)
]
2
−
℘
(
z
)
−
℘
(
w
)
.
{\displaystyle \wp (z+w)={\frac {1}{4}}\left[{\frac {\wp '(z)-\wp '(w)}{\wp (z)-\wp (w)}}\right]^{2}-\wp (z)-\wp (w).}
As well as the duplication formula:[19]
℘
(
2
z
)
=
1
4
[
℘
″
(
z
)
℘
′
(
z
)
]
2
−
2
℘
(
z
)
.
{\displaystyle \wp (2z)={\frac {1}{4}}\left[{\frac {\wp ''(z)}{\wp '(z)}}\right]^{2}-2\wp (z).}
These formulas also have a geometric interpretation, if one looks at the elliptic curve
C
¯
g
2
,
g
3
C
{\displaystyle {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}
together with the mapping
φ
:
C
/
Λ
→
C
¯
g
2
,
g
3
C
{\displaystyle {\varphi }:\mathbb {C} /\Lambda \to {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}
as in the previous section.
The group structure of
(
C
/
Λ
,
+
)
{\displaystyle (\mathbb {C} /\Lambda ,+)}
translates to the curve
C
¯
g
2
,
g
3
C
{\displaystyle {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}
and can be geometrically interpreted there:
The sum of three pairwise different points
a
,
b
,
c
∈
C
¯
g
2
,
g
3
C
{\displaystyle a,b,c\in {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}
is zero if and only if they lie on the same line in
P
C
2
{\displaystyle \mathbb {P} _{\mathbb {C} }^{2}}
.[20]
This is equivalent to:
det
(
1
℘
(
u
+
v
)
−
℘
′
(
u
+
v
)
1
℘
(
v
)
℘
′
(
v
)
1
℘
(
u
)
℘
′
(
u
)
)
=
0
,
{\displaystyle \det \left({\begin{array}{rrr}1&\wp (u+v)&-\wp '(u+v)\\1&\wp (v)&\wp '(v)\\1&\wp (u)&\wp '(u)\\\end{array}}\right)=0,}
where
℘
(
u
)
=
a
{\displaystyle \wp (u)=a}
,
℘
(
v
)
=
b
{\displaystyle \wp (v)=b}
and
u
,
v
∉
Λ
{\displaystyle u,v\notin \Lambda }
.
[21]
Typography [ edit ]
The Weierstrass's elliptic function is usually written with a rather special, lower case script letter ℘, which was Weierstrass's own notation introduced in his lectures of 1862–1863.[footnote 1]
In computing, the letter ℘ is available as \wp
in TeX . In Unicode the code point is U+2118 ℘ SCRIPT CAPITAL P (℘, ℘ ), with the more correct alias weierstrass elliptic function .[footnote 2] In HTML , it can be escaped as ℘
.
See also [ edit ]
^
This symbol was also used in the version of Weierstrass's lectures published by Schwarz in the 1880s. The first edition of A Course of Modern Analysis by E. T. Whittaker in 1902 also used it.[22]
^
The Unicode Consortium has acknowledged two problems with the letter's name: the letter is in fact lowercase, and it is not a "script" class letter, like U+1D4C5 𝓅 MATHEMATICAL SCRIPT SMALL P , but the letter for Weierstrass's elliptic function.
Unicode added the alias as a correction.[23] [24]
References [ edit ]
^ a b Hulek, Klaus. (2012), Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen (in German) (2., überarb. u. erw. Aufl. 2012 ed.), Wiesbaden: Vieweg+Teubner Verlag, p. 8, ISBN 978-3-8348-2348-9
^ Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 259, ISBN 978-3-540-32058-6
^ Jeremy Gray (2015), Real and the complex: a history of analysis in the 19th century (in German), Cham, p. 71, ISBN 978-3-319-23715-2 {{citation }}
: CS1 maint: location missing publisher (link )
^ Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 294, ISBN 978-3-540-32058-6
^ Ablowitz, Mark J.; Fokas, Athanassios S. (2003). Complex Variables: Introduction and Applications . Cambridge University Press. p. 185. doi :10.1017/cbo9780511791246 . ISBN 978-0-521-53429-1 .
