Cyclicity and generation of points mod $$p$$ on elliptic curves.(English)Zbl 0731.14011

Let $$E$$ be an elliptic curve defined over $${\mathbb Q}$$ and, for each prime $$p$$ for which $$E$$ has good reduction, let $$E({\mathbb F}_ p)$$ be the group of rational points on the reduction of $$E$$ modulo $$p$$. First Serre raised the question how often this group is cyclic and proved, assuming the generalized Riemann hypothesis (GRH), that the set of primes with this property has a positive density.
Using results of sieve theory the authors succeed in proving a similar result without the assumption of the GRH. This result is a generalization of a previous result of the second author [J. Number Theory 16, 147–168 (1983; Zbl 0526.12010)].
Now let $$\Gamma$$ be a free subgroup of $$E({\mathbb Q})$$ of rank $$r$$ and define $$S(\Gamma)=\{p\text{ prime}\mid E({\mathbb F}_ p)=\Gamma_ p\},$$ where $$\Gamma_ p$$ is the reduction of $$\Gamma$$ mod $$p$$ and, if $$E$$ has CM by an imaginary quadratic field $$K$$, $$S'(\Gamma)=\{p\text{ prime}\mid E({\mathbb F}_ p)=\Gamma_ p \text{ and } p \text{ is inert in } K\}$$. Under the assumption of the GRH and if $$r$$ is sufficiently large then $$S(\Gamma)$$, $$S'(\Gamma)$$ have certain densities.
This result is also an improvement of the authors’ previous work in Compos. Math. 58, 13–44 (1986; Zbl 0598.14018). The whole problem is related to Artin’s primitive root conjecture for elliptic curves [cf. S. Lang and H. Trotter, Bull. Am. Math.Soc. 83, 289–292 (1977; Zbl 0345.12008)].

MSC:

 14G05 Rational points 11G05 Elliptic curves over global fields 11N36 Applications of sieve methods
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References:

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