23 Weierstrass Elliptic and Modular Functions Applications 23.19 Interrelations 23.21 Physical Applications

§23.20 Mathematical Applications

Keywords:: Weierstrass elliptic functions, modular functions
Permalink:: http://dlmf.nist.gov/23.20

§23.20(i) Conformal Mappings
§23.20(ii) Elliptic Curves
§23.20(iii) Factorization
§23.20(iv) Modular and Quintic Equations
§23.20(v) Modular Functions and Number Theory

§23.20(i) Conformal Mappings

Notes:: See McKean and Moll (1999, §2.8).
Keywords:: Weierstrass elliptic functions, conformal mapping
Permalink:: http://dlmf.nist.gov/23.20.i

¶ Rectangular Lattice

The boundary of the rectangle , with vertices 0, $\omega_{1}$ , $\omega_{1}+\omega_{3}$ , $\omega_{3}$ , is mapped strictly monotonically by $\mathop{\wp\/}\nolimits$ onto the real line with $0\to\infty$ , $\omega_{1}\to e_{1}$ , $\omega_{1}+\omega_{3}\to e_{2}$ , $\omega_{3}\to e_{3}$ , $0\to-\infty$ . There is a unique point $z_{0}\in[\omega_{1},\omega_{1}+\omega_{3}]\cup[\omega_{1}+\omega_{3},\omega_{3}]$ such that $\mathop{\wp\/}\nolimits\!\left(z_{0}\right)=0$ . The interior of is mapped one-to-one onto the lower half-plane.

¶ Rhombic Lattice

Keywords:: Weierstrass elliptic functions, conformal mapping, modular functions

The two pairs of edges $[0,\omega_{1}]\cup[\omega_{1},2\omega_{3}]$ and $[2\omega_{3},2\omega_{3}-\omega_{1}]\cup[2\omega_{3}-\omega_{1},0]$ of are each mapped strictly monotonically by $\mathop{\wp\/}\nolimits$ onto the real line, with $0\to\infty$ , $\omega_{1}\to e_{1}$ , $2\omega_{3}\to-\infty$ ; similarly for the other pair of edges. For each pair of edges there is a unique point $z_{0}$ such that $\mathop{\wp\/}\nolimits\!\left(z_{0}\right)=0$ .

The interior of the rectangle with vertices 0, $\omega_{1}$ , $2\omega_{3}$ , $2\omega_{3}-\omega_{1}$ is mapped two-to-one onto the lower half-plane. The interior of the rectangle with vertices 0, $\omega_{1}$ , $\frac{1}{2}\omega_{1}+\omega_{3}$ , $\frac{1}{2}\omega_{1}-\omega_{3}$ is mapped one-to-one onto the lower half-plane with a cut from $e_{3}$ to $\mathop{\wp\/}\nolimits\!\left(\frac{1}{2}\omega_{1}+\omega_{3}\right)\>(=% \mathop{\wp\/}\nolimits\!\left(\frac{1}{2}\omega_{1}-\omega_{3}\right))$ . The cut is the image of the edge from $\frac{1}{2}\omega_{1}+\omega_{3}$ to $\frac{1}{2}\omega_{1}-\omega_{3}$ and is not a line segment.

For examples of conformal mappings of the function $\mathop{\wp\/}\nolimits\!\left(z\right)$ , see Abramowitz and Stegun (1964, pp. 642–648, 654–655, and 659–60).

For conformal mappings via modular functions see Apostol (1990, §2.7).

§23.20(ii) Elliptic Curves

Keywords:: Mordell’s theorem, Weierstrass elliptic functions, coordinate systems, elliptic curves, projective coordinates
Referenced by:: §22.18(iv), §23.1, §23.1, ¶ ‣ §23.22(ii)
Permalink:: http://dlmf.nist.gov/23.20.ii

An algebraic curve that can be put either into the form

23.20.1 $C:y^{2}=x^{3}+ax+b,$

Symbols:: : real part of , : imaginary part of , : elliptic curve, : constant and : constant
Permalink:: http://dlmf.nist.gov/23.20.E1
Encodings:: TeX, pMML, png

or equivalently, on replacing by and by (projective coordinates), into the form

23.20.2 $C:y^{2}z=x^{3}+axz^{2}+bz^{3},$

Symbols:: : real part of , : imaginary part of , : complex, : elliptic curve, : constant and : constant
Permalink:: http://dlmf.nist.gov/23.20.E2
Encodings:: TeX, pMML, png

is an example of an elliptic curve (§22.18(iv)). Here and are real or complex constants.

Points on the curve can be parametrized by $x=\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ , $2y={\mathop{\wp\/}\nolimits^{{\prime}}}\!\left(z;g_{2},g_{3}\right)$ , where $g_{2}=-4a$ and $g_{3}=-4b$ : in this case we write . The curve is made into an abelian group (Macdonald (1968, Chapter 5)) by defining the zero element as the point at infinity, the negative of by , and generally $P_{1}+P_{2}+P_{3}=0$ on the curve iff the points $P_{1}$ , $P_{2}$ , $P_{3}$ are collinear. It follows from the addition formula (23.10.1) that the points $P_{j}=P(z_{j})$ , , have zero sum iff $z_{1}+z_{2}+z_{3}\in\mathbb{L}$ , so that addition of points on the curve corresponds to addition of parameters $z_{j}$ on the torus $\Complex\setmod\mathbb{L}$ ; see McKean and Moll (1999, §§2.11, 2.14).

