Study of group orders of elliptic curves
Let \(E\) be an elliptic curve over the rational numbers \(\mathbb{Q}\). If \(E\) has good reduction at prime \(p\), the group \(E(\mathbb{F}_p)\) over the finite field \(\mathbb{F}_p\) is a finite group, and hence, a torsion group. Recall that the finite group injects into \(\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/m\mathbb{Z}\) for some positive integer \(m\), and denote by \(f(x,E)\) the number of primes \(p\) of good reduction up to a positive real number \(x\), for which \(E(\mathbb{F}_p)\) is cyclic. In 1978, Serre proved under the GRH that \(f(x,E) \sim C_E \text{Li}(x)\) where \(C_E\) is a constant depending on \(E\) and \(\text{Li}(x)=\int_2^x (1/\log(t))\,dt\), and in 1980, Ram Murty unconditionally proved the result for \(E\) with complex multiplication. In 1990, Ram Murty and Rajiv Gupta unconditionally proved the interesting result that \(E(\mathbb{F}_p)\) is cyclic for infinitely many primes \(p\) if and only if \(E\) has an irrational \(2\)-division point. Clearly, if \(\#\,E(\mathbb{F}_p)\) is square-free, then the group is cyclic. The author proves in this paper under review that ``under the GRH, the set of primes \(p\) such that \(E(\mathbb{F}_p)\) is a cyclic group of non-square-free order has positive density, provided that \(\text{Gal}(\mathbb{Q}(E[q])/(\mathbb{Q}) \cong \text{GL}_2(\mathbb{F}_q)\) for any prime \(q\).'' The simplest case of square-free order is the prime order. In 1988, Koblitz conjectured that the number of primes \(p\) up to \(x\) such that \(E(\mathbb{F}_p)\) has prime order is asymptotic to \(C_E\, x/(\log x)^2\), and the author proves in this paper that \textit{if \(E\) has no complex multiplication, then the natural density of primes \(p\) such that \(E(\mathbb{F}_p)\) has prime order is zero.} Based on numerical results for \(E : y^2+y = x^3-x\), which are contained in the paper, the author introduces a refined conjecture: Suppose that \(E(\mathbb{Q})\) is torsion-free, and that \(E\) has no complex multiplication. Then, the number of primes \(p\) up to \(x\) such that \(\#\, E(\mathbb{F}_p)\) has exactly \(k\) prime divisors is asymptotic to \(C_{E,k}\, x(\log \log x)^{k-1} / (\log x)^2\) where \(C_E\) is a positive constant.
scientific article; zbMATH DE number 5577335