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arXiv · 1807.05255

Extremal primes of elliptic curves without complex multiplication

C. David·A. Gafni·A. Malik·N. Prabhu·C. L. Turnage-Butterbaugh
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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.

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C. David, A. Gafni, A. Malik, N. Prabhu, C. L. Turnage-Butterbaugh. 2018-07-13. Extremal primes of elliptic curves without complex multiplication. https://arxiv.org/abs/1807.05255

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