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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 is 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.

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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

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} /\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 )} . With that definition, we have ℘ ( z , ω 1 , ω 2 ) = ω 1 − 2 ℘ ( z / ω 1 , ω 2 / ω 1 ) {\displaystyle \wp (z,\omega _{1},\omega _{2})=\omega _{1}^{-2}\wp (z/\omega _{1},\omega _{2}/\omega _{1})} .

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 a homogeneous function in that:
℘ ( λ z , λ ω 1 , λ ω 2 ) = λ − 2 ℘ ( z , ω 1 , ω 2 ) . {\displaystyle \wp (\lambda z,\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-2}\wp (z,\omega _{1},\omega _{2}).}
  • ℘ {\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}} .7 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.8

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 equation9 ℘ ′ 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.10

Invariants

The coefficients of the above differential equation g 2 {\displaystyle g_{2}} and g 3 {\displaystyle 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 {\displaystyle g_{2}} and g 3 {\displaystyle g_{3}} are homogeneous functions of degree − 4 {\displaystyle -4} and − 6 {\displaystyle -6} . That is11 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 {\displaystyle g_{2}} and g 3 {\displaystyle 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:12 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 {\displaystyle g_{2}} and g 3 {\displaystyle 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:13 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 m ( k ) := ∑ d ∣ k d m {\displaystyle \sigma _{m}(k):=\sum _{d\mid {k}}d^{m}} is the divisor function and q = e π i τ {\displaystyle q=e^{\pi i\tau }} is the nome.

Modular discriminant

The modular discriminant Δ {\displaystyle \Delta } is defined as the discriminant of the characteristic polynomial of the differential equation ℘ ′ 2 ( z ) = 4 ℘ 3 ( z ) − g 2 ℘ ( z ) − g 3 {\displaystyle \wp '^{2}(z)=4\wp ^{3}(z)-g_{2}\wp (z)-g_{3}} as follows: 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 {\displaystyle 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 a d − b c = 1 {\displaystyle ad-bc=1} .14

Note that Δ = ( 2 π ) 12 η 24 {\displaystyle \Delta =(2\pi )^{12}\eta ^{24}} where η {\displaystyle \eta } is the Dedekind eta function.15

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.16

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.17 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:18 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:19 ℘ ( 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} )} .20 This also provides a very rapid algorithm for computing ℘ ( z , τ ) {\displaystyle \wp (z,\tau )} .

Relation to elliptic curves

See also: Elliptic curve § Elliptic curves over the complex numbers

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 } ∪ { O } ⊂ C 2 ∪ P 1 ( C ) = 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 \{O\}\subset \mathbb {C} ^{2}\cup \mathbb {P} _{1}(\mathbb {C} )=\mathbb {P} _{2}(\mathbb {C} ).}

where O {\displaystyle O} is a point lying on the line at infinity P 1 ( C ) {\displaystyle \mathbb {P} _{1}(\mathbb {C} )} . For this cubic there exists no rational parameterization, if Δ ≠ 0 {\displaystyle \Delta \neq 0} .21 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 '} :22

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})} .23

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 theorem

The addition theorem states24 that if z , w , {\displaystyle z,w,} and z + w {\displaystyle z+w} do not belong to Λ {\displaystyle \Lambda } , then det [ 1 ℘ ( z ) ℘ ′ ( z ) 1 ℘ ( w ) ℘ ′ ( w ) 1 ℘ ( z + w ) − ℘ ′ ( z + w ) ] = 0. {\displaystyle \det {\begin{bmatrix}1&\wp (z)&\wp '(z)\\1&\wp (w)&\wp '(w)\\1&\wp (z+w)&-\wp '(z+w)\end{bmatrix}}=0.} This states that the points P = ( ℘ ( z ) , ℘ ′ ( z ) ) , {\displaystyle P=(\wp (z),\wp '(z)),} Q = ( ℘ ( w ) , ℘ ′ ( w ) ) , {\displaystyle Q=(\wp (w),\wp '(w)),} and R = ( ℘ ( z + w ) , − ℘ ′ ( z + w ) ) {\displaystyle R=(\wp (z+w),-\wp '(z+w))} are collinear, the geometric form of the group law of an elliptic curve.

This can be proven25 by considering constants A , B {\displaystyle A,B} such that ℘ ′ ( z ) = A ℘ ( z ) + B , ℘ ′ ( w ) = A ℘ ( w ) + B . {\displaystyle \wp '(z)=A\wp (z)+B,\quad \wp '(w)=A\wp (w)+B.} Then the elliptic function ℘ ′ ( ζ ) − A ℘ ( ζ ) − B {\displaystyle \wp '(\zeta )-A\wp (\zeta )-B} has a pole of order three at zero, and therefore three zeros whose sum belongs to Λ {\displaystyle \Lambda } . Two of the zeros are z {\displaystyle z} and w {\displaystyle w} , and thus the third is congruent to − z − w {\displaystyle -z-w} .

