Timeline for Smart elliptic curve rational point search given Reg*#Sha
Current License: CC BY-SA 3.0
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| Feb 4, 2012 at 20:20 | comment | added | Dror Speiser | My comment is not in regards to constructing a search algorithm, but an algorithm mimicking the case of finding a fundamental unit in a real quadratic field. In that case, given $R_Kh_K$, we get the unit in log-polynomial time in the discriminant - much faster than a search algorithm. Thus, it would be interesting to construct an analogue Arakelov theory for elliptic curves. But, as it does in the field case, this would probably prove the finiteness of Sha - not that you need finiteness of Sha for a search algorithm. Hence, it must be hard. | |
| Feb 4, 2012 at 18:44 | comment | added | Noam D. Elkies | The techniques described in my ANTS-IV paper (arXiv.org/pdf/math/0005139v1.pdf) yield a subexponential-time algorithm. Still, the Heegner-point technique is better in the usual case of rank 1. #Sha can be ignored because it's a square: if #Sha = $n^2$ then Reg*#Sha is the height of $n$ times a generator, so find that point and then saturate to find $n$ and the generator. (This also happens for the Heegner-point technique; see for instance my ANTS-I paper.) | |
| Feb 4, 2012 at 14:29 | history | answered | Joe Silverman | CC BY-SA 3.0 |