In terms of the addition law can be expressed , ; otherwise $(x_{1},y_{1})+(x_{2},y_{2})=(x_{3},y_{3})$ , where

23.20.3

$x_{3}=m^{2}-x_{1}-x_{2},$

$y_{3}=-m(x_{3}-x_{1})-y_{1},$

Symbols:: : integer, : real part of and : imaginary part of
Permalink:: http://dlmf.nist.gov/23.20.E3
Encodings:: TeX, TeX, pMML, pMML, png, png

and

23.20.4 $m=\begin{cases}(3x_{1}^{2}+a)/(2y_{1}),&P_{1}=P_{2},\\ (y_{2}-y_{1})/(x_{2}-x_{1}),&P_{1}\neq P_{2}.\end{cases}$

Symbols:: : integer, : real part of , : imaginary part of , : constant and : points
Permalink:: http://dlmf.nist.gov/23.20.E4
Encodings:: TeX, pMML, png

If $a,b\in\Real$ , then intersects the plane $\Real^{2}$ in a curve that is connected if $\Delta\equiv 4a^{3}+27b^{2}>0$ ; if $\Delta<0$ , then the intersection has two components, one of which is a closed loop. These cases correspond to rhombic and rectangular lattices, respectively. The addition law states that to find the sum of two points, take the third intersection with of the chord joining them (or the tangent if they coincide); then its reflection in the -axis gives the required sum. The geometric nature of this construction is illustrated in McKean and Moll (1999, §2.14), Koblitz (1993, §§6, 7), and Silverman and Tate (1992, Chapter 1, §§3, 4): each of these references makes a connection with the addition theorem (23.10.1).

If $a,b\in\Rational$ , then by rescaling we may assume $a,b\in\Integer$ . Let denote the set of points on that are of finite order (that is, those points for which there exists a positive integer with ), and let be the sets of points with integer and rational coordinates, respectively. Then $\emptyset\subseteq T\subseteq I\subseteq K\subseteq C$ . Both are subgroups of , though may not be. always has the form $T\times\Integer^{r}$ (Mordell’s Theorem: Silverman and Tate (1992, Chapter 3, §5)); the determination of , the rank of , raises questions of great difficulty, many of which are still open. Both and are finite sets. must have one of the forms $\Integer\setmod(n\Integer)$ , $1\leq n\leq 10$ or , or $(\Integer\setmod(2\Integer))\times(\Integer\setmod(2n\Integer))$ , $1\leq n\leq 4$ . To determine , we make use of the fact that if $(x,y)\in T$ then $y^{2}$ must be a divisor of $\Delta$ ; hence there are only a finite number of possibilities for . Values of are then found as integer solutions of $x^{3}+ax+b-y^{2}=0$ (in particular must be a divisor of $b-y^{2}$ ). The resulting points are then tested for finite order as follows. Given , calculate , , by doubling as above. If any of these quantities is zero, then the point has finite order. If any of , , is not an integer, then the point has infinite order. Otherwise observe any equalities between , , , , and their negatives. The order of a point (if finite and not already determined) can have only the values 3, 5, 6, 7, 9, 10, or 12, and so can be found from , , , , , , or . If none of these equalities hold, then has infinite order.

For extensive tables of elliptic curves see Cremona (1997, pp. 84–340).

§23.20(iii) Factorization

Keywords:: Weierstrass elliptic functions, cryptography, factorization, primality testing
Permalink:: http://dlmf.nist.gov/23.20.iii

§27.16 describes the use of primality testing and factorization in cryptography. For applications of the Weierstrass function and the elliptic curve method to these problems see Bressoud (1989) and Koblitz (1999).

§23.20(iv) Modular and Quintic Equations

Keywords:: modular equations, modular functions, quintic equations
Permalink:: http://dlmf.nist.gov/23.20.iv

The modular equation of degree , prime, is an algebraic equation in $\alpha=\mathop{\lambda\/}\nolimits\!\left(p\tau\right)$ and $\beta=\mathop{\lambda\/}\nolimits\!\left(\tau\right)$ . For and with $u=\alpha^{{1/4}}$ , $v=\beta^{{1/4}}$ , the modular equation is as follows:

23.20.5 $v^{8}(1+u^{8})=4u^{4},$

Symbols:: , and : degree
Permalink:: http://dlmf.nist.gov/23.20.E5
Encodings:: TeX, pMML, png

23.20.6 $u^{4}-v^{4}+2uv(1-u^{2}v^{2})=0,$

Symbols:: , and : degree
Permalink:: http://dlmf.nist.gov/23.20.E6
Encodings:: TeX, pMML, png

23.20.7 ${u^{6}-v^{6}+5u^{2}v^{2}(u^{2}-v^{2})+4uv(1-u^{4}v^{4})=0},$

Symbols:: , and : degree
Referenced by:: §23.20(iv)
Permalink:: http://dlmf.nist.gov/23.20.E7
Encodings:: TeX, pMML, png

23.20.8 $(1-u^{8})(1-v^{8})=(1-uv)^{8},$

Symbols:: , and : degree
Permalink:: http://dlmf.nist.gov/23.20.E8
Encodings:: TeX, pMML, png

For further information, including the application of (23.20.7) to the solution of the general quintic equation, see Borwein and Borwein (1987, Chapter 4).

§23.20(v) Modular Functions and Number Theory

Keywords:: Weierstrass elliptic functions, modular functions, number theory
Permalink:: http://dlmf.nist.gov/23.20.v

For applications of modular functions to number theory see §27.14(iv) and Apostol (1990). See also Silverman and Tate (1992), Serre (1973, Part 2, Chapters 6, 7), Koblitz (1993), and Cornell et al. (1997).

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