Alternative form

The addition theorem can be put into the alternative form, for z , w , z − w , z + w ∉ Λ {\displaystyle z,w,z-w,z+w\not \in \Lambda } :26 ℘ ( 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:27 ℘ ( 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).}

Proofs

1. This can be proven as follows.28 If one looks at the elliptic curve C g 2 , g 3 C {\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }} a line λ = { ( x , y ) ∈ C 2 : y = m x + q } {\displaystyle \lambda =\{(x,y)\in \mathbb {C} ^{2}:y=mx+q\}} intersects it in three points: C g 2 , g 3 C ∩ λ = { P , Q , R } {\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }\cap \lambda =\{P,Q,R\}} . By the group law, these points are P = ( ℘ ( u ) , ℘ ′ ( u ) ) Q = ( ℘ ( v ) , ℘ ′ ( v ) ) {\displaystyle P=(\wp (u),\wp '(u))\quad Q=(\wp (v),\wp '(v))\quad } R = ( ℘ ( u + v ) , − ℘ ′ ( u + v ) ) {\displaystyle R=(\wp (u+v),-\wp '(u+v))} with u , v ∉ Λ {\displaystyle u,v\notin \Lambda } . From the formula of a secant line we have m = y P − y Q x P − x Q = ℘ ′ ( u ) − ℘ ′ ( v ) ℘ ( u ) − ℘ ( v ) {\displaystyle m={\frac {y_{P}-y_{Q}}{x_{P}-x_{Q}}}={\frac {\wp '(u)-\wp '(v)}{\wp (u)-\wp (v)}}} letting C g 2 , g 3 C = λ {\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }=\lambda } we have the equation ( m x + q ) 2 = 4 x 3 − g 2 x − g 3 {\displaystyle (mx+q)^{2}=4x^{3}-g_{2}x-g_{3}} which becomes 4 x 3 − m 2 x 2 − ( 2 m q + g 2 ) x − g 3 − q 2 = 0 {\displaystyle 4x^{3}-m^{2}x^{2}-(2mq+g_{2})x-g_{3}-q^{2}=0} using Vieta's formulas one obtains: x P + x Q + x R = m 2 4 {\displaystyle x_{P}+x_{Q}+x_{R}={\frac {m^{2}}{4}}} which provides the wanted formula ℘ ( u + v ) + ℘ ( u ) + ℘ ( v ) = 1 4 [ ℘ ′ ( u ) − ℘ ′ ( v ) ℘ ( u ) − ℘ ( v ) ] 2 {\displaystyle \wp (u+v)+\wp (u)+\wp (v)={\frac {1}{4}}\left[{\frac {\wp '(u)-\wp '(v)}{\wp (u)-\wp (v)}}\right]^{2}}

2. A second proof from Akhiezer's book29 is the following:

if f {\displaystyle f} is arbitrary elliptic function then: f ( u ) = c ∏ i = 1 n σ ( u − a i ) σ ( u − b i ) c ∈ C {\displaystyle f(u)=c\prod _{i=1}^{n}{\frac {\sigma (u-a_{i})}{\sigma (u-b_{i})}}\quad c\in \mathbb {C} }

where σ {\displaystyle \sigma } is one of the Weierstrass functions and a i , b i {\displaystyle a_{i},b_{i}} are the respective zeros and poles in the period parallelogram. We then let a function k ( u , v ) = ℘ ( u ) − ℘ ( v ) {\displaystyle k(u,v)=\wp (u)-\wp (v)} From the previous lemma we have: k ( u , v ) = ℘ ( u ) − ℘ ( v ) = c σ ( u + v ) σ ( u − v ) σ ( u ) 2 {\displaystyle k(u,v)=\wp (u)-\wp (v)=c{\frac {\sigma (u+v)\sigma (u-v)}{\sigma (u)^{2}}}}

From some calculations one can find that c = 1 σ ( v ) 2 ⟹ ℘ ( u ) − ℘ ( v ) = σ ( u + v ) σ ( u − v ) σ ( u ) 2 σ ( v ) 2 {\displaystyle c={\frac {1}{\sigma (v)^{2}}}\implies \wp (u)-\wp (v)={\frac {\sigma (u+v)\sigma (u-v)}{\sigma (u)^{2}\sigma (v)^{2}}}}

By definition the Weierstrass Zeta function: d d z ln ⁡ σ ( z ) = ζ ( z ) {\displaystyle {\frac {d}{dz}}\ln \sigma (z)=\zeta (z)} therefore we logarithmicly differentiate both sides obtaining: ℘ ′ ( u ) ℘ ( u ) − ℘ ( v ) = ζ ( u + v ) − 2 ζ ( u ) − ζ ( u − v ) {\displaystyle {\frac {\wp '(u)}{\wp (u)-\wp (v)}}=\zeta (u+v)-2\zeta (u)-\zeta (u-v)} Once again by definition ζ ′ ( z ) = − ℘ ( z ) {\displaystyle \zeta '(z)=-\wp (z)} thus by differentiating once more on both sides and rearranging the terms we obtain

