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

Let E/Q be a non-CM elliptic curve. Write p + 1 - ap(E) for the number of Fp-points of the reduction of E modulo a prime p of good reduction. We prove the following: (i) under the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH), the number of primes p < x with |ap(E)| a prime is bounded above by C1(E) x (log x)2 ; (ii) under GRH, the number of primes p < x with |ap(E)| the product of at most 4 distinct primes, counted without multiplicity, is bounded below by C2(E) x (log x)2 ; (iii) under GRH, the number of primes p < x with |ap(E)| the product of at most 5 distinct primes, counted with multiplicity, is bounded below by C3(E) x (log x)2 ; (iv) under GRH, Artin's Holomorphy Conjecture, and a Pair Correlation Conjecture for Artin L-functions, the number of primes p < x with |ap(E)| the product of at most 2 distinct primes, counted with multiplicity, is bounded below by C4(E) x (log x)2. The constants Ci(E), 1 < i < 4, are factors of an explicit constant C(E) that appears in the conjecture #{p < x : |ap(E)| is prime} & SIM;C(E) x (log x)2.& COPY; 2023 Elsevier Inc. All rights reserved.