Extremal primes of elliptic curves without complex multiplication

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

doi:10.1090/proc/14748

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Extremal primes of elliptic curves without complex multiplication

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

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https://doi.org/10.1090/proc/14748
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Zitation

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.

Zitierlink: https://hdl.handle.net/21.11116/0000-0007-FE39-C

Zusammenfassung

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.