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

Abstract

Let E be an elliptic curve defined over Q and without complex multiplication. For a prime p of good reduction for E, we write #Ep(Fp) = p + 1 − ap(E) for the number of Fp-rational points of the reduction Ep of E modulo p. Under the Generalized Riemann Hypothesis (GRH), we study the primes p for which the integer |ap(E)| is a prime. In particular, we prove the following results: (i) the number of primes p < x for which |ap(E)| is a prime is bounded from above by C 1 (E) x (log x) 2 for some constant C 1 (E); (ii) the number of primes p < x for which |ap(E)| is the product of at most 4 distinct primes, counted without multiplicity, is bounded from below by C 2 (E) x (log x) 2 for some constant C 2 (E); (iii) the number of primes p < x for which |ap(E)| is the product of at most 5 distinct primes, counted with multiplicity, is bounded from below by C 3 (E) 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 |ap(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 |ap(E)| is the product of at most 2 distinct primes, counted with multiplicity, is bounded from below by C 4 (E) x (log x) 2 for some constant C 4 (E). The constants C i (E), 1 ≤ i ≤ 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 : |ap(E)| is prime} ∼ C(E) x (log x) 2 .