Details

In the original article [1], two quartic families of elliptic curves,

and

are related by the birational transformation

.

After additional changes of variables, we obtain the Hamiltonian forms

,

.

The elliptic function accomplishes time parameterization of both surfaces. For the purposes of this Demonstration, a second-order approximation will certainly suffice,

,

with the nome calculated by the Mathematica built-in function EllipticNomeQ.

Solution of relative to is guaranteed up to scale according to the pole-and-zero structure over (Cf. [1], section 20). Comparison with the standard solution in terms of Jacobi elliptic functions verifies this assertion [3]. It is also possible to compare with numerical solutions to the equations of motion; however, discrete analysis makes for even better validation and deeper insight.

A third Hamiltonian surface is obtained by applying a quadratic shear,

.

from quartic to cubic ,

.

Generally, shear transformations are canonical or symplectic, both of which preserve Hamilton's equations of motion and leave time differentials invariant. The Hamiltonian does not conform to the short Weierstrass equation associated with cubic anharmonic oscillation [4]. Nevertheless, a chord-and-tangent addition rule is easy to derive using a simple procedure [5, 6]. For chords, we obtain

,

,

and for tangents,

,

.

Sometimes we need to watch out for hidden points at infinity, but usually these equations are good enough to iterate along the time axis. Assume that point generates an infinite sequence of unique points. We calculate

by tangent, and all subsequent

by chords. The iteration notation is consistent with cryptography literature [6]. Point iteration is not limited to the chord-and-tangent construction. Addition rules with hyperbolic and cubic intersection geometries exist for quartic surfaces and [7, 8]. These are unified rules, which avoid case splitting.

Iterating point addition along each Hamiltonian, we see linear action along the time axis. This is the property that Edwards refers to as a group homomorphism. Comparing , and , we also see that iterations , coincide only on the odd coset . These observations can be proven rigorously using only the built-in Mathematica function PolynomialReduce, as suggested by Hales [9]. Ultimately, we take the points sets , as discrete solutions along , and . Equivalence of discrete solutions immediately implies equivalence of continuous solutions, though only up to a rescaling of the period parallelogram,

.

The real periods and are also complete elliptic integrals

,

,

taken along surfaces and . Both integrals evaluate in terms of Gaussian hypergeometric function [10, 11].

References

[1] H. M. Edwards, "A Normal Form for Elliptic Curves." Bulletin of the American Mathematical Society, 44, 2007 pp. 393–422. doi:10.1090/S0273-0979-07-01153-6.

[2] B. Klee. "Edwards's Solution of Pendulum Oscillation" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/EdwardssSolutionOfPendulumOscillation.

[3] W. P. Reinhardt and P. L. Walker, "Jacobian Elliptic Functions", NIST Digital Library of Mathematical Functions, Chapter 22. dlmf.nist.gov/22.

[4] B. Klee. "Weierstrass Solution of Cubic Anharmonic Oscillation" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/WeierstrassSolutionOfCubicAnharmonicOscillation.

[5] J. H. Silverman and J. T. Tate, Rational Points on Elliptic Curves, Cham: Springer, 2015. doi:10.1007/978-3-319-18588-0.

[6] C. Costello. "Pairings for Beginners." (Apr 4, 2018) craigcostello.com.au/pairing.

[7] C. Arene, T. Lange, M. Naehrig and C. Ritzenthaler. "Faster Computation of the Tate Pairing."arxiv.org/abs/0904.0854.

[8] D. J. Bernstein and T. Lange. "Explicit-Formulas Database." (Apr 4, 2018) hyperelliptic.org/EFD.

[9] T. Hales. "The Group Law for Edwards Curves." arxiv.org/abs/1610.05278.

[10] N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences. "a(n) = binomial(2n, n)^2." oeis.org/A002894.

[11] N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences. "a(n) = (4*n)! /((2*n)!*n!^2)." oeis.org/A000897.