Divisibility Biases in the Orders of Elliptic Curve Reductions

2026/06/23 by Sung Min Lee, Nara Sheen
Mathematics · #math.NT

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Abstract

Let E be an elliptic curve over the rationals. In 2004, Cojocaru proved, using the Chebotarev density theorem, that the set of primes p ≤ x for which m divides #E_p(\mathbbF_p) has a natural density. In 2009, Banks and Shparlinski proved an averaged version of this result over families of elliptic curves. In this article, we give a more explicit analysis of these densities. In particular, we show that, for Serre curves, the density of primes p for which m | #E_p(\mathbbF_p) is approximately 1/φ(m), and is always greater than 1/m for every m ≥ 2. Thus, the orders #E_p(\mathbbF_p) exhibit a bias toward divisibility by m. Finally, based on Jones' method, we prove that the average of the individual m-divisibility densities coincides with the average density proposed by Banks and Shparlinski.

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