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

  • Chantal David EMAIL logo and Jie Wu
Published/Copyright: January 19, 2012
Forum Mathematicum
From the journal Forum Mathematicum Volume 24 Issue 1

Abstract.

Let E be an elliptic curve over without complex multiplication. For each prime p of good reduction, let |E(p)| be the order of the group of points of the reduced curve over p. According to a conjecture of Koblitz, there should be infinitely many such primes p such that |E(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 ). We prove in this paper that, under the GRH, there are at least 2.778CE twin x/(logx)2 primes p such that |E(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 CE twin 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(p)| is prime. Finally, we remark that our results still hold under an hypothesis weaker than the GRH.

Keywords.: Applications of sieve methods; elliptic curves
Received: 2008-12
Revised: 2010-02-24
Published Online: 2012-01-19
Published in Print: 2012-January

© 2012 by Walter de Gruyter Berlin Boston

Articles in the same Issue

  1. Prelims
  2. The Lerch zeta function I. Zeta integrals
  3. The Lerch zeta function II. Analytic continuation
  4. Generalized Ramanujan conjecture over general imaginary quadratic fields
  5. Almost prime values of the order of elliptic curves over finite fields
  6. Manifolds associated with (2)n+1-colored regular graphs
  7. A Steinness criterion for Stein fibrations
  8. Crown-free highly arc-transitive digraphs
  9. Pitt's inequality and the fractional Laplacian: Sharp error estimates
  10. On affine group actions on Stein manifolds
https://doi.org/10.1515/form.2011.051
Keywords for this article
Applications of sieve methods; elliptic curves

Articles in the same Issue

  1. Prelims
  2. The Lerch zeta function I. Zeta integrals
  3. The Lerch zeta function II. Analytic continuation
  4. Generalized Ramanujan conjecture over general imaginary quadratic fields
  5. Almost prime values of the order of elliptic curves over finite fields
  6. Manifolds associated with (2)n+1-colored regular graphs
  7. A Steinness criterion for Stein fibrations
  8. Crown-free highly arc-transitive digraphs
  9. Pitt's inequality and the fractional Laplacian: Sharp error estimates
  10. On affine group actions on Stein manifolds
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