# Existence of infinitely many Heegner points that are divisible by $p^{n}$ in $K_{\lambda}$

Let $$E$$ be an elliptic curve over $$\mathbb{Q}$$, $$p>2$$ a prime and $$n,s\in\mathbb{N}$$.

For $$j\in\{1,...,s\}$$ let $$n_{j}\in\mathbb{N}$$ be a natural number which may or may not be coprime to $$p$$.

Let $$K/\mathbb{Q}$$ be an imaginary quadratic extension so that the Heegner hypothesis holds (i.e. the primes dividing $$N$$ split).

The Heegner point of conductor $$c$$ is denoted $$y_{c}$$ and defined over $$K[c]$$, the ray class field of conductor $$c$$. It is given by a map $$(\mathbb {C}/\mathcal O_c\rightarrow\mathbb C/\mathfrak N_c)$$ where $$\mathcal O_c= \mathbb Z+ c\mathcal O_K$$, $$\mathfrak N_c=\mathfrak N\cap\mathcal O_c$$ and $$N$$ splits as $$\mathfrak N\overline{\mathfrak N}$$.

If $$l$$ is a prime, let $$K[c]_{\lambda}$$ be the localization of $$K[c]$$ at any prime $$\lambda$$ above $$l$$.

Question 1:

Are there infinitely many primes $$q$$ so that:

(1.) $$q\nmid p\prod_{j=1}^{s}n_{j}$$

(2.) For any $$j\in\{1,...,s\}$$ the Heegner point $$y_{n_{j}q}$$ is divisible by $$p^{n}$$ in $$K[n_jq]_{\pi}$$.

Question 2:

Let $$l\neq p$$. Are there infinitely many primes $$q$$ so that:

(1.) $$q\nmid p\lambda\prod_{j=1}^{s}n_{j}$$

(2.) For any $$j\in\{1,...,s\}$$ the Heegner point $$y_{n_{j}q}$$ is divisible by $$p^{n}$$ in $$K[n_jq]_{\lambda}$$.

• Did you mean somewhere $\alpha \in K$ such that $\{ x \in K, x O_K \subset \mathbb{Z}+\alpha \mathbb{Z}\}=c O_K$, and evaluating the modular j function $j(\alpha)$ to generate the ray class field ? – reuns Dec 30 '17 at 0:01