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Let $p > 3$ be a prime, let $E/\mathbb{F}_p$ be a supersingular curve, let $N > \sqrt{p}$ be a prime dividing $p + 1$, and let $\mu_N$ denote the multiplicative group of $N$th roots of unity in $\overline{\mathbb{F}_p}$.

(a) Prove that $\mu_N \subseteq \mathbb{F}_{p^2}$

(b) Prove that $\#E(\mathbb{F}_{p^2}) = (p + 1)^2$ and $E[N] \subseteq E(\mathbb{F}_{p^2})$

Jyrki Lahtonen
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  • This is rather difficult to read. Could you please try MathJax: https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference ? – Angina Seng Nov 09 '17 at 22:08
  • it's not easier for you to write with $\LaTeX$? – Darío A. Gutiérrez Nov 09 '17 at 23:43
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    $\mathbb{F}p(\zeta_N) = \mathbb{F}{p^f}$ where $f = \text{ord}(p \bmod N)$. That $N^2 \ |\ #E(\mathbb{F}{p^2})$ and $N$ is prime implies $E[N] \subset E(\mathbb{F}{p^2})$ – reuns Nov 10 '17 at 11:56
  • I edited the question a bit based on some educated guessing. Please check. What tools/results have you learned about? – Jyrki Lahtonen Nov 10 '17 at 11:59

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