Elliptic Curves and the Hopf Fibration

Abstract

By combining tools from different areas of mathematics, we obtain 3D visualizations of elliptic curves over different fields that faithfully capture the underlying algebra and geometry.

Figures (14)

• Figure 1: The elliptic curves $y^2=x^3+3x$ over $\mathbb{F}_{625}$ (orange), and $y^2=x^3+3$ over $\mathbb{F}_{2401}$ (blue).
• Figure 2: The curve $y=x^2$ over $\mathbb{R}$ (green), over $\mathbb{C}$ (blue: two views, projecting to three dimensions by deleting the imaginary component of $x$ or $y$), and over finite fields: $\mathbb{F}_7$ (purple), $\mathbb{F}_{37}$ (red).
• Figure 3: A circle three ways: (1,2) an implicit / parametric plane curve in $\mathbb{R}^2$ (left); (3) an ill-defined real number in $\mathbb{R}/2\pi\mathbb{Z}$ as a periodic point in $\mathbb{R}$ or element of a fundamental domain (right).
• Figure 4: Equation-based illustrations of elliptic curves, for the curves $y^2 = x^3 -x$ (red) and $y^2 = x^3 + 1$ (green). These illustrations obscure the underlying torus geometry.
• Figure 5: A point on an elliptic curve: three views (1) as lattice (2) in fundamental domain (3) rolled up. The same elliptic curves as the previous illustration.