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Reference describing supersingular elliptic curves over algebraically closed field in characteristic 2

MathOverflow Asked by up-too-high on December 1, 2021

I’m looking for a reference for the fact that over an algebraically closed field of characteristic two, there is (essentially) only one supersingular elliptic curve.

This fact appears on Wikipedia, and there may be an exercise in a book of Silverman, but it would be great to cite a book or paper.

Thanks!

2 Answers

This is stated and proven in Section 8.2 of Max Deuring's Die Typen der Multiplikatorenringe elliptischer Funktionenkörper., Abh. Math. Semin. Hansische Univ. 14, 197-272 (1941). ZBL67.0107.01 (the case $p = 2$ starts at the bottom of Page 252).

Answered by Ben Smith on December 1, 2021

You could get it by brute force - the supersingular j-invariants have to lie in $mathbb{F}_{2^2}$, so you can just check each of them.

Answered by Sarah Arpin on December 1, 2021

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