Elliptic Curves over Finite Fields
A gallery of elliptic curves over finite fields — each panel showing the points of a different curve over a different finite field.
with Nadir Hajouji
- Medium Gallery prints
- Technique Custom path tracer
- Dimensions 36 × 36 in
- Year 2025
Exhibitions
The Mathematics
By a classical theorem of Deuring, the 𝔽ₚ-points of an elliptic curve can be realized inside a companion curve defined over the complex numbers. Uniformizing that characteristic-zero curve as a torus ℂ/Λ, the finite-field points settle into a lattice — a picture that brings out arithmetic symmetries hidden in the raw scatter of points.
Technique
We compute an explicit characteristic-zero lift of the curve, together with a lift of the Frobenius map, and recover the finite-field points as the fixed points of powers of Frobenius acting on ℂ/Λ. To make this torus visible, we use the work of Pinkall to produce a conformal embedding of the characteristic-zero curve into ℝ³: we find a suitable curve on the 2-sphere and take its preimage under the Hopf fibration, giving a flat, conformally embedded torus in the 3-sphere, which we then stereographically project into ℝ³. The result is rendered with a custom path tracer.