Frobenius distributions in short intervals for CM elliptic curves

JOURNAL OF NUMBER THEORY

卷:188

Article

Agwu, Anthony¹  Harris, Phillip²  James, Kevin³  Kannan, Siddarth⁴  Li, Huixi³ 

[1] UMBC, Dept Math & Stat, Baltimore, MD 21250 USA

[2] Univ Illinois, Dept Math, Urbana, IL 61820 USA

[3] Clemson Univ, Dept Math Sci, Clemson, SC 29634 USA

[4] Pomona Coll, Dept Math, Claremont, CA 91711 USA

关键词: Elliptic curve;    Trace of Frobenius;    Frobenius distributions;    Sato-Tate conjecture;    Lang-Trotter conjecture;    Complex multiplication;    Elliptic champion prime;    Elliptic trailing prime;   

DOI  :  10.1016/j.jnt.2018.01.007

来源: Elsevier

【 摘 要 】

For an elliptic curve E/Q, Hasse's theorem asserts that #E(F-p) = p + 1 - a(p), where vertical bar a(p)vertical bar <= 2 root p. Assuming that E has complex multiplication, we establish asymptotics for primes p for which a(p) is in subintervals of the Hasse interval [-2 root p, 2 root p] of measure o(root p). In particular, given a function f = o(1) satisfying some mild conditions, we provide counting functions for primes p where vertical bar a(p)vertical bar is an element of(2 root p(1 - f (p)), 2 root p), and for primes where a(p) is an element of(2 root p(c - f (p)), 2c root p), where c is an element of(0, 1) is a constant. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free