^ a b c d e Apostol, Tom M. (1976), Modular functions and Dirichlet series in number theory (in German), New York: Springer-Verlag, p. 11, ISBN 0-387-90185-X
^ Apostol, Tom M. (1976). Modular functions and Dirichlet series in number theory . New York: Springer-Verlag. p. 14. ISBN 0-387-90185-X . OCLC 2121639 .
^ Apostol, Tom M. (1976), Modular functions and Dirichlet series in number theory (in German), New York: Springer-Verlag, p. 14, ISBN 0-387-90185-X
^ Apostol, Tom M. (1990). Modular functions and Dirichlet series in number theory (2nd ed.). New York: Springer-Verlag. p. 20. ISBN 0-387-97127-0 . OCLC 20262861 .
^ Apostol, Tom M. (1976). Modular functions and Dirichlet series in number theory . New York: Springer-Verlag. p. 50. ISBN 0-387-90185-X . OCLC 2121639 .
^ Chandrasekharan, K. (Komaravolu), 1920- (1985). Elliptic functions . Berlin: Springer-Verlag. p. 122. ISBN 0-387-15295-4 . OCLC 12053023 . {{cite book }}
: CS1 maint: multiple names: authors list (link ) CS1 maint: numeric names: authors list (link )
^ Busam, Rolf (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 270, ISBN 978-3-540-32058-6
^ Apostol, Tom M. (1976), Modular functions and Dirichlet series in number theory (in German), New York: Springer-Verlag, p. 13, ISBN 0-387-90185-X
^ K. Chandrasekharan (1985), Elliptic functions (in German), Berlin: Springer-Verlag, p. 33, ISBN 0-387-15295-4
^ Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers . New York: McGraw–Hill. p. 721. LCCN 59014456 .
^ Reinhardt, W. P.; Walker, P. L. (2010), "Weierstrass Elliptic and Modular Functions" , in Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions , Cambridge University Press, ISBN 978-0-521-19225-5 , MR 2723248 .
^ Hulek, Klaus. (2012), Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen (in German) (2., überarb. u. erw. Aufl. 2012 ed.), Wiesbaden: Vieweg+Teubner Verlag, p. 12, ISBN 978-3-8348-2348-9
^ Hulek, Klaus. (2012), Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen (in German) (2., überarb. u. erw. Aufl. 2012 ed.), Wiesbaden: Vieweg+Teubner Verlag, p. 111, ISBN 978-3-8348-2348-9
^ a b Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 286, ISBN 978-3-540-32058-6
^ Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 287, ISBN 978-3-540-32058-6
^ Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 288, ISBN 978-3-540-32058-6
^ teika kazura (2017-08-17), The letter ℘ Name & origin? , MathOverflow , retrieved 2018-08-30
^ "Known Anomalies in Unicode Character Names" . Unicode Technical Note #27 . version 4. Unicode, Inc. 2017-04-10. Retrieved 2017-07-20 .
^ "NameAliases-10.0.0.txt" . Unicode, Inc. 2017-05-06. Retrieved 2017-07-20 .
Abramowitz, Milton ; Stegun, Irene Ann , eds. (1983) [June 1964]. "Chapter 18" . Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 627. ISBN 978-0-486-61272-0 . LCCN 64-60036 . MR 0167642 . LCCN 65-12253 .
N. I. Akhiezer , Elements of the Theory of Elliptic Functions , (1970) Moscow, translated into English as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island ISBN 0-8218-4532-2
Tom M. Apostol , Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York ISBN 0-387-97127-0 (See chapter 1.)
K. Chandrasekharan, Elliptic functions (1980), Springer-Verlag ISBN 0-387-15295-4
Konrad Knopp , Funktionentheorie II (1947), Dover Publications; Republished in English translation as Theory of Functions (1996), Dover Publications ISBN 0-486-69219-1
Serge Lang , Elliptic Functions (1973), Addison-Wesley, ISBN 0-201-04162-6
E. T. Whittaker and G. N. Watson , A Course of Modern Analysis , Cambridge University Press , 1952, chapters 20 and 21
External links [ edit ]