23 Weierstrass Elliptic and Modular FunctionsApplications23.19 Interrelations23.21 Physical Applications

- §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

The boundary of the rectangle $R$, with vertices $0$, ${\omega}_{1}$, ${\omega}_{1}+{\omega}_{3}$, ${\omega}_{3}$, is mapped strictly monotonically by $\mathrm{\wp}$ onto the real line with $0\to \mathrm{\infty}$, ${\omega}_{1}\to {e}_{1}$, ${\omega}_{1}+{\omega}_{3}\to {e}_{2}$, ${\omega}_{3}\to {e}_{3}$, $0\to -\mathrm{\infty}$. There is a unique point ${z}_{0}\in \left[{\omega}_{1},{\omega}_{1}+{\omega}_{3}\right]\cup \left[{\omega}_{1}+{\omega}_{3},{\omega}_{3}\right]$ such that $\mathrm{\wp}\left({z}_{0}\right)=0$. The interior of $R$ is mapped one-to-one onto the lower half-plane.

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

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 $\mathrm{\wp}\left(\frac{1}{2}{\omega}_{1}+{\omega}_{3}\right)\phantom{\rule{veryverythickmathspace}{0ex}}\left(=\mathrm{\wp}\left(\frac{1}{2}{\omega}_{1}-{\omega}_{3}\right)\right)$. 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 $\mathrm{\wp}\left(z\right)$, see Abramowitz and Stegun (1964, pp. 642–648, 654–655, and 659–60).

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

An algebraic curve that can be put either into the form

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

or equivalently, on replacing $x$ by $x/z$ and $y$ by $y/z$ (projective coordinates), into the form

23.20.2 | $$C:{y}^{2}z={x}^{3}+ax{z}^{2}+b{z}^{3},$$ | ||

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

Points $P=\left(x,y\right)$ on the curve can be parametrized by $x=\mathrm{\wp}\left(z;{g}_{2},{g}_{3}\right)$, $2y={\mathrm{\wp}}^{\prime}\left(z;{g}_{2},{g}_{3}\right)$, where ${g}_{2}=-4a$ and ${g}_{3}=-4b$: in this case we write $P=P\left(z\right)$. The curve $C$ is made into an abelian group (Macdonald (1968, Chapter 5)) by defining the zero element $o=\left(0,1,0\right)$ as the point at infinity, the negative of $P=\left(x,y\right)$ by $-P=\left(x,-y\right)$, 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 ${P}_{j}=P\left({z}_{j}\right)$, $j=1,2,3$, have zero sum iff ${z}_{1}+{z}_{2}+{z}_{3}\in \mathbb{L}$, so that addition of points on the curve $C$ corresponds to addition of parameters ${z}_{j}$ on the torus $\mathrm{\u2102}/\mathbb{L}$; see McKean and Moll (1999, §§2.11, 2.14).

In terms of $\left(x,y\right)$ the addition law can be expressed $\left(x,y\right)+o=\left(x,y\right)$, $\left(x,y\right)+\left(x,-y\right)=o$; otherwise $\left({x}_{1},{y}_{1}\right)+\left({x}_{2},{y}_{2}\right)=\left({x}_{3},{y}_{3}\right)$, where

23.20.3 | ${x}_{3}$ | $={m}^{2}-{x}_{1}-{x}_{2},$ | ||

${y}_{3}$ | $=-m\left({x}_{3}-{x}_{1}\right)-{y}_{1},$ | |||

and

23.20.4 | $$m=\{\begin{array}{cc}\left(3{x}_{1}^{2}+a\right)/\left(2{y}_{1}\right),\hfill & {P}_{1}={P}_{2},\hfill \\ \left({y}_{2}-{y}_{1}\right)/\left({x}_{2}-{x}_{1}\right),\hfill & {P}_{1}\ne {P}_{2}.\hfill \end{array}$$ | ||

If $a,b\in \mathrm{\mathbb{R}}$, then $C$ intersects the plane ${\mathrm{\mathbb{R}}}^{2}$ in a curve that is connected if $\mathrm{\Delta}\equiv 4{a}^{3}+27{b}^{2}>0$; if $$, 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 $C$ of the chord joining them (or the tangent if they coincide); then its reflection in the $x$-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 $a,b\in \mathrm{\mathbb{Q}}$, then by rescaling we may assume $a,b\in \mathrm{\mathbb{Z}}$.
Let $T$ denote the set of points on $C$ that are of finite order (that is,
those points $P$ for which there exists a positive integer $n$ with $nP=o$),
and let $I,K$ be the sets of points with integer and
rational coordinates, respectively. Then
$\mathrm{\varnothing}\subseteq T\subseteq I\subseteq K\subseteq C$.
Both $T,K$ are subgroups of $C$, though $I$ may not be. $K$ always has the form
$T\times {\mathrm{\mathbb{Z}}}^{r}$ (*Mordell’s Theorem*:
Silverman and Tate (1992, Chapter 3, §5)); the determination of $r$, the
rank of $K$, raises questions of
great difficulty, many of which are still open. Both $T$ and $I$ are finite sets.
$T$ must have one of the forms $\mathrm{\mathbb{Z}}/\left(n\mathrm{\mathbb{Z}}\right)$, $1\le n\le 10$ or
$n=12$, or $\left(\mathrm{\mathbb{Z}}/\left(2\mathrm{\mathbb{Z}}\right)\right)\times \left(\mathrm{\mathbb{Z}}/\left(2n\mathrm{\mathbb{Z}}\right)\right)$,
$1\le n\le 4$. To determine $T$, we make use of the fact that if
$\left(x,y\right)\in T$ then ${y}^{2}$ must be a divisor of $\mathrm{\Delta}$; hence there are only a
finite number of possibilities for $y$. Values of $x$ are then found as integer
solutions of ${x}^{3}+ax+b-{y}^{2}=0$ (in particular $x$ must be a divisor of
$b-{y}^{2}$). The resulting points are then tested for finite order as follows.
Given $P$, calculate $2P$, $4P$, $8P$ by doubling as above. If any of these
quantities is zero, then the point has finite order. If any of $2P$, $4P$,
$8P$ is not an integer, then the point has infinite order. Otherwise observe any
equalities between $P$, $2P$, $4P$, $8P$, 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 $2P=-P$, $4P=-P$,
$4P=-2P$, $8P=P$, $8P=-P$, $8P=-2P$, or $8P=-4P$. If none of these
equalities hold, then $P$ has infinite order.

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

The *modular equation* of degree $p$, $p$ prime, is an algebraic equation
in $\alpha =\lambda \left(p\tau \right)$ and $\beta =\lambda \left(\tau \right)$. For
$p=2,3,5,7$ and with $u={\alpha}^{1/4}$, $v={\beta}^{1/4}$, the modular
equation is as follows:

23.20.5 | $${v}^{8}\left(1+{u}^{8}\right)=4{u}^{4},$$ | ||

$p=2$, | |||

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

$p=3$, | |||

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

$p=5$, | |||

23.20.8 | $$\left(1-{u}^{8}\right)\left(1-{v}^{8}\right)={\left(1-uv\right)}^{8},$$ | ||

$p=7$. | |||