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  Extremal primes of elliptic curves without complex multiplication

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.

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Item Permalink: https://hdl.handle.net/21.11116/0000-0007-FE39-C Version Permalink: https://hdl.handle.net/21.11116/0000-0007-FE3A-B
Genre: Journal Article

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 Creators:
David, C., Author
Gafni, A., Author
Malik, A., Author
Prabhu, N., Author
Turnage-Butterbaugh, C. L.1, Author           
Affiliations:
1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201              

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Free keywords: Mathematics, Number Theory
 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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Language(s): eng - English
 Dates: 2019-07-012020
 Publication Status: Issued
 Pages: 15
 Publishing info: -
 Table of Contents: -
 Rev. Type: Peer
 Identifiers: arXiv: 1807.05255
DOI: 10.1090/proc/14748
 Degree: -

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Title: Proceedings of the American Mathematical Society
  Abbreviation : Proc. Amer. Math. Soc.
Source Genre: Journal
 Creator(s):
Affiliations:
Publ. Info: American Mathematical Society
Pages: - Volume / Issue: 148 (3) Sequence Number: - Start / End Page: 929 - 943 Identifier: -