§23.20 Mathematical Applications

An algebraic curve that can be put either into the form

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

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

Points $P=(x,y)$ on the curve can be parametrized by $x=\mathrm{\wp }(z;g_2,g_3)$, $2y={\mathrm{\wp }}^{\prime }(z;g_2,g_3)$, where $g_2=-4a$ and $g_3=-4b$: in this case we write $P=P(z)$. The curve $C$ is made into an abelian group (Macdonald (1968, Chapter 5)) by defining the zero element $o=(0,1,0)$ as the point at infinity, the negative of $P=(x,y)$ by $-P=(x,-y)$, 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 $P_j=P(z_j)$, $j=1,2,3$, have zero sum iff $z_1{z}_2{z}_3\in 𝕃$, so that addition of points on the curve $C$ corresponds to addition of parameters $z_j$ on the torus $ℂ/𝕃$; see McKean and Moll (1999, §§2.11, 2.14).

In terms of $(x,y)$ the addition law can be expressed $(x,y)o=(x,y)$, $(x,y)(x,-y)=o$; otherwise $(x_1,y_1)(x_2,y_2)=(x_3,y_3)$, where

and

If $a,b\in ℝ$, then $C$ intersects the plane ${ℝ}^2$ in a curve that is connected if $\mathrm{\Delta }\equiv 4a^{3}27b^2>0$; 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 $a,b\in \mathbb{Q}$, then by rescaling we may assume $a,b\in \mathbb{Z}$. 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 $T\times {\mathbb{Z}}^r$ (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 $\mathbb{Z}/(n\mathbb{Z})$, $1\le n\le 10$ or $n=12$, or $(\mathbb{Z}/(2\mathbb{Z}))\times (\mathbb{Z}/(2n\mathbb{Z}))$, $1\le n\le 4$. To determine $T$, we make use of the fact that if $(x,y)\in T$ 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 $x^{3}axb-y^2=0$ (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).

§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).

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:

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).

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).