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Primitive points on elliptic curves. (English) Zbl 0598.14018

One of the results of this paper is a conditional proof of an elliptic analogue of Artin’s well-known conjecture on primitive roots. Thus let \(E\) be an elliptic curve defined over \(\mathbb Q\), and fix a point a in the group \(E(\mathbb Q)\) of points of \(E\) defined over \(\mathbb Q\). Let \(N_ a(x)\) denote the number of rational primes \(p\leq x\) for which the reduction of \(a\pmod p\) generates the group \(E(\mathbb F_ p)\) of points of \(E\pmod p\). Assuming the Riemann hypothesis for the Dedekind zeta functions of certain number fields related to \(E\) and \(a\), the authors prove that when \(E\) has complex multiplication by the full ring of integers of a complex quadratic field \(k\), then
\[ N^*_ a(x)=C_ E(a)(x/\log x)+O(x(\log \log x)/(\log x)^ 2) \] as \(x\to \infty\). Here the * indicates that only those \(p\) which split in \(k\) are being counted, and \(C_ E(a)\) is a certain constant consistent with that appearing in the corresponding conjecture for \(N_ a(x)\) made by S. Lang and H. Trotter [Bull. Am. Math. Soc. 83, 289–292 (1977; Zbl 0345.12008)]. It is also shown that \(C_ E(a)>0\) in certain cases; namely if 2 or 3 are inert in \(k\), or if \(k=\mathbb Q(\sqrt{-11})\). The corresponding conditional results for the classical Artin conjecture were proved by C. Hooley [J. Reine Angew. Math. 225, 209–220 (1967; Zbl 0221.10048)] using sieve methods. The authors’ proof appears to be on similar lines, at least after the formulation of an appropriate divisibility criterion; however, the reviewer is unable to comment on any of the details.
The rest of the paper treats the more general problem when the group generated by \(a\) in \(E(\mathbb Q)\) is replaced by an arbitrary free subgroup \(\Gamma\) of \(E(\mathbb Q)\). Let \(N_{\Gamma}(x)\) be the analogous counting function for those \(p\) such that the reduction of \(\Gamma\pmod p\) is \(E(\mathbb F_ p)\). Again assuming a suitable generalized Riemann hypothesis, it is shown that if \(E\) has no complex multiplication then \[ N_{\Gamma}(x)\sim C_ E(\Gamma)(x/\log x)\quad\text{as}\;x\to \infty, \] provided the rank of \(\Gamma\) is at least 18. If there is complex multiplication a similar conditional result holds for \(N^*_{\Gamma}(x)\) provided the rank is at least 10.
Finally some unconditional results are proved; for example, if \(E\) has complex multiplication and \(\Gamma\) has rank at 6, then \[ N^*_{\Gamma}(x)\gg x/(\log x)^ 2\quad\text{as}\;x\to \infty. \] Analogous lower bounds in the classical case were first proved by the authors [Invent. Math. 78, 127–130 (1984; Zbl 0549.10037)]; see also some recent work of D. R. Heath-Brown [Q. J. Math., Oxf., II. Ser. 37, 27–38 (1986; Zbl 0586.10025)].
Reviewer: D. W. Masser

MSC:

11G05 Elliptic curves over global fields
11G15 Complex multiplication and moduli of abelian varieties
11N69 Distribution of integers in special residue classes
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References:

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