On the average exponent of elliptic curves modulo p

2012/03/20 by Tristan Freiberg, Freiberg, Tristan, Pär Kurlberg +1
Computer Science · Mathematics · #11G05 #11N36 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1203.4382

openalex publication_date 2012/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given an elliptic curve E/Q and a prime p at which E has good reduction, let e_p be the exponent of the group E_p(F_p) of F_p-rational points on the reduction of E modulo p. Under the Generalized Riemann Hypothesis (GRH) for the Dedekind zeta functions of the division fields of E, we show that there is a certain constant c_E, depending on E and satisfying 0 < c_E < 1, such that e_p/#E_p(F_p) is equal to c_E on average. In the case where E has complex multiplication (CM) the result holds without GRH. If E is a non-CM curve we show that c_E is equal to a rational number depending on E times a universal constant c = ∏_q 1 - q³/(q²-1)(q⁵-1) = 0.899..., the product being over all primes q.

Citations

• Elliptic curve cryptosystems
• The average exponent of elliptic curves modulo p

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