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On the cyclicity of the group of \({\mathbb F}_p\)-rational points of non-CM elliptic curves. (English) Zbl 1038.11034

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 \(p\). 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 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 \(C_E=\sum_{k\geq1}\mu(k)/[L_k:\mathbb{Q}]\).
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 [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 [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 [R. Gupta and 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 \(\operatorname{Re}(s)>3/4\), 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.

MSC:

11G05 Elliptic curves over global fields
11G20 Curves over finite and local fields
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References:

[1] A. C. Cojocaru, Cyclicity of Elliptic Curves Modulo \(p\); A. C. Cojocaru, Cyclicity of Elliptic Curves Modulo \(p\)
[2] Gupta, R.; Murty, M. R., Cyclicity and generation of points modulo \(p\) on elliptic curves, Invent. Math., 101, 225-235 (1990) · Zbl 0731.14011
[3] Hooley, C., Applications of Sieve Methods to the Theory of Numbers (1976), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0327.10044
[4] Kraus, A., Une remarque sur les points de torsion des courbes elliptiques, C. R. Acad. Sci. Paris Sér. I, 321, 1143-1146 (1995) · Zbl 0862.11037
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[6] Lagarias, J.; Odlyzko, A., (Fröhlich, A., Effective versions of the Chebotarev density theorem in Algebraic Number Fields (1977), Academic Press: Academic Press New York), 409-464
[7] Lang, S.; Trotter, H., Primitive points on elliptic curves, Bull Amer. Math. Soc., 83, 289-292 (1977) · Zbl 0345.12008
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[13] Serre, J.-P., Propriétés Galoisiennes des points d’ordre fini des courbes elliptiques, In-vent. Math., 15, 259-331 (1972) · Zbl 0235.14012
[14] J.-P. Serre, Reésumé des cours de 1977-1978, Annuaire du Collège de France 1978, pp. 67-70, in; J.-P. Serre, Reésumé des cours de 1977-1978, Annuaire du Collège de France 1978, pp. 67-70, in
[15] J.-P. Serre, Quelques applications du théorème de densité de Chebotarev, Publ. Math. I. H. E. S.; J.-P. Serre, Quelques applications du théorème de densité de Chebotarev, Publ. Math. I. H. E. S.
[16] Silverman, J. H., The Arithmetic of Elliptic Curves (1986), Springer-Verlag: Springer-Verlag Berlin · Zbl 0585.14026
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