The average exponent of elliptic curves modulo p

2012/06/26 by Jie Wu, Wu, Jie
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1206.5929

openalex publication_date 2012/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E be an elliptic curve defined over \mathbb Q. For a prime p of good reduction for E, denote by e_p the exponent of the reduction of E modulo p. Under GRH, we prove that there is a constant C_E∈ (0, 1) such that (1)/(π(x)) ∑_{p≤ x} e_p = 1/2 C_E x + O_E(x^5/6 (log x)^4/3) for all x≥ 2, where the implied constant depends on E at most. When E has complex multiplication, the same asymptotic formula with a weaker error term O_E(1/(log x)^1/14) is established unconditionally. These improve some recent results of Freiberg and Kurlberg.

Citations

• Elliptic curve cryptosystems

Cited by

• On the average exponent of elliptic curves modulo p

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