Weierstrass elliptic function

Class of mathematical functions

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 P function

Symbol for Weierstrass ℘ {\displaystyle \wp } -function

Model of Weierstrass ℘ {\displaystyle \wp } -function

Motivation

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

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

  • ℘ {\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

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

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

The real part of the invariant g3 as a function of the square of the nome q on the unit disk.
The imaginary part of the invariant g3 as a function of the square of the nome q on the unit disk.

The coefficients of the above differential equation g2 and g3 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 g2 and g3 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} , g2 and g3 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 g2 and g3 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

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 e1, e2 and e3

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

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

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

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

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

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 weierstrass elliptic function.[footnote 2] In HTML, it can be escaped as &weierp;.

Character information
Preview ℘
Unicode name SCRIPT CAPITAL P / WEIERSTRASS ELLIPTIC FUNCTION
Encodings decimal hex
Unicode 8472 U+2118
UTF-8 226 132 152 E2 84 98
Numeric character reference &#8472; &#x2118;
Named character reference &weierp;, &wp;

See also

Footnotes

  1. ^ 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]
  2. ^ 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

  1. ^ 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
  2. ^ Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 259, ISBN 978-3-540-32058-6
  3. ^ 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)
  4. ^ Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 294, ISBN 978-3-540-32058-6
  5. ^ 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.
  6. ^ 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
  7. ^ 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.
  8. ^ 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
  9. ^ 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.
  10. ^ 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.
  11. ^ 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)
  12. ^ Busam, Rolf (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 270, ISBN 978-3-540-32058-6
  13. ^ 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
  14. ^ K. Chandrasekharan (1985), Elliptic functions (in German), Berlin: Springer-Verlag, p. 33, ISBN 0-387-15295-4
  15. ^ Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw–Hill. p. 721. LCCN 59014456.
  16. ^ 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.
  17. ^ 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
  18. ^ 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
  19. ^ 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
  20. ^ Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 287, ISBN 978-3-540-32058-6
  21. ^ Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 288, ISBN 978-3-540-32058-6
  22. ^ teika kazura (2017-08-17), The letter ℘ Name & origin?, MathOverflow, retrieved 2018-08-30
  23. ^ "Known Anomalies in Unicode Character Names". Unicode Technical Note #27. version 4. Unicode, Inc. 2017-04-10. Retrieved 2017-07-20.
  24. ^ "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

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