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

Abstract

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(logx)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(logx)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(logx)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(logx)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}∼C(E)x(logx)2.