Extremal primes for elliptic curves

For an elliptic curve E/QE/Q, we define an extremal prime for E to be a prime p of good reduction such that the trace of Frobenius of E at p   is ±⌊2p⌋, i.e., maximal or minimal in the Hasse interval. Conditional on the Riemann Hypothesis for certain Hecke L  -functions, we prove that if End(E)=OKEnd(E)=OK, where K   is an imaginary quadratic field of discriminant ≠−3,−4≠−3,−4, then the number of extremal primes ≤X for E   is asymptotic to X3/4/log⁡XX3/4/log⁡X. We give heuristics for related conjectures.