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Extremal primes for elliptic curves without complex multiplication


Authors: C. David, A. Gafni, A. Malik, N. Prabhu and C. L. Turnage-Butterbaugh
Journal: Proc. Amer. Math. Soc. 148 (2020), 929-943
MSC (2010): Primary 11G05, 11N05
DOI: https://doi.org/10.1090/proc/14748
Published electronically: August 28, 2019
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Abstract: Fix an elliptic curve $ E$ over $ \mathbb{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, i.e., $ a_p(E) = \pm \left [ 2 \sqrt {p} \right ]$. Assuming that all the symmetric power $ L$-functions associated to $ E$ have analytic continuation for all $ s \in \mathbb{C}$ and satisfy the expected functional equation and the Generalized Riemann Hypothesis, we provide upper bounds for the number of extremal 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, and refine certain intermediate estimates taking advantage of the fact that extremal primes are less probable than primes where $ a_p(E)$ is fixed because of the Sato-Tate distribution.


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Additional Information

C. David
Affiliation: Department of Mathematics, Concordia University, 1455 de Maisonneuve West, Montreal, Quebec H3G 1M8, Canada
Email: chantal.david@concordia.ca

A. Gafni
Affiliation: Department of Mathematics, The University of Mississippi, Hume Hall 305, University, Mississippi 38677
Email: ayla.gafni@gmail.com

A. Malik
Affiliation: Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854
Email: amita.malik@rutgers.edu

N. Prabhu
Affiliation: Department of Mathematics and Statistics, Queen’s University, 48 University Avenue, Kingston, Ontario K7L 3N6, Canada
Email: neha.prabhu@queensu.ca

C. L. Turnage-Butterbaugh
Affiliation: Department of Mathematics and Statistics, Carleton College, 1 North College Street, Northfield, Minnesota 55057
Email: cturnageb@carleton.edu

DOI: https://doi.org/10.1090/proc/14748
Received by editor(s): January 18, 2019
Received by editor(s) in revised form: June 18, 2019, and June 24, 2019
Published electronically: August 28, 2019
Communicated by: Amanda Folsom
Article copyright: © Copyright 2019 American Mathematical Society