Timeline for Order of reduction of infinite order rational point on an Elliptic Curve

6 events

when toggle format	what		by	license	comment
Nov 13, 2015 at 16:37	answer	added	Joe Silverman		timeline score: 1
Sep 25, 2015 at 13:57	answer	added	David Lampert		timeline score: 1
Sep 25, 2015 at 13:43	comment	added	David Lampert		Thanks for good comments. I'll write Felipe Voloch's as an answer.
Sep 25, 2015 at 0:52	comment	added	Vesselin Dimitrov		Already in $\mathbb{G}_m$ the answer is clearly negative, for the same reason that Felipe gave you. The statement on positive density can be proved conditionally on GRH. One can in fact say a lot more in the elliptic case: the group $E(\mathbb{F}_p)$ is the cyclic group $\langle P \mod{p} \rangle$ for a definite positive density of primes $p$, unless there is an obvious global obstruction ($P$ being divisible over $\mathbb{Q}$ or the rational torsion being non-cyclic). Those are elliptic variants of Artin's conjecture; Alina Cojocaru has obtained various interesting results in this direction.
Sep 25, 2015 at 0:36	comment	added	Felipe Voloch		No. The primes that split in the field $\mathbb{Q}(Q)$, where $2Q=P$ are counterexamples. One expects a positive density of such primes, but not density one, if $P$ is not divisible over the rationals.
Sep 25, 2015 at 0:23	history	asked	David Lampert	CC BY-SA 3.0