Extremal primes for elliptic curves without complex multiplication

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Fix an elliptic curve E E over Q \mathbb {Q} . An extremal prime for E E is a prime p p of good reduction such that the number of rational points on E E modulo p p is maximal or minimal in relation to the Hasse bound, i.e., a p ( E ) = ± [ 2 p ] a_p(E) = \pm \left [ 2 \sqrt {p} \right ] . Assuming that all the symmetric power L L -functions associated to E E have analytic continuation for all s ∈ C s \in \mathbb {C} and satisfy the expected functional equation and the Generalized Riemann Hypothesis, we provide upper bounds for the number of extremal primes when E 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 ) a_p(E) is fixed because of the Sato-Tate distribution.