Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 083, 13 pages      arXiv:1707.02893      https://doi.org/10.3842/SIGMA.2017.083
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

Twists of Elliptic Curves

Max Kronberg a, Muhammad Afzal Soomro b and Jaap Top a
a) Johan Bernoulli Institute for Mathematics and Computer Science, Nijenborgh 9, 9747 AG Groningen, The Netherlands
b) Quaid-e-Awam University of Engineering, Science & Technology (QUEST), Sakrand Road, Nawabshah, Sindh, Pakistan

Received July 10, 2017, in final form October 23, 2017; Published online October 25, 2017

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.

Key words: elliptic curve; twist; automorphisms; Galois cohomology.

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