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23 Weierstrass Elliptic and Modular FunctionsApplications

§23.20 Mathematical Applications

Contents

§23.20(i) Conformal Mappings

Rectangular Lattice

The boundary of the rectangle R, with vertices 0, ω1, ω1+ω3, ω3, is mapped strictly monotonically by onto the real line with 0, ω1e1, ω1+ω3e2, ω3e3, 0-. There is a unique point z0[ω1,ω1+ω3][ω1+ω3,ω3] such that (z0)=0. The interior of R is mapped one-to-one onto the lower half-plane.

Rhombic Lattice

The two pairs of edges [0,ω1][ω1,2ω3] and [2ω3,2ω3-ω1][2ω3-ω1,0] of R are each mapped strictly monotonically by onto the real line, with 0, ω1e1, 2ω3-; similarly for the other pair of edges. For each pair of edges there is a unique point z0 such that (z0)=0.

The interior of the rectangle with vertices 0, ω1, 2ω3, 2ω3-ω1 is mapped two-to-one onto the lower half-plane. The interior of the rectangle with vertices 0, ω1, 12ω1+ω3, 12ω1-ω3 is mapped one-to-one onto the lower half-plane with a cut from e3 to (12ω1+ω3)(=(12ω1-ω3)). The cut is the image of the edge from 12ω1+ω3 to 12ω1-ω3 and is not a line segment.

For examples of conformal mappings of the function (z), 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

An algebraic curve that can be put either into the form

23.20.1 C:y2=x3+ax+b,

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

23.20.2 C:y2z=x3+axz2+bz3,

is an example of an elliptic curve22.18(iv)). Here a and b are real or complex constants.

Points P=(x,y) on the curve can be parametrized by x=(z;g2,g3), 2y=(z;g2,g3), where g2=-4a and g3=-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 P1+P2+P3=0 on the curve iff the points P1, P2, P3 are collinear. It follows from the addition formula (23.10.1) that the points Pj=P(zj), j=1,2,3, have zero sum iff z1+z2+z3𝕃, so that addition of points on the curve C corresponds to addition of parameters zj on the torus /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 (x1,y1)+(x2,y2)=(x3,y3), where

23.20.3 x3 =m2-x1-x2,
y3 =-m(x3-x1)-y1,

and

23.20.4 m={(3x12+a)/(2y1),P1=P2,(y2-y1)/(x2-x1),P1P2.

If a,b, then C intersects the plane 2 in a curve that is connected if Δ4a3+27b2>0; if Δ<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 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, then by rescaling we may assume a,b. 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 TIKC. Both T,K are subgroups of C, though I may not be. K always has the form T×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 /(n), 1n10 or n=12, or (/(2))×(/(2n)), 1n4. To determine T, we make use of the fact that if (x,y)T then y2 must be a divisor of Δ; hence there are only a finite number of possibilities for y. Values of x are then found as integer solutions of x3+ax+b-y2=0 (in particular x must be a divisor of b-y2). 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).

§23.20(iii) Factorization

§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

The modular equation of degree p, p prime, is an algebraic equation in α=λ(pτ) and β=λ(τ). For p=2,3,5,7 and with u=α1/4, v=β1/4, the modular equation is as follows:

23.20.5 v8(1+u8)=4u4,
p=2,
23.20.6 u4-v4+2uv(1-u2v2)=0,
p=3,
23.20.7 u6-v6+5u2v2(u2-v2)+4uv(1-u4v4)=0,
p=5,
23.20.8 (1-u8)(1-v8)=(1-uv)8,
p=7.

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

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).