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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!

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    $\begingroup$ Why not cite the exercise in Silverman's book AEC? It is exercise 5.7. Use Theorem 3.1(iii). $\endgroup$
    – KConrad
    Commented Jul 7, 2020 at 18:26

2 Answers 2

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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.

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  • $\begingroup$ Another explicit approach: Let $E: y^2 + (a_1x + a_3)y = x^3 + a_2x^2 + a_4x + a_6$ be an elliptic curve over $\mathbb{F}_2[a_1,a_2,a_3,a_4,a_6]$. The $2$-division polynomial $\Psi_2(x)$ is $a_1^2x^2 + a_3^2$, and $j(E) = a_1^{12}/(a_1^3(\cdots) + a_3^4)$. Now $E$ is supersingular if and only if $\Psi_2$ has no roots (over the algebraic closure), which is precisely when $a_1 = 0$, and then this forces $j(E) = 0$. $\endgroup$
    – Ben Smith
    Commented Oct 7, 2020 at 12:18
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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).

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