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In p.192 The Arithmetic of Elliptic Curves, the proposition3.1 is described when $m$ is relatively prime to $p=char(k)$. How is this fact when $p|m$?

For example,Example3.3.2.

Let $E/\mathbb{Q}$ be the elliptic curve

\begin{equation} E:y^2= x^3+3. \end{equation} And the discriminant $\mathbb{\Delta}=-2^4\cdot3^5 \neq0$ mod $p\geq5$.

Then $|\tilde{E}(\mathbb{F}_5)|=6$ , $|\tilde{E}(\mathbb{F}_7)|=13$.

From these calculations, this book concludes that $E(\mathbb{Q})$ has no nontrivial torsion.

However , according to the proposition 3.1 ,I think that this proves only when m is relatively prime 5 or 7.

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  • $\begingroup$ what is proposition 3.1? $\endgroup$ – Lord Shark the Unknown Aug 14 '17 at 19:21
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    $\begingroup$ Isn't the point that the torsion subgroup would inject into both the reduced curves (because they have good reduction), and since $6$ and $13$ are relatively prime there can't be any torsion. $\endgroup$ – sharding4 Aug 14 '17 at 19:22

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