Neuberg cubics over finite fields

TL;DR

This paper extends the concept of Neuberg cubics from classical triangle geometry to finite fields, revealing new invariants for elliptic curves and analyzing tangent conics over finite fields.

Contribution

It introduces Neuberg cubics in finite field settings and explores their properties, providing explicit examples and new invariants for elliptic curves.

Findings

Neuberg cubics can be defined over finite fields.

Tangent conics for Weierstrass cubics are either identical or disjoint.

Explicit example over _{23} demonstrates the theory.

Abstract

The framework of universal geometry allows us to consider metrical properties of affine views of elliptic curves, even over finite fields. We show how the Neuberg cubic of triangle geometry extends to the finite field situation and provides interesting potential invariants for elliptic curves, focussing on an explicit example over $\mathbb{F}_{23}$. We also prove that tangent conics for a Weierstrass cubic are identical or disjoint.