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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 $K[n_jq]_{\lambda}$.

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  • $\begingroup$ 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 ? $\endgroup$ – reuns Dec 30 '17 at 0:01

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