Almost prime values of the order of elliptic curves over finite fields

TL;DR

This paper proves that, assuming GRH, a positive proportion of primes p yield elliptic curve groups over finite fields with at most 8 prime factors, advancing understanding of elliptic curve group orders and their prime factorizations.

Contribution

It provides explicit bounds and improves previous results on the distribution of elliptic curve group orders with few prime factors, linking to the twin prime conjecture for elliptic curves.

Findings

At least 2.778 C_E^{twin} x / (log x)^2 primes p have |E(F_p)| with ≤8 prime factors.

Improves bounds on the number of primes p with prime order |E(F_p)|.

Results hold under weaker hypotheses than GRH.

Abstract

Let $E$ be an elliptic curve over $\Q$ without complex multiplication, and which is not isogenous to a curve with non-trivial rational torsion. For each prime $p$ of good reduction, let $|E(\F_p)|$ be the order of the group of points of the reduced curve over $\F_p$. We prove in this paper that, under the GRH, there are at least $2.778 C_E^{\rm twin} x / (\log{x})^2$ primes $p$ such that $|E(\F_p)|$ has at most 8 prime factors, counted with multiplicity. This improves previous results of Steuding & Weng and Murty & Miri. This is also the first result where the dependence on the conjectural constant $C_E^{\rm twin}$ appearing in the twin prime conjecture for elliptic curves (also known as Koblitz's conjecture) is made explicit. This is achieved by sieving a slightly different sequence than the one used by previous authors. By sieving the same sequence and using Selberg's linear sieve, we can also improve the constant appearing in the upper bound for the number of primes $p$ such that $|E(\F_p)|$ is prime. Finally, we remark that our results still hold under an hypothesis weaker than the GRH.