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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.

• Motivation

• Definition

• Properties

• Laurent expansion

• Differential equation

• Invariants

• Modular discriminant

• The constants e 1 , e 2 and e 3

• Relation to Jacobi's elliptic functions

• Relation to Jacobi's theta functions

• Relation to elliptic curves

• Addition theorem

• Alternative form

• Typography

• See also

• Footnotes

• References

• External links

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.cite-ref-5-1-0[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 ψ : R / 2 π 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 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 Λ {\displaystyle \mathbb {C} /\Lambda } , which is topologically equivalent to a torus.cite-ref-2[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.cite-ref-3[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.cite-ref-4[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.cite-ref-5[5]

Let ω ω 1 , ω 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 Λ := Z ω ω 1 + Z ω ω 2 := { m ω ω 1 + n ω 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 ω 1 , ω ω 2 ) := ℘ ℘ ( z ) = 1 z 2 + ∑ ∑ λ λ ∈ ∈ Λ Λ ∖ ∖ { 0 } ( 1 ( z − − λ λ ) 2 − − 1 λ 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 Λ {\displaystyle \mathbb {C} /\Lambda } .

It is common to use 1 {\displaystyle 1} and τ 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 ω 1 {\textstyle \omega _{1}} maps the lattice Z ω ω 1 + Z ω ω 2 {\displaystyle \mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}} isomorphically onto the lattice Z + Z τ τ {\displaystyle \mathbb {Z} +\mathbb {Z} \tau } with τ τ = ω ω 2 ω ω 1 {\textstyle \tau ={\tfrac {\omega _{2}}{\omega _{1}}}} . Because − − τ τ {\displaystyle -\tau } can be substituted for τ τ {\displaystyle \tau } , without loss of generality we can assume τ τ ∈ ∈ H {\displaystyle \tau \in \mathbb {H} } , and then define ℘ ℘ ( z , τ τ ) := ℘ ℘ ( z , 1 , τ τ ) {\displaystyle \wp (z,\tau ):=\wp (z,1,\tau )} . With that definition, we have ℘ ℘ ( z , ω ω 1 , ω ω 2 ) = ω ω 1 − − 2 ℘ ℘ ( z / ω ω 1 , ω ω 2 / ω ω 1 ) {\displaystyle \wp (z,\omega _{1},\omega _{2})=\omega _{1}^{-2}\wp (z/\omega _{1},\omega _{2}/\omega _{1})} .

• ℘ ℘ {\displaystyle \wp } is a meromorphic function with a pole of order 2 at each period λ λ {\displaystyle \lambda } in Λ Λ {\displaystyle \Lambda } .

• ℘ ℘ {\displaystyle \wp } is a homogeneous function in that:

℘ ℘ ( λ λ z , λ λ ω ω 1 , λ λ ω ω 2 ) = λ λ − − 2 ℘ ℘ ( z , ω ω 1 , ω 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 Λ {\displaystyle z\in \mathbb {C} \setminus \Lambda } , which can be seen in the following way:

℘ ℘ ( − − z ) = 1 ( − − z ) 2 + ∑ ∑ λ λ ∈ ∈ Λ Λ ∖ ∖ { 0 } ( 1 ( − − z − − λ λ ) 2 − − 1 λ λ 2 ) = 1 z 2 + ∑ ∑ λ λ ∈ ∈ Λ Λ ∖ ∖ { 0 } ( 1 ( z + λ λ ) 2 − − 1 λ λ 2 ) = 1 z 2 + ∑ ∑ λ λ ∈ ∈ Λ Λ ∖ ∖ { 0 } ( 1 ( z − − λ λ ) 2 − − 1 λ 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 ∈ ∈ 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:cite-ref-1-6-0[6] ℘ ℘ ′ ( z ) = − − 2 ∑ ∑ λ λ ∈ ∈ Λ Λ 1 ( z − − λ 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 ω 1 {\displaystyle \omega _{1}} and ω ω 2 {\displaystyle \omega _{2}} .cite-ref-1-6-1[6] This means: ℘ ℘ ( z + ω ω 1 ) = ℘ ℘ ( z ) = ℘ ℘ ( z + ω ω 2 ) , and ℘ ℘ ′ ( z + ω ω 1 ) = ℘ ℘ ′ ( z ) = ℘ ℘ ′ ( z + ω 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 λ ) = ℘ ℘ ( z ) {\displaystyle \wp (z+\lambda )=\wp (z)} and ℘ ℘ ′ ( z + λ λ ) = ℘ ℘ ′ ( z ) {\displaystyle \wp '(z+\lambda )=\wp '(z)} for all λ λ ∈ ∈ Λ Λ {\displaystyle \lambda \in \Lambda } .

