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

Chantal David (1) , J Wu (2)

1 Concordia University [Montreal]
2 IECN - Institut Élie Cartan de Nancy

Let E be an elliptic curve over Q without complex multiplication. 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. According to a conjecture of Koblitz, there should be infinitely many such primes p such that |E(F p)| is prime, unless there are some local obstructions predicted by the conjecture. Suppose that E is a curve without local obstructions (which is the case for most elliptic curves over Q). We prove in this paper that, under the GRH, there are at least 2.778C twin E 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 [20, 21] and Miri & Murty [15]. This is also the first result where the dependence on the conjectural constant C twin E appearing in Koblitz's conjecture (also called the twin prime conjecture for elliptic curves) is made explicit. This is achieved by sieving a slightly different sequence than the one of [20] and [15]. By sieving the same sequence and using Selberg's linear sieve, we can also improve the constant of Zywina [24] 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.

Domaines: Théorie des nombres [math.NT]

Chantal David, J Wu. Almost prime values of the order of elliptic curves over finite fields. Forum Mathematicum, 2012, ⟨10.1515/form.2011.051⟩. ⟨hal-01278383⟩