Extremal primes of elliptic curves without complex multiplication
Abstract
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. Assuming that all the symmetric power L-functions associated to E are automorphic and satisfy the Generalized Riemann Hypothesis, we give the first non-trivial upper bounds for the number of such 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 (arXiv:1305.5283) and refine certain intermediate estimates taking advantage of the fact that extremal primes have a very small Sato-Tate measure.
Cite
@article{arxiv.1807.05255,
title = {Extremal primes of elliptic curves without complex multiplication},
author = {C. David and A. Gafni and A. Malik and N. Prabhu and C. L. Turnage-Butterbaugh},
journal= {arXiv preprint arXiv:1807.05255},
year = {2019}
}
Comments
11 pages, to appear in the Proceedings of the American Mathematical Society