3
$\begingroup$

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!

Improve this question
$\endgroup$
1
  • 2
    $\begingroup$ Why not cite the exercise in Silverman's book AEC? It is exercise 5.7. Use Theorem 3.1(iii). $\endgroup$
    – KConrad
    Jul 7, 2020 at 18:26

2 Answers 2

Reset to default
3
$\begingroup$

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.

Improve this answer
$\endgroup$
1
  • $\begingroup$ 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
    Oct 7, 2020 at 12:18
2
$\begingroup$

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).

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.