科研之友
微信
新浪微博
Facebook
分享链接
摘要
Let E be an elliptic curve over Q without complex multiplication. For each prime p of good reduction, let vertical bar E(F-p)vertical bar 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 vertical bar E(F-p)vertical bar 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(E)(twin)x/(log x)(2) primes p such that vertical bar E(F-p)vertical bar 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-E(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 vertical bar E(F-p)vertical bar I is prime. Finally, we remark that our results still hold under an hypothesis weaker than the GRH.
- 出版日期2012-1
- 单位山东师范大学; INRIA
