Fixed $a_p=p+1-\#E(\mathbb{F}_p)$ and $a_p \ne 0$ on an elliptic curve infinitely often for fixed curve over the rationals?

Question

In this and this question we show that if $p=27a^2+27a+7$ is prime, then the order of the elliptic curve $y^2=x^3+2$ modulo $p$ is either $p$ or $p+2$.

Q1 Can we unconditionally show that the order is $p$ or $p+2$ for infinitely many primes?

Let $a_4,a_6$ be integers and define the elliptic curve $E : y^2=x^3+a_4 x + a_6$. For prime $p$ define $a_p=p+1-\#E(\mathbb{F}_p)$

Let $a_4,a_6$ and $a_p \ne 0$ be fixed.

Q2 Are there choices of $a_4,a_6$ and $a_p \ne 0$ for which the above definitions hold for infinitely many primes $p$?

We have $|a_p| < 2 \sqrt{p}$, so assuming it behaves like a random number the probability is about $\frac{1}{2\sqrt{p}}$ which agrees with the quadratic density in Q1.

We know that given $a_p$ we can find $a_4,a_6$, but we want them fixed.

In case of positive answer:

Q3 Are there other constructions with $a_p=1$ (order $p$)? This might give factoring algorithm for numbers of special form.

Answer 1

I believe there is not a single elliptic curve $E$ such that $a_p(E)$ is known to take any fixed nonzero value infinitely often.

This is (part of) the Lang-Trotter conjecture, which is wide open.

Prior work on the Lang-Trotter conjecture may be helpful to you, e.g. Lang-Trotter Conjecture for CM Elliptic Curves by Daqing Wan and Ping Xi which applies in great generality the same idea of specific values of $a_p$ occurring at prime values of quadratic polynomials.