TY - JOUR

T1 - Extremal primes for elliptic curves

AU - James, Kevin

AU - Tran, Brandon

AU - Trinh, Minh Tam

AU - Wertheimer, Phil

AU - Zantout, Dania

N1 - Funding Information:
This work began while the second, third and fourth authors were undergraduate participants in the 2012 NSF sponsored REU in Computational Algebraic Geometry, Combinatorics and Number Theory at Clemson University. The fifth author served as a graduate mentor and the first author was a supervisor of the program. We all wish to thank the host institution Clemson University and the National Science Foundation , who funded this REU through DMS-1156761 . We also wish to thank the anonymous referee for his or her helpful comments.
Publisher Copyright:
© 2016 Elsevier Inc.

PY - 2016/7/1

Y1 - 2016/7/1

N2 - 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.

AB - 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.

KW - Distribution of primes

KW - Elliptic curves

KW - Frobenius distributions

KW - Lang-Trotter conjecture

KW - Trace of Frobenius

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U2 - 10.1016/j.jnt.2016.01.009

DO - 10.1016/j.jnt.2016.01.009

M3 - Article

AN - SCOPUS:84959570707

VL - 164

SP - 282

EP - 298

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -