Extremal primes for elliptic curves without complex multiplication

Fix an elliptic curve E over Q. An extremal prime for E is a prime p of good reduction such that the number of rational points on E modulo p is maximal or minimal in relation to the Hasse bound, i.e., a_p(E) = ±[2^√p^]. Assuming that all the symmetric power L-functions associated to E have analytic continuation for all s ∈ C and satisfy the expected functional equation and the Generalized Riemann Hypothesis, we provide upper bounds for the number of extremal primes when E is a curve without complex multiplication. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in the work of Rouse and Thorner, and refine certain intermediate estimates taking advantage of the fact that extremal primes are less probable than primes where a_p(E) is fixed because of the Sato-Tate distribution.