# Item

ITEM ACTIONSEXPORT

Released

Journal Article

#### Extremal primes of elliptic curves without complex multiplication

##### External Ressource

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

(Publisher version)

##### Fulltext (public)

arXiv:1807.05255.pdf

(Preprint), 199KB

proc14748_AM.pdf

(Any fulltext), 435KB

##### Supplementary Material (public)

There is no public supplementary material available

##### Citation

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

##### 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.