Let r := min { | λ λ | : 0 ≠ ≠ λ λ ∈ ∈ 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 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.cite-ref-1-6-2[6]

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 equationcite-ref-1-6-3[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.cite-ref-1-6-4[6]

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 Λ {\displaystyle \Lambda } they can be viewed as functions in ω ω 1 {\displaystyle \omega _{1}} and ω 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 iscite-ref-0-7-0[7] g 2 ( λ λ ω ω 1 , λ λ ω ω 2 ) = λ λ − − 4 g 2 ( ω ω 1 , ω ω 2 ) {\displaystyle g_{2}(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-4}g_{2}(\omega _{1},\omega _{2})} g 3 ( λ λ ω ω 1 , λ λ ω ω 2 ) = λ λ − − 6 g 3 ( ω ω 1 , ω ω 2 ) {\displaystyle g_{3}(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-6}g_{3}(\omega _{1},\omega _{2})} for λ λ ≠ ≠ 0 {\displaystyle \lambda \neq 0} .

If ω ω 1 {\displaystyle \omega _{1}} and ω ω 2 {\displaystyle \omega _{2}} are chosen in such a way that Im ⁡ ⁡ ( ω ω 2 ω 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 τ = ω ω 2 ω ω 1 {\displaystyle \tau ={\tfrac {\omega _{2}}{\omega _{1}}}} . One has:cite-ref-2-8-0[8] g 2 ( 1 , τ τ ) = ω ω 1 4 g 2 ( ω ω 1 , ω ω 2 ) , {\displaystyle g_{2}(1,\tau )=\omega _{1}^{4}g_{2}(\omega _{1},\omega _{2}),} g 3 ( 1 , τ τ ) = ω ω 1 6 g 3 ( ω ω 1 , ω 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 τ ) := g 2 ( 1 , τ τ ) {\displaystyle g_{2}(\tau ):=g_{2}(1,\tau )} and g 3 ( τ τ ) := g 3 ( 1 , τ τ ) . {\displaystyle g_{3}(\tau ):=g_{3}(1,\tau ).} As functions of τ 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:cite-ref-9[9] g 2 ( τ τ ) = 4 3 π π 4 [ 1 + 240 ∑ ∑ k = 1 ∞ ∞ σ 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 τ ) = 8 27 π π 6 [ 1 − − 504 ∑ ∑ k = 1 ∞ ∞ σ 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 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 π i τ τ {\displaystyle q=e^{\pi i\tau }} is the nome.

The modular discriminant Δ 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 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 Δ ( a τ τ + b c τ τ + d ) = ( c τ τ + d ) 12 Δ Δ ( τ 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} .cite-ref-10[10]

Note that Δ Δ = ( 2 π π ) 12 η η 24 {\displaystyle \Delta =(2\pi )^{12}\eta ^{24}} where η η {\displaystyle \eta } is the Dedekind eta function.cite-ref-11[11]

For the Fourier coefficients of Δ Δ {\displaystyle \Delta } , see Ramanujan tau function.

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 ω 1 2 ) {\displaystyle e_{1}\equiv \wp \left({\frac {\omega _{1}}{2}}\right)} e 2 ≡ ≡ ℘ ℘ ( ω ω 2 2 ) {\displaystyle e_{2}\equiv \wp \left({\frac {\omega _{2}}{2}}\right)} e 3 ≡ ≡ ℘ ℘ ( ω ω 1 + ω 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 Λ {\displaystyle \Lambda } and not on its generators.cite-ref-12[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 Δ {\displaystyle \Delta } does not vanish on the upper half plane.cite-ref-13[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:cite-ref-14[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 λ ( τ τ ) = e 3 − − e 2 e 1 − − e 2 , τ τ = ω ω 2 ω ω 1 . {\displaystyle \lambda (\tau )={\frac {e_{3}-e_{2}}{e_{1}-e_{2}}},\quad \tau ={\frac {\omega _{2}}{\omega _{1}}}.}

For numerical work, it is often convenient to calculate the Weierstrass elliptic function in terms of Jacobi's elliptic functions.

The basic relations are:cite-ref-15[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}}}.}

The function ℘ ℘ ( z , τ τ ) = ℘ ℘ ( z , 1 , ω ω 2 / ω ω 1 ) {\displaystyle \wp (z,\tau )=\wp (z,1,\omega _{2}/\omega _{1})} can be represented by Jacobi's theta functions: ℘ ℘ ( z , τ τ ) = ( π π θ θ 2 ( 0 , q ) θ θ 3 ( 0 , q ) θ θ 4 ( π π z , q ) θ θ 1 ( π π z , q ) ) 2 − − π π 2 3 ( θ θ 2 4 ( 0 , q ) + θ 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 π i τ τ {\displaystyle q=e^{\pi i\tau }} is the nome and τ τ {\displaystyle \tau } is the period ratio ( τ τ ∈ ∈ H ) {\displaystyle (\tau \in \mathbb {H} )} .cite-ref-16[16] This also provides a very rapid algorithm for computing ℘ ℘ ( z , τ τ ) {\displaystyle \wp (z,\tau )} .

