Timeline for Recent progress toward Birch and Swinnerton-Dyer conjecture
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 12, 2023 at 13:11 | comment | added | Will Sawin | @plm I don't think purely geometric conjectures about cycles will imply arithmetic conjectures about BSD, but I do think BSD can be expressed in cycle-theoretic language as a special cade of general conjectures about algebraic cycles like the Bloch-Kato conjecture. | |
| Jul 12, 2023 at 8:10 | comment | added | plm | @StanleyYaoXiao, thank you. Does it look like there could be a result akin to Gross-Zagier for higher rank elliptic curves ? I have to learn abour all that. But i mean, if the method is so central, it cannot be so specific not to have spawnlings in nearby settings. Is there a Gross-Zagier formula for higher-dimensional abelian varieties or more general motives ? Any good entry points and recent survey on that circle of ideas ? | |
| Jul 12, 2023 at 8:03 | comment | added | plm | @WillSawin, thank you. Do you think that BSD follows from a conjunction of standard conjectures on algebraic cycles in a broad sense ? I have to learn about Lichtenbaum et al's formalism, and don't know the type of methods that are expected to be required for his program or the proof of Bloch-Kato's Tamagawa number conjecture, nor the amount of overlap between the 2, eg if one subsumes the other. Any good reference or gut feeling for that ? :) | |
| Jul 12, 2023 at 7:49 | comment | added | plm | I haven't looked at Bhargava et al's work, do analytic/probabilistic methods seem capable of leading to complete proofs of BSD ? Or even just of almost sure-BSD ? | |
| May 29, 2017 at 16:31 | comment | added | Will Sawin | @StanleyYaoXiao I would say this is not quite correct. I think instead the difficulty in the $r \geq 2$ case is the lack of a construction of a Heegner point, or rather that the Heegner point is torsion. The fundamental difficulty in BSD is probably in finding rational points rather than in proving that they are nontorsion once you've found them. It's similar to the Tate conjecture, Hodge conjecture, etc. where I would say the fundamental difficulty is just finding algebraic cycles. | |
| May 26, 2017 at 13:35 | comment | added | Stanley Yao Xiao | The difference between the $r \geq 2$ case and the $r \leq 1$ case is the lack of the Gross-Zagier formula, which relates the analytic rank and the algebraic (Mordell-Weil) rank of an elliptic curve, which is the essence of the conjecture. | |
| May 26, 2017 at 11:49 | history | edited | Myshkin | CC BY-SA 3.0 |
added 243 characters in body
|
| May 26, 2017 at 11:04 | history | answered | Myshkin | CC BY-SA 3.0 |