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Let $E$ be an eliptic curve over a finite field $\Bbb F_q$ of order $q$.

Let $m$ be an integer relatively prime to $q$.

Why is $E[n](\overline{\Bbb F}_q)\cong\Bbb Z_n\times\Bbb Z_n$?

It's obvious from Lagrange theorem that it is divisible by $\Bbb Z_n$, but why is the number of left cosets also $n$?

Noam
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  • Are you familiar with the division polynomials of elliptic curves? See also, e.g. Silverman or any text on elliptic curves really. The necessary assumption $\gcd(n,q)=1$ is an oddity of positive characteristic. – Jyrki Lahtonen Jul 18 '21 at 13:35
  • Wikipedia may be more useful than that single post here. Anyway, knowing the degree of the division polynomials gives (barring the exceptional inseparable cases) the number of $n$ torsion points, and lets us conclude. – Jyrki Lahtonen Jul 18 '21 at 13:39
  • The easiest way to understand this is to consider what happens over $\mathbb{C}$. In this case, by well-known results in complex analysis, every elliptic curve $E$ is isomorphic to $\mathbb{C}/\Lambda$ for some lattice $\Lambda$. Simple geometry then shows that the $n$-torsion points in $\mathbb{C}/\Lambda$ have group structure $\mathbb{Z}_n \times \mathbb{Z}_n$. What's NOT easy to see is why the geometry of $E$ over $\mathbb{F}_p$ must match that of $\mathbb{C}$. But at least the picture over $\mathbb{C}$ gives you a starting point for understanding what happens over $\mathbb{F}_p$. – djao Jul 19 '21 at 01:55
  • thanks @JyrkiLahtonen for the direction,I'll be back once I finish comprehensing the division polynomials. here's a simple explanation of what the division polynomials are: https://math.stackexchange.com/questions/3184689/what-is-l-th-division-polynomial-of-e – Noam Jul 19 '21 at 19:03

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