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 Δ ≠ ≠ 0 {\displaystyle \Delta \neq 0} .cite-ref-5-1-1[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 '} :cite-ref-17[17]

φ φ ( ℘ ℘ , ℘ ℘ ′ ) : C / Λ Λ → → C ¯ ¯ g 2 , g 3 C , z ↦ ↦ { [ ℘ ℘ ( z ) : ℘ ℘ ′ ( z ) : 1 ] z ∉ ∉ Λ Λ [ 0 : 1 : 0 ] z ∈ ∈ Λ 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 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 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 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 ω 1 + Z ω ω 2 {\displaystyle \mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}} , such that

g 2 = g 2 ( ω ω 1 , ω ω 2 ) {\displaystyle g_{2}=g_{2}(\omega _{1},\omega _{2})} and g 3 = g 3 ( ω ω 1 , ω ω 2 ) {\displaystyle g_{3}=g_{3}(\omega _{1},\omega _{2})} .cite-ref-18[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.

The addition theorem statescite-ref-19[19] that if z , w , {\displaystyle z,w,} and z + w {\displaystyle z+w} do not belong to Λ 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 provencite-ref-20[20] 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 ζ ) − − A ℘ ℘ ( ζ 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 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} .

The addition theorem can be put into the alternative form, for z , w , z − − w , z + w ∉ Λ Λ {\displaystyle z,w,z-w,z+w\not \in \Lambda } :cite-ref-3-21-0[21] ℘ ℘ ( 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:cite-ref-3-21-1[21] ℘ ℘ ( 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).}

This can be proven from the addition theorem shown above. The points P = ( ℘ ℘ ( u ) , ℘ ℘ ′ ( u ) ) , Q = ( ℘ ℘ ( v ) , ℘ ℘ ′ ( v ) ) , {\displaystyle P=(\wp (u),\wp '(u)),Q=(\wp (v),\wp '(v)),} and R = ( ℘ ℘ ( u + v ) , − − ℘ ℘ ′ ( u + v ) ) {\displaystyle R=(\wp (u+v),-\wp '(u+v))} are collinear and lie on the curve y 2 = 4 x 3 − − g 2 x − − g 3 {\displaystyle y^{2}=4x^{3}-g_{2}x-g_{3}} . The slope of that line is 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)}}.} So x = x P = ℘ ℘ ( u ) {\displaystyle x=x_{P}=\wp (u)} , x = x Q = ℘ ℘ ( v ) {\displaystyle x=x_{Q}=\wp (v)} , and x = x R = ℘ ℘ ( u + v ) {\displaystyle x=x_{R}=\wp (u+v)} all satisfy a cubic ( m x + q ) 2 = 4 x 3 − − g 2 x − − g 3 , {\displaystyle (mx+q)^{2}=4x^{3}-g_{2}x-g_{3},} where q {\displaystyle q} is a constant. This 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.} Thus 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}.}

A direct proof is as follows.cite-ref-22[22] Any elliptic function f {\displaystyle f} can be expressed as: f ( u ) = c ∏ ∏ i = 1 n σ σ ( u − − a i ) σ 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 σ {\displaystyle \sigma } is the Weierstrass sigma function and a i , b i {\displaystyle a_{i},b_{i}} are the respective zeros and poles in the period parallelogram. Considering the function ℘ ℘ ( u ) − − ℘ ℘ ( v ) {\displaystyle \wp (u)-\wp (v)} as a function of u {\displaystyle u} , we have ℘ ℘ ( u ) − − ℘ ℘ ( v ) = c σ σ ( u + v ) σ σ ( u − − v ) σ σ ( u ) 2 . {\displaystyle \wp (u)-\wp (v)=c{\frac {\sigma (u+v)\sigma (u-v)}{\sigma (u)^{2}}}.} Multiplying both sides by u 2 {\displaystyle u^{2}} and letting u → → 0 {\displaystyle u\to 0} , we have 1 = − − c σ σ ( v ) 2 {\displaystyle 1=-c\sigma (v)^{2}} , so c = − − 1 σ σ ( v ) 2 ⟹ ⟹ ℘ ℘ ( u ) − − ℘ ℘ ( v ) = − − σ σ ( u + v ) σ σ ( u − − v ) σ σ ( u ) 2 σ 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 σ ( z ) = ζ ζ ( z ) {\displaystyle {\frac {d}{dz}}\ln \sigma (z)=\zeta (z)} therefore we logarithmically differentiate both sides with respect to u {\displaystyle u} obtaining: ℘ ℘ ′ ( u ) ℘ ℘ ( u ) − − ℘ ℘ ( v ) = ζ ζ ( u + v ) − − 2 ζ ζ ( u ) − − ζ 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 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).}

