CM elliptic curves and primes captured by quadratic polynomials

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with complex multiplication. For a prime $p$, some formulas for $a_p = p + 1 \sharp E(\mathbb{F}_p)$ are given in terms of the binomial coefficients. We show that the equality $a_p = r$ holds for some fixed integer $r$ if and only if a certain quadratic polynomial represents the prime $p$. In particular, for $E \colon y^2 = x^3 + x, a_p = 2$ holding for an odd prime $p$ if and only if $p$ is of the form $n^2 + 1$ and for $E \colon y^2 = x^3 - 11x + 14, a_p = 2$ holding for an odd prime $p$ if and only if $p$ is of the form $(4n)^2 + 1; a_p = -2$ holding for an odd prime $p$ if and only if $p$ is of the form $(4n + 2)^2 + 1$. In some CM cases the Lang-Trotter conjecture and the Hardy-Littlewood conjecture are equivalent.