Let $E$ be an elliptic curve defined over an algebraic number field $K$ and $X$ be a cyclic subgroup of $E(K)$ of order $p^n$, then we have the isogeny $E\rightarrow E/X$ and the dual isogeny $ E/X\rightarrow E$.


I want to ask why the kernel of the dual isogeny $ E/X\rightarrow E$ is also a cyclic group of order $p^n$. I have read most of Silverman's book(arithmetic of elliptic curves), any reference or proof is appreciated. Thanks in advance.



Kernel of dual isogeny of elliptic curve
Let $\phi$ be the map $$\phi:E \rightarrow E/X$$ then we have to use $\phi\circ \hat{\phi} = [p^{n}]$, and so there's an exact sequence given by $$0\rightarrow X \rightarrow E[p^{n}] \rightarrow \text{ker}\hat{\phi} \rightarrow 0 .$$ This gives an exact sequence $$0\rightarrow \mathbb{Z}\_{p^n} \rightarrow \mathbb{Z}\_{p^{n}}^2\rightarrow \text{ker}\hat{\phi} \rightarrow 0. $$


Now just pick a generator of $\mathbb{Z}\_{p^n}^2$ that isn't in $\mathbb{Z}\_{p^n}$ and its image should generate $\text{ker}\hat{\phi}$.


