Timeline for Finiteness of Obstruction to a Local-Global Principle
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
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| Oct 29, 2009 at 22:48 | comment | added | Jonah Sinick | I don't have any idea of how one would treat the case with analytic rank > 1 - maybe it's not reasonable to expect an analogous approach in this case. Still, this different way of computing Sha is interesting even in the rank 0 and 1 cases in light of the fact that most elliptic curves over Q seem to have analytic rank 0 or 1. It would be interesting to identify a large and natural family of projective varieties over Q with the property that Hilbert's 10th problem is decidable for this class. (This would be analogous to Gromov's discovery of hyperbolic groups in relation to the word problem.) | |
| Oct 29, 2009 at 14:20 | comment | added | Pete L. Clark | Ah, I see what you are saying. Yes, if the full BSD is given to you, you have a different way to compute the order of Sha. In the case where the analytic rank is zero, this seems to have a priori bounded running time. But if the analytic rank is greater than 1, how are you going to compute it (rigorously) without computing the Mordell-Weil rank instead? Anyway, I am far from an expert in the algorithmic aspects here. I hope someone else will weigh in. | |
| Oct 29, 2009 at 8:20 | comment | added | Jonah Sinick | Thanks for your response. What I had in mind in referring to an effective bound on Sha and an a priori bound on running time is the strong from of the Birch and Swinnerton-Dyer conjecture - for example, I have the impression that in the analytic rank zero case it's possible to compute the size of Sha by computing the central critical value of the L-function attached to the elliptic curve to high precision (with running time bounded a priori) using the fact that elliptic curves are modular. But maybe I'm just confused. | |
| Oct 29, 2009 at 7:58 | vote | accept | Jonah Sinick | ||
| Oct 29, 2009 at 7:26 | history | answered | Pete L. Clark | CC BY-SA 2.5 |