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 pair $(\mathbb {C}/ If $\lambda $ is a prime, let $K_{c,\lambda}$ be the localization of $K[c]$ at any prime above $\lambda$.

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 $K_{n_{j}q,p}$.

Question 2:

Let $\lambda\neq p $ be a prime. 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 $K_{n_{j}q,\lambda}$.