Extremal primes for elliptic curves without complex multiplication

Abstract:

Fix an elliptic curve $E$ over $\mathbb {Q}$. 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., $a_p(E) = \pm \left [ 2 \sqrt {p} \right ]$. Assuming that all the symmetric power $L$-functions associated to $E$ have analytic continuation for all $s \in \mathbb {C}$ and satisfy the expected functional equation and the Generalized Riemann Hypothesis, we provide upper bounds for the number of extremal primes when $E$ 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.