Timeline for Heights of multiples of rational points on elliptic curves
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| May 15, 2016 at 12:59 | vote | accept | WhatsUp | ||
| May 13, 2016 at 14:47 | comment | added | WhatsUp | My principle aim is actually to effectively determine all such $n$, for a big bound like $M = 1e6$. Is it possible, or something known to be hard? But this effective bound on the number also looks interesting to me! | |
| May 13, 2016 at 14:42 | comment | added | Joe Silverman | @WhatsUp Do you want to determine all $n$, or do you want to bound the number of such $n$? The latter can be done, since there are effective qualitative versions of Roth's theorem. In other words, I think that one can find an (effective) constant $C=C(E,P)$ so that for all $M\ge2$, $$\#\{n : d(nP)\le M\} \le C\sqrt{\log M}.$$ This should follow from the results in my paper "A quantitative version of Siegel's theorem", J. Reine Angew. Math. 378 (1987), 60-100. | |
| May 13, 2016 at 14:29 | comment | added | WhatsUp | Thank you for the answer. It is now clear to me that the problem has something essential to do with diophantine approximations. The question I have in mind is the following: given a bound $M$, is it possible to determine all the $n$ such that $nP$ has denominator smaller than $M$? Of course the effective method of Baker gives a theoretical "Yes" answer, but since the expected number of $n$ is just $\sqrt(\log(M))$, I wonder if this can be done simpler, by proving e.g. that the sequence $d_n$ is strictly increasing for $n$ larger than an effectively computable bound. | |
| May 13, 2016 at 1:00 | history | edited | Joe Silverman | CC BY-SA 3.0 |
added 2 characters in body
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| May 13, 2016 at 0:24 | history | answered | Joe Silverman | CC BY-SA 3.0 |