# 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?

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.

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

• This question seems related to the question of extremal primes. For CM curves the exact distribution is proved by James (RIP) and Pollack, see sciencedirect.com/science/article/pii/S0022314X16302670 Commented Jun 27 at 15:25
• Looking at the paper, it seems that they rely on the fact that such extremal primes correspond to Gaussian primes in certain narrow sectors, which is can be handled because of the strong equidistribution of roots of quadratic congruences. The problem you ask seems to be roughly equivalent to Landau's conjecture on prime values of $n^2 + k$ for a fixed integer $k$. The latter is basically impossible with current technology. Commented Jun 27 at 19:48
• @StanleyYaoXiao Thanks. Did the paper prove that there are no other sporadic solutions except primes of certain form?
– joro
Commented Jun 28 at 11:26

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.
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.