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. We give the first non-trivial upper bounds, both unconditional and on GRH, for the number of such primes when $E$ is a curve without complex multiplication. In order to obtain this bound, we estimate the joint distribution of the fractional part of $\sqrt{p}$ and primes in conjugacy classes of certain Galois groups. Adapting and refining a result by Rouse and Thorner arXiv:1305.5283, a sharper bound for extremal primes is obtained if we assume that all the symmetric power $L$-functions associated to $E$ are automorphic and satisfy GRH. 查看全文>>