Almost prime values of the order of elliptic curves over finite fields (Q2882748)

Let \(E\) be an elliptic curve over \(\mathbb{Q}\) without complex multiplication. Under a certain hypothesis, David and Wu prove new asymptotic estimates for the number of primes \(p<x\) of good reduction such that \(|E(\mathbb{F}_p)|\) (the order of the reduced curve over the finite field \(\mathbb{F}_p\)) has a limited number of prime factors.NEWLINENEWLINEMotivation for the results is given by Koblitz's conjecture, which is discussed in Sections 1 and 2 of the article. Koblitz's conjecture is an analogue of the twin prime conjecture, and states NEWLINE\[NEWLINE \pi^{\mathrm{twin}}_{E} (x) := \left| \left\{ p \leq x : p \nmid N_E, |E(\mathbb{F}_p)| \mathrm{ is prime} \right\} \right| \sim \frac{C^{\mathrm{twin}}_{E} x}{(\log x) ^2} NEWLINE\]NEWLINE as \(x \rightarrow \infty\), where \(C^{\mathrm{twin}}_{E}\) is an explicit constant. The first result of David and Wu is the lower bound NEWLINE\[NEWLINE \left| \left\{ p \leq x : (|E(\mathbb{F}_p)|,M_E)=1, |E(\mathbb{F}_p)|=P_r \right\} \right| \geq \frac{1.323}{1-\theta} \frac{C^{\mathrm{twin}}_{E} x}{(\log x) ^2} NEWLINE\]NEWLINE for \(x\) sufficiently large. Here, \(M_E\) is an integer depending on \(E\) which is specified in the paper, and \(P_r\) denotes any number with at most \(r\) prime factors, counted with multiplicity. The integer \(r\) depends on the parameter \(\theta\) which satisfies \(1/2 \leq \theta < 1\). The theorem is conditional on the hypothesis that any Dedekind zeta function or Artin \(L\)-function is non-vanishing on the region \(\mathrm{Re}(s)>\theta\), an assumption which is weaker than the GRH. In the case \(r=8\), the theorem improves on a result from \textit{J. Steuding} and \textit{A. Weng} [Acta Arith. 117, No. 4, 341--352 (2005); erratum ibid. 119, No. 4, 407--408 (2005; Zbl 1080.11065)]. The authors also prove an upper bound for \(\pi^{\mathrm{twin}}_{E}(x)\) which improves on one from \textit{D. Zywina} [``The large sieve and Galois representations'', \url{arXiv:0812.2222}] although we must point out that Zywina's result holds over a general number field.NEWLINENEWLINEThe proof is quite involved; it uses an explicit form of the Chebotarev Density Theorem for the number fields generated over \(\mathbb{Q}\) by the \(n\)-torsion points of \(E\), discussed in Section 3 of the paper. This is combined with a version of Greaves' weighted sieve of dimension 1 (recalled from \textit{H. Halberstam} and \textit{H.-E. Richert} [Elementary and analytic theory of numbers, Banach Cent. Publ. 17, 183--215 (1985; Zbl 0592.10041)]) given in Section 4, followed by the proof of the first main result. In Section 5, the authors combine asymptotics that they have already proved with Selberg's linear sieve, to establish the bound on \(\pi^{\mathrm{twin}}_{E}(x)\).