Lemma 3.2.1 in Baker, González-Jiménez, González, Poonen, "Finiteness theorems for modular curves of genus at least 2", Amer. J. Math. 127 (2005), 1325–1387.
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I don't understand why "all supersingular elliptic curves over $\bar{F_p}$ are isogenous".
Could anyone help me?
This problem arises from Supersingular elliptic curves and their "functorial" structure over F_p^2.
Moreover, I found some similar question: Isogenies between elliptic curves and their endomorphism rings and Isogenies between supersingular elliptic curves.
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HaomengXu
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1One way to prove this is to show that the rational Dieudonne module of a supersingular elliptic curve has to be an isocrystal of the form (depending on how you normalize things) $\begin{pmatrix}1 & 0 \ p & 1\end{pmatrix}$, and then use Tate's isogeny theorem. – Alex Youcis Apr 15 '23 at 14:26
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1Crossposted to MO – Viktor Vaughn Apr 15 '23 at 22:10
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1(Now answered on https://mathoverflow.net/questions/444846/why-all-supersingular-elliptic-curves-over-bar-mathbbf-p-are-isogenous) – Watson Apr 16 '23 at 17:55