On the cyclicity of the group of \({\mathbb F}_p\)-rational points of non-CM elliptic curves
Let \(E\) be an elliptic curve over \(\mathbb{Q}\), \(p\) a prime of good reduction, \(\tilde{E}\) the reduction of \(E\) modulo \(p\). If \(E(\mathbb{Q})\) has positive rank and we fix a point \(a\) of infinite order in \(E(\mathbb{Q})\), then the analogue of Artin's conjecture is to determine the density of prime numbers \(p\) such that \(p\) is a prime of good reduction and \(\widetilde{E}(\mathbb{F}_p)\) is cyclic generated by the \(a\) modulo \(p\). \textit{J.-P. Serre} proved [Résumé des cours de 1977--78, Collège de France, Œuvres Vol. III, Springer-Verlag, 465--468 (1986; Zbl 0849.01049)] that \textit{C. Hooley}'s conditional method [Applications of sieve methods to the theory of numbers, Cambridge University Press (1976; Zbl 0327.10044)] for proving Artin's conjecture on primitive roots can be adapted to show that the latter set has a density. More precisely, given a prime number \(q\), let \(E[q]\) be the subgroup of \(q\)-torsion points of \(E\) and \(L_q=\mathbb{Q}(E[q])\), \(L_1=\mathbb{Q}\), for every square-free integer \(k\geq1\), \(L_k=\prod_{q \mid q}L_q\) and \(f(x,\mathbb{Q})=\#\{p\leq x \mid p\) is of good reduction, \(\widetilde{E}(\mathbb{F}_p)\) is cyclic\(\}\), then, under the assumption that the Generalized Riemann Hypothesis (GRH) holds for the zeta-function of \(L_k\), \[ f(x,\mathbb{Q})=C_E \text{li}(x)+O(x\log(\log(x))/(\log(x))^2), \] where \(C_E=\sum_{k\geq1}\mu(k)/[L_k:\mathbb{Q}]\). \textit{M. R. Murty} showed [Sieve methods, exponential sums and applications in number theory, Cambridge University Press, 325--344 (199; Zbl 0931.11018)] that \(C_E\neq0\) if \(E[2]\nsubseteq E(\mathbb{Q})\). He also showed [\textit{M. R. Murty}, J. Number Theory 16, 147--168 (1983; Zbl 0526.12010)] that the use of GRH can be suppressed if it is assumed that the elliptic curve has complex multiplication (CM). In [\textit{M. R. Murty}, Proc. Indian Acad. Sci., Math. Sci. 97, 247--250 (1987; Zbl 0654.14018)] he showed unconditionally for certain elliptic curves without CM the existence of infinitely many primes \(p\) for which \(\widetilde{E}(\mathbb{F}_p)\) is cyclic. Furthermore, together with Gupta he also showed unconditionally [\textit{R. Gupta} and \textit{M. R. Murty}, Invent. Math. 101, 225--235 (1990; Zbl 0731.14011)] that the necessary and sufficient condition for \(\widetilde{E}(\mathbb{F}_p)\) to be cyclic for infinitely many primes \(p\) is \(E[2]\nsubseteq E(\mathbb{Q})\). In this case they obtained : \(\#\{p\leq x\mid p\) of good reduction, \(\widetilde{E}(\mathbb{F}_p)\) cyclic\(\}\ll x/\log^2(x)\). In the current paper the author shows that if \(E/\mathbb{Q}\) is an elliptic curve without CM and the zeta-fuctions of all \(L_k\) does not vanish on \(\Re(s)>3/4\), then \[ f(x,\mathbb{Q})=C_E\text{li}(x)+O(x\log(\log(x))/(\log(x))^2). \] The author does a careful analysis on which steps the latter vanishing hypothesis is necessary, and it turns out that if the goal is just \(f(x,\mathbb{Q})\sim C_E\text{li}(x)\), then this hypothesis is just necessary at one step, the others being unconditional. As in the case of Serre's result, the main tool is Chebotarev's density theorem.
scientific article; zbMATH DE number 1886723
scientific article