Timeline for Is there an analog of the Birch/Swinnerton-Dyer conjecture for abelian varieties in higher dimensions?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 21, 2023 at 9:42 | comment | added | Sebastien Palcoux | What about this new preprint by Emmanuel Lecouturier and Jun Wang? | |
| Sep 26, 2014 at 5:03 | comment | added | NAME_IN_CAPS | Bloch calls the generalised rank equality for smooth projective algebraic varieties a "Recurring Fantasy" (which ranks above Idle Speculation) in eudml.org/doc/152633 | |
| Sep 26, 2014 at 1:11 | comment | added | Joe Silverman | @FilippoAlbertoEdoardo It would indeed be useful for someone to give a brief description of the generalizations (I am not qualified to do so), but I did mention in the first paragraph of my answer that they exist, in particular as formulated by Tate, Beilinson, and (as you say) Bloch and Kato. | |
| Sep 26, 2014 at 0:20 | comment | added | Joseph O'Rourke | @FilippoAlbertoEdoardo: I must admit I was largely ignorant of the more general varieties. Feel free to post information in that direction; we would all learn from it. | |
| Sep 25, 2014 at 22:37 | comment | added | Filippo Alberto Edoardo | In your question, you ask about Abelian Varieties and on that direction Joe Slverman's answer is indeed very complete (and accepted, as I see). But "most" of BSD has been generalized to more general varieties, without the need of any group structure, mainly by Bloch and Kato. So I wonder whether you are intentionally focusing on Abelian Varieties or if you were interested in generalizations to any sort of higher domensional gadhets. | |
| Sep 25, 2014 at 11:58 | comment | added | anon | Yes, the generalization to abelian varieties was due to Tate (in his February 1966 Bourbaki talk). Now, it would not be so difficult to make the generalization, but at the time few understood heights on abelian varieties. And only Tate was capable of showing that the statement is compatible with isogenies, which is convincing evidence that he got the factors correct. | |
| Sep 25, 2014 at 10:49 | vote | accept | Joseph O'Rourke | ||
| Sep 25, 2014 at 1:44 | comment | added | Joseph O'Rourke | PDF download of 1965 Tate paper | |
| Sep 25, 2014 at 0:15 | comment | added | Felipe Voloch | @JoeSilverman I am not sure. Ask him, next time you see him. | |
| Sep 25, 2014 at 0:03 | comment | added | Joe Silverman | @FelipeVoloch Is the abelian variety generalization due to Tate? It may well be, I must admit that I do not know the history. | |
| Sep 25, 2014 at 0:02 | answer | added | Joe Silverman | timeline score: 28 | |
| Sep 24, 2014 at 23:47 | comment | added | Felipe Voloch | Yes. J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. Séminaire Bourbaki, Vol. 9, Exp. No. 306,(1965) 415–440 | |
| Sep 24, 2014 at 23:39 | history | asked | Joseph O'Rourke | CC BY-SA 3.0 |