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Journal Article

Extremal primes of elliptic curves without complex multiplication


Turnage-Butterbaugh,  C. L.
Max Planck Institute for Mathematics, Max Planck Society;

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David, C., Gafni, A., Malik, A., Prabhu, N., & Turnage-Butterbaugh, C. L. (2020). Extremal primes of elliptic curves without complex multiplication. Proceedings of the American Mathematical Society, 148(3), 929-943. doi:10.1090/proc/14748.

Cite as: http://hdl.handle.net/21.11116/0000-0007-FE39-C
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.