Timeline for Surjectivity of reduction maps of elliptic curves over Q

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Nov 21, 2012 at 2:57	comment	added	Felipe Voloch		I was thinking about the cyclic case. As in Maarten's answer, the non-cyclic situation is trickier. If $P$ is a generator and doesn't map to a generator, then either the reduction has a non-cyclic group (and the prime $p$ splits in the field adjoining the relevant torsion) or the reduction is cyclic of larger order and the image of $P$ is divisible by the index and you can detect that by how the prime $p$ behaves in the extension obtained by dividing $P$ by the index.
Nov 21, 2012 at 0:50	comment	added	R.P.		Hi Felipe, I don't mind getting all these answers, they've been very helpful :) I've checked out the reference to Gupta-Murty, which does indeed prove what you say. Rank $\ge 6$ is large though, so I'll check for improvements later. (Although if Maarten Derickx's calculation proves correct, I guess some assumption on the size of $E(\mathbf{Q})$ on top of rank $>0$ is necessary.) I don't understand your solution to 3., do you mean to start by picking an isogeny that maps to the reduced curve? If so, I fail to see how you can get anything from that unless the isogeny lifts to $\mathbf{Q}$.
Nov 20, 2012 at 18:41	history	answered	Felipe Voloch	CC BY-SA 3.0