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.cite-ref-24[footnote 1] 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.cite-ref-27[footnote 2] In HTML, it can be escaped as &weierp;.

| Preview | ℘ | ℘ |

|---|---|---|

| Unicode name | SCRIPT CAPITAL P / WEIERSTRASS ELLIPTIC FUNCTION | 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; | &weierp;, &wp; |

• Weierstrass functions

• Jacobi elliptic functions

• Lemniscate elliptic functions

cite-note-24footnote 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.cite-ref-23[23]

cite-note-27footnote 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.cite-ref-25[24]cite-ref-26[25]

cite-note-5-11. ↑ citerefhulek-klaus-2012Hulek, 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

cite-note-22. ↑ citerefrolf-busam2006Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 259, ISBN 978-3-540-32058-6

cite-note-33. ↑ citerefjeremy-gray2015Jeremy 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)

cite-note-44. ↑ citerefrolf-busam2006Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 294, ISBN 978-3-540-32058-6

cite-note-55. ↑ citerefablowitzfokas2003Ablowitz, 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.

cite-note-1-66. ↑ citerefapostol-tom-m-1976Apostol, Tom M. (1976), Modular functions and Dirichlet series in number theory (in German), New York: Springer-Verlag, p. 11, ISBN 0-387-90185-X

cite-note-0-77. ↑ citerefapostol-tom-m-1976Apostol, Tom M. (1976). Modular functions and Dirichlet series in number theory. New York: Springer-Verlag. p. 14. ISBN 0-387-90185-X. OCLC 2121639.

cite-note-2-88. ↑ citerefapostol-tom-m-1976Apostol, Tom M. (1976), Modular functions and Dirichlet series in number theory (in German), New York: Springer-Verlag, p. 14, ISBN 0-387-90185-X

cite-note-99. ↑ citerefapostol-tom-m-1990Apostol, 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.

cite-note-1010. ↑ citerefapostol1976Apostol, Tom M. (1976). Modular functions and Dirichlet series in number theory. New York: Springer-Verlag. p. 50. ISBN 0-387-90185-X. OCLC 2121639.

cite-note-1111. ↑ citerefchandrasekharan-k-komaravolu-1920-1985Chandrasekharan, 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)

cite-note-1212. ↑ citerefbusam2006Busam, Rolf (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 270, ISBN 978-3-540-32058-6

cite-note-1313. ↑ citerefapostol1976Apostol, Tom M. (1976), Modular functions and Dirichlet series in number theory (in German), New York: Springer-Verlag, p. 13, ISBN 0-387-90185-X

cite-note-1414. ↑ citerefk-chandrasekharan1985K. Chandrasekharan (1985), Elliptic functions (in German), Berlin: Springer-Verlag, p. 33, ISBN 0-387-15295-4

cite-note-1515. ↑ citerefkorn-ga-korn-tm1961Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw–Hill. p. 721. LCCN 59014456.

cite-note-1616. ↑ citerefreinhardtwalker2010Reinhardt, 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.

cite-note-1717. ↑ citerefhulek-klaus-2012Hulek, 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

cite-note-1818. ↑ citerefhulek-klaus-2012Hulek, 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

cite-note-1919. ↑ citerefwatsonwhittaker1927Watson; Whittaker (1927), A course in modern analysis (4 ed.), Cambridge University Press, pp. 440–441

cite-note-2020. ↑ citerefwatsonwhittaker1927Watson; Whittaker (1927), A course in modern analysis (4 ed.), Cambridge University Press, pp. 440–441

cite-note-3-2121. ↑ citerefrolf-busam2006Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 286, ISBN 978-3-540-32058-6

cite-note-2222. ↑ citerefakhiezer1990Akhiezer (1990), Elements of the theory of elliptic functions, AMS, pp. 40–41

cite-note-2323. ↑ citerefteika-kazura2017teika kazura (2017-08-17), The letter ℘ Name & origin?, MathOverflow, retrieved 2018-08-30

cite-note-2524. ↑ "Known Anomalies in Unicode Character Names". Unicode Technical Note #27. version 4. Unicode, Inc. 2017-04-10. Retrieved 2017-07-20.

cite-note-2625. ↑ "NameAliases-10.0.0.txt". Unicode, Inc. 2017-05-06. Retrieved 2017-07-20.

• citerefabramowitzstegun1983Abramowitz, 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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