Timeline for Dwork's use of p-adic analysis in algebraic geometry
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 22, 2010 at 11:19 | comment | added | Felipe Voloch | Satoh's algorithm is indeed related although, for elliptic curves, the relevant p-adic theory was known before Dwork. | |
| Mar 22, 2010 at 9:09 | comment | added | Dror Speiser | Would Satoh's algorithm for computing zeta functions, that relies on Dwork's work on deformations, be related to p-adic analysis? It's different than Kedlaya's work, but still uses p-adic geometry stuff that I don't understand. | |
| Mar 22, 2010 at 0:43 | comment | added | Pete L. Clark | @Marty: Good question. In my answer (which was written independently of Felipe's answer and Bjorn's comment) I also mentioned Kedlaya's p-adic work. But I implied that his methods give not only a p-adic proof of Weil conjectures but also "Weil II". (This has since been fixed.) Upon a closer look, it seems that he proves some but not all of p-adic Weil II. I would indeed like to hear an expert weigh in on what remains to be done. | |
| Mar 22, 2010 at 0:41 | comment | added | Emerton | Kedlaya built on the foundational work of Berthelot and others, and was also very much inspired by Laumon's proof in the $\ell$-adic context via the Fourier transform. Of course, Kedlaya added a lot himself, and here it is also worth mentioning that his underlying techniques (the $p$-adic analysis of Frobenius) are squarely in the tradition of Dwork. | |
| Mar 22, 2010 at 0:35 | comment | added | Marty | I'm upvoting this answer, because I think that the work of Kedlaya is most relevant to the question. But I also think this answer should be expanded to include more history on Monsky-Washnitzer cohomology, etc.. My impression is that Kedlaya's work is strong enough to prove the Weil conjectures with purely p-adic methods, but I'm not expert enough to say who else contributed to this before Kedlaya. Also, I do not know whether p-adic methods are strong enough to prove everything in Weil II. Perhaps an expert can expand on this answer? | |
| Mar 22, 2010 at 0:29 | history | answered | Felipe Voloch | CC BY-SA 2.5 |