Let $E/$ℚ be an elliptic curve and $P$ ∈ $E($ℚ$)$ a rational point of infinite order. Does the reduction of $P$ mod $p$ generate a maximal cyclic subgroup of $E(\mathbb{F}$_{$p$}$)$ for almost all primes $p$?

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# Order of reduction of infinite order rational point on an Elliptic Curve

### 2 Answers

However, it is true that $$f_p=\text{order of $\tilde P$ in $\tilde E(\mathbb F_p)$}$$ cannot be "too small, too often." For example, for every $\epsilon>0$, the series $$ \sum_{p~\text{prime}} \frac{\log p}{p\cdot f_p^\epsilon} $$ converges. (More precisely, the series is ${}\le 3\epsilon^{-1}+O(1)$ as $\epsilon\to0$.)

No. There is a positive density of primes that split in ℚ$(Q, E[2])$ (where $2Q=P$) and excluding the finitely many primes for which reduction of $E[2]$ isn't injective. For such primes any maximal cyclic subgroup of $E(\mathbb{F}$_{$p$}$)$ has even order so reduction of $P$ can't be a generator (since reduction of $P$ is 2-divisible).

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However, it is true that $$f_p=\text{order of $\tilde P$ in $\tilde E(\mathbb F_p)$}$$ cannot be "too small, too often." For example, for every $\epsilon>0$, the series $$ \sum_{p~\text{prime}} \frac{\log p}{p\cdot f_p^\epsilon} $$ converges. (More precisely, the series is ${}\le 3\epsilon^{-1}+O(1)$ as $\epsilon\to0$.)

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Internat. J. Math.7(1996), 373-391. Thank you for the kind words about my books. There are also many papers by Murty and others on the elliptic analogue of Artin's conjecture. If you google "elliptic artin conjecture", you'll find some references that will lead you to the literature on the subject. - Joe Silverman Nov 13 at 17:36