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I have some questions concerning the definition of Heegner points.

Let $K=\mathbb{Q}(\sqrt{D})$ with $D<0$, and $E/\mathbb{Q}$ be an elliptic curve of conductor $N$. Assume that every prime $l\mid N$ splits completely in $K$. Why does this hypothesis imply that there is an ideal $\mathcal{N}\subset \mathcal{O}_K$ such that $\mathcal{O}_K/\mathcal{N}\cong \mathbb{Z}/N\mathbb{Z}$?

Let $n\geq 1$ be an integer coprime to $N$ and $\mathcal{O}_n=\mathbb{Z}+n\mathcal{O}_K$ the order of index $n$ in $\mathcal{O}_K$. I checked that the $\mathcal{O}_n$-module $\mathcal{N}_n:=\mathcal{N}\cap \mathcal{O}_n$ is proper, hence invertible. But why do we have that $\mathcal{O}_n/\mathcal{N}_n\cong \mathbb{Z}/N\mathbb{Z}$?

Thus, $\mathcal{N}_n^{-1}/\mathcal{O}_n\cong \mathbb{Z}/N\mathbb{Z}$. By modularity, the isogeny $\phi:\mathbb{C}/\mathcal{O}_n\longrightarrow \mathbb{C}/\mathcal{N}_n^{-1}$ ($\ker \phi\cong \mathbb{Z}/N\mathbb{Z}$) defines a complex point $x_n$ on the curve $X_0(N)$. My final question is: why does $x_n$ belong to $X_0(N)(K_n)$, where $K_n$ is the ring class field of conductor $n$?

Thank you!

defacto
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    For your first question, this has nothing to do with elliptic curves. If $\mathfrak{p}_j$ is a prime of $O_K$ such that $|O_K/\mathfrak{p}_j|=p_j$ is prime then $O_K/\mathfrak{p}_j^r\cong \Bbb{Z}/(p_j^r)$ (as groups and rings). The $p_j^r$ are the prime powers divisors of $N$, using your "split completely" assumption. If $O$ is an order in $O_K$ and $I$ is an invertible $O$-ideal then $O/I^r\cong O_K/(IO_K)^r$. – reuns Jun 14 '22 at 19:33
  • Sorry of course I meant if $\mathfrak{p}_j$ is unramified, which is the case from the "split completely" assumption. – reuns Jun 15 '22 at 01:23
  • @reuns Thanks. Then $\mathcal{O}_K/\mathcal{N}_n\mathcal{O}_K\cong \mathcal{O}_K/\mathcal{N}$ and the result follows. Where can I check the proof that $\mathcal{O}/I^r\cong \mathcal{O}_K/(I\mathcal{O}_K)^r$? – defacto Jun 15 '22 at 11:28
  • For question 2), see Prop 1.23 in Francesca Gala. Heegner points on X0(N). For question 1) see https://math.stackexchange.com/questions/2447788 – Watson Jun 16 '22 at 14:01

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