− ℘ ( u + v ) = − ℘ ( u ) + 1 2 ℘ ″ ( v ) [ ℘ ( u ) − ℘ ( v ) ] − ℘ ′ ( u ) [ ℘ ′ ( u ) − ℘ ′ ( v ) ] [ ℘ ( u ) − ℘ ( v ) ] 2 {\displaystyle -\wp (u+v)=-\wp (u)+{\frac {1}{2}}{\frac {\wp ''(v)[\wp (u)-\wp (v)]-\wp '(u)[\wp '(u)-\wp '(v)]}{[\wp (u)-\wp (v)]^{2}}}}

Knowing that ℘ ″ {\displaystyle \wp ''} has the following differential equation 2 ℘ ″ = 12 ℘ 2 − g 2 {\displaystyle 2\wp ''=12\wp ^{2}-g_{2}} and rearranging the terms one gets the wanted formula ℘ ( u + v ) = 1 4 [ ℘ ′ ( u ) − ℘ ′ ( v ) ℘ ( u ) − ℘ ( v ) ] 2 − ℘ ( u ) − ℘ ( v ) . {\displaystyle \wp (u+v)={\frac {1}{4}}\left[{\frac {\wp '(u)-\wp '(v)}{\wp (u)-\wp (v)}}\right]^{2}-\wp (u)-\wp (v).}

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 1862–1863.30 It should not be confused with the normal mathematical script letters P: 𝒫 and 𝓅.

In computing, the letter ℘ is available as \wp in TeX. In Unicode the code point is U+2118 ℘ SCRIPT CAPITAL P (&weierp;, &wp;), with the more correct alias weierstrass elliptic function.31 In HTML, it can be escaped as &weierp;.

Character information
Preview℘
Unicode nameSCRIPT CAPITAL P /WEIERSTRASS ELLIPTIC FUNCTION
Encodingsdecimalhex
Unicode8472U+2118
UTF-8226 132 152E2 84 98
Numeric character reference&#8472;&#x2118;
Named character reference&weierp;, &wp;

See also

Footnotes

Wikimedia Commons has media related to Weierstrass's elliptic functions.

References

  1. 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 978-3-8348-2348-9 978-3-8348-2348-9 ↩

  2. Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 259, ISBN 978-3-540-32058-6 978-3-540-32058-6 ↩

  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) 978-3-319-23715-2 ↩

  4. Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 294, ISBN 978-3-540-32058-6 978-3-540-32058-6 ↩

  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. 978-0-521-53429-1 ↩

  6. 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 0-387-90185-X ↩

  7. 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 0-387-90185-X ↩

  8. 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 0-387-90185-X ↩

  9. 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 0-387-90185-X ↩

  10. 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 0-387-90185-X ↩

  11. 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. 0-387-90185-X ↩

  12. 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 0-387-90185-X ↩

  13. 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. 0-387-97127-0 ↩

  14. 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. 0-387-90185-X ↩

  15. 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) 0-387-15295-4 ↩

  16. Busam, Rolf (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 270, ISBN 978-3-540-32058-6 978-3-540-32058-6 ↩

  17. 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 0-387-90185-X ↩

  18. K. Chandrasekharan (1985), Elliptic functions (in German), Berlin: Springer-Verlag, p. 33, ISBN 0-387-15295-4 0-387-15295-4 ↩

  19. Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw–Hill. p. 721. LCCN 59014456. /wiki/Theresa_M._Korn ↩

  20. 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. 978-0-521-19225-5 ↩

  21. 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 978-3-8348-2348-9 978-3-8348-2348-9 ↩

  22. 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 978-3-8348-2348-9 978-3-8348-2348-9 ↩

  23. 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 978-3-8348-2348-9 978-3-8348-2348-9 ↩

  24. Watson; Whittaker (1927), A course in modern analysis (4 ed.), Cambridge University Press, pp. 440–441 ↩

  25. Watson; Whittaker (1927), A course in modern analysis (4 ed.), Cambridge University Press, pp. 440–441 ↩

  26. Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 286, ISBN 978-3-540-32058-6 978-3-540-32058-6 ↩

  27. Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 286, ISBN 978-3-540-32058-6 978-3-540-32058-6 ↩

  28. Watson; Whittaker (1927), A course in modern analysis (4 ed.), Cambridge University Press, pp. 440–441 ↩

  29. Akhiezer's book Elements of the theory of elliptic functions https://www.ams.org/books/mmono/079/mmono079-endmatter.pdf https://www.ams.org/books/mmono/079/mmono079-endmatter.pdf ↩

  30. 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.[24] /wiki/A_Course_of_Modern_Analysis ↩

  31. 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.[25][26] /wiki/Unicode_Consortium ↩