Timeline for Applications of $p$-adic Hodge theory
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 25, 2015 at 14:42 | answer | added | Oli Gregory | timeline score: 7 | |
| Dec 27, 2014 at 4:11 | comment | added | tracing | @Kevin.lijh: You won't make me angry! I don't mean to read through their works; but you can read the titles and introductions to their papers. This is a sensible and standard way to learn what people are doing in mathematics. E.g. if you look at Taylor's papers, you will see his main focus is on proving theorems relating Galois reps. and automorphic forms, and that Sato--Tate was one big application. If you look at Kisin's papers, you