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 -