On the average exponent of elliptic curves modulo p

TL;DR

This paper investigates the average behavior of the exponent of elliptic curves modulo p, establishing a constant ratio to the group size under GRH, with unconditional results for CM curves.

Contribution

It provides a new average result for the exponent of elliptic curves modulo p, including unconditional proofs for CM curves and conditional results under GRH for non-CM curves.

Findings

For CM curves, the average ratio is c_E without GRH.

For non-CM curves, c_E is a rational multiple of a universal constant.

The universal constant c is approximately 0.899.

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 = \prod_q {1 - q^3/(q^2-1)(q^5-1)} = 0.899..., the product being over all primes q.