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Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and without complexmultiplication. 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 $p$. Underthe Generalized Riemann Hypothesis (GRH), we study the primes $p$ for which theinteger $|a_p(E)|$ 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 boundedfrom 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 4distinct 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, countedwith multiplicity, is bounded from below by $C_3(E) \frac{x}{(\log x)^2}$ forsome positive constant $C_3(E) > 0$. Under GRH, we also prove the convergenceof the sum of the reciprocals of the primes $p$ for which $|a_p(E)|$ is aprime. Furthermore, under GRH, together with Artin's Holomorphy Conjecture anda Pair Correlation Conjecture for Artin L-functions, we prove that the numberof primes $p < x$ for which $|a_p(E)|$ is the product of at most 2 distinctprimes, 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 ofanother explicit constant $C(E)$ that appears in the conjecture that $\#\{p <x: |a_p(E)| \ \text{is prime}\} \sim C(E) \frac{x}{(\log x)^2}$.

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