Extremal primes for elliptic curves

JOURNAL OF NUMBER THEORY

卷:164

Article

James, Kevin¹  Brandon Tran³  Minh-Tam Trinh⁴  Wertheimer, Phil²  Zantout, Dania¹ 

[1] Clemson Univ, Dept Math Sci, Box 340975, Clemson, SC 29634 USA

[2] Univ Maryland, Dept Math, College Pk, MD 20742 USA

[3] MIT, Dept Math, Cambridge, MA 02142 USA

[4] Univ Chicago, Dept Math, Chicago, IL 60637 USA

关键词: Frobenius distributions;    Trace of Frobenius;    Distribution of primes;    Elliptic curves;    Lang-Trotter conjecture;   

DOI  :  10.1016/j.jnt.2016.01.009

来源: Elsevier

【 摘 要 】

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 +/-[2 root p], 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) = O-K, where K is an imaginary quadratic field of discriminant not equal -3,-4, then the number of extremal primes <= X for E is asymptotic to X-3/4/ log X. We give heuristics for related conjectures. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free