Extremal primes for elliptic curves

Kevin James, Brandon Tran, Minh Tam Trinh, Phil Wertheimer, Dania Zantout

Research output: Contribution to journalArticlepeer-review


For an elliptic curve E/Q, we define an extremal prime for E to be a prime p of good reduction such that the trace of Frobenius of E at p is ±⌊2p⌋, i.e., maximal or minimal in the Hasse interval. Conditional on the Riemann Hypothesis for certain Hecke L-functions, we prove that if End(E)=OK, where K is an imaginary quadratic field of discriminant ≠-3, -4, then the number of extremal primes ≤X for E is asymptotic to X3/4/log X. We give heuristics for related conjectures.

Original languageEnglish (US)
Pages (from-to)282-298
Number of pages17
JournalJournal of Number Theory
StatePublished - Jul 1 2016


  • Distribution of primes
  • Elliptic curves
  • Frobenius distributions
  • Lang-Trotter conjecture
  • Trace of Frobenius

ASJC Scopus subject areas

  • Algebra and Number Theory


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