Twists of Elliptic Curves

TL;DR

This paper extends the theory of elliptic curve twists to all characteristics, clarifying the Galois cohomology correspondence and providing illustrative examples.

Contribution

It generalizes the existing theory of elliptic curve twists to include characteristics 2 and 3, using Galois cohomology explicitly.

Findings

Extended twist theory to all characteristics

Explicit Galois cohomology correspondence

Illustrative examples provided

Abstract

In this note we extend the theory of twists of elliptic curves as presented in various standard texts for characteristic not equal to two or three to the remaining characteristics. For this, we make explicit use of the correspondence between the twists and the Galois cohomology set $H^1\big(\operatorname{G}_{\overline{K}/K}, \operatorname{Aut}_{\overline{K}}(E)\big)$. The results are illustrated by examples.