On Edwards curves in differential equations

We study ordinary differential equations related with the Edwards curves x 2 + y 2 = 1 + δ x 2 y 2 over the field of real numbers with negative δ . We show that the Edwards curves can be parameterized by means of solutions of a linear-cubic oscillator depending upon the parameter δ , and the solutions can be expressed in terms of the Jacobian elliptic functions. The parameterizations obtained allow us to interpret the Edwards addition law on the Edwards curves similarly as the addition of the classical circle group can be interpreted using the addition formulas for trigonometric sine and cosine functions. We pay attention to the Edwards curve with δ = − 2 , which can be parameterized also by means of solutions of a quintic oscillator; these solutions also can be expressed in terms of the Jacobian elliptic functions. This parameterization alow us to define the point addition on the Edwards curve with δ = − 2 that differs from the Edwards addition law.