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 - 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

SN - 0022-314X

VL - 164

SP - 282

EP - 298

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -