Extremal primes for elliptic curves without complex multiplication

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

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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, i.e., ap(E) = ±[2p]. Assuming that all the symmetric power L-functions associated to E have analytic continuation for all s ∈ 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 ap(E) is fixed because of the Sato-Tate distribution.

Original languageEnglish (US)
Pages (from-to)929-943
Number of pages15
JournalProceedings of the American Mathematical Society
Issue number3
StatePublished - 2020

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics


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