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Zbl 0598.14018
Gupta, Rajiv; Murty, M. Ram
(Ram Murty, M.)
Primitive points on elliptic curves. (English)
[J] Compos. Math. 58, 13-44 (1986). ISSN 0010-437X; ISSN 1570-5846/e

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 $\Bbb Q$, and fix a point a in the group $E(\Bbb Q)$ of points of $E$ defined over $\Bbb Q$. Let $N\sb a(x)$ denote the number of rational primes $p\le x$ for which the reduction of $a\pmod p$ generates the group $E(\Bbb F\sb p)$ 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 $$ N\sp*\sb a(x)=C\sb E(a)(x/\log x)+O(x(\log \log x)/(\log x)\sp 2) $$ as $x\to \infty$. Here the * indicates that only those $p$ which split in $k$ are being counted, and $C\sb E(a)$ is a certain constant consistent with that appearing in the corresponding conjecture for $N\sb a(x)$ made by {\it S. Lang} and {\it H. Trotter} [Bull. Am. Math. Soc. 83, 289--292 (1977; Zbl 0345.12008)]. It is also shown that $C\sb E(a)>0$ in certain cases; namely if 2 or 3 are inert in $k$, or if $k=\Bbb Q(\sqrt{-11})$. The corresponding conditional results for the classical Artin conjecture were proved by {\it 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. \par The rest of the paper treats the more general problem when the group generated by $a$ in $E(\Bbb Q)$ is replaced by an arbitrary free subgroup $\Gamma$ of $E(\Bbb Q)$. Let $N\sb{\Gamma}(x)$ be the analogous counting function for those $p$ such that the reduction of $\Gamma\pmod p$ is $E(\Bbb F\sb p)$. Again assuming a suitable generalized Riemann hypothesis, it is shown that if $E$ has no complex multiplication then $$N\sb{\Gamma}(x)\sim C\sb E(\Gamma)(x/\log x)\quad\text{as}\ x\to \infty,$$ provided the rank of $\Gamma$ is at least 18. If there is complex multiplication a similar conditional result holds for $N\sp*\sb{\Gamma}(x)$ provided the rank is at least 10. \par Finally some unconditional results are proved; for example, if $E$ has complex multiplication and $\Gamma$ has rank at 6, then $$N\sp*\sb{\Gamma}(x)\gg x/(\log x)\sp 2\quad\text{as}\ x\to \infty.$$ 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 {\it D. R. Heath-Brown} [Q. J. Math., Oxf., II. Ser. 37, 27--38 (1986; Zbl 0586.10025)].
[D. W. Masser]

MSC 2000:: *11G05 Elliptic curves over global fields
11G15 Complex multiplication and moduli of abelian varieties
11N69 Distribution of integers in special residue classes

Keywords: elliptic analogue of Artin's conjecture; primitive roots; complex multiplication

Citations: Zbl 0345.12008; Zbl 0221.10048; Zbl 0549.10037; Zbl 0586.10025

Cited in: Zbl 1205.11063 Zbl 1078.11039 Zbl 0883.11041 Zbl 0731.14011

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