Almost prime values of the order of elliptic curves over finite fields
type: Item
instance of: scholarly article
title: Almost prime values of the order of elliptic curves over finite fields
zbMATH Open document ID: 1257.11086
author: Chantal David
author: Jie Wu
published in: Forum Mathematicum
publication date: 2012-05-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/0812.2860
Mathematics Subject Classification ID: 11N36
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 6031460
zbMATH Keywords: elliptic curves
zbMATH Keywords: sieve methods
zbMATH Keywords: Koblitz's conjecture
MaRDI profile type: Publication
OpenAlex ID: W2085192447
arXiv ID: 0812.2860
cites work: Reductions of an elliptic curve with almost prime orders
cites work: Frobenius fields for elliptic curves
cites work: The functions of the linear sieve
cites work: Modular Forms and the Chebotarev Density Theorem
cites work: Galois properties of points of finite order of elliptic curves
cites work: Some effective cases of the Brauer-Siegel theorem
cites work: On the number of prime divisors of the order of elliptic curves modulo p
DOI: 10.1515/FORM.2011.051
review text: 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…
description: scientific article; zbMATH DE number 6031460