# Kernel of dual isogeny of elliptic curve

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.

• Even better. The Weil pairing shows that the kernel of the dual isogeny is a Galois module isomorphic to $\mu_{p^n}$. – Chris Wuthrich Dec 16 '20 at 10:41

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 $$\text{ker}\hat{\phi}$$.
• Thanks, the first exact sequence should be $0\rightarrow\ker\phi$... – user169802 Dec 15 '20 at 23:47