Non-CM elliptic curves with infinitely many almost prime Frobenius traces

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and without complex multiplication. For a prime $p$ of good reduction for $E$, we write $\#E_p(\mathbb{F}_p) = p + 1 - a_p(E)$ for the number of $\mathbb{F}_p$-rational points of the reduction $E_p$ of $E$ modulo $p$. Under the Generalized Riemann Hypothesis (GRH), we study the primes $p$ for which the integer $|a_p(E)|$ is a prime. In particular, we prove the following results: (i) the number of primes $p < x$ for which $|a_p(E)|$ is a prime is bounded from above by $C_1(E) \frac{x}{(\log x)^2}$ for some constant $C_1(E)$; (ii) the number of primes $p < x$ for which $|a_p(E)|$ is the product of at most 4 distinct primes, counted without multiplicity, is bounded from below by $C_2(E) \frac{x}{(\log x)^2}$ for some constant $C_2(E)$; (iii) the number of primes $p < x$ for which $|a_p(E)|$ is the product of at most 5 distinct primes, counted with multiplicity, is bounded from below by $C_3(E) \frac{x}{(\log x)^2}$ for some positive constant $C_3(E) > 0$. Under GRH, we also prove the convergence of the sum of the reciprocals of the primes $p$ for which $|a_p(E)|$ is a prime. Furthermore, under GRH, together with Artin's Holomorphy Conjecture and a Pair Correlation Conjecture for Artin L-functions, we prove that the number of primes $p < x$ for which $|a_p(E)|$ is the product of at most 2 distinct primes, counted with multiplicity, is bounded from below by $C_4(E) \frac{x}{(\log x)^2}$ for some constant $C_4(E)$. The constants $C_i(E)$, $1 \leq i \leq 4$, are defined explicitly in terms of $E$ and are factors of another explicit constant $C(E)$ that appears in the conjecture that $\#\{p < x: |a_p(E)| \ \text{is prime}\} \sim C(E) \frac{x}{(\log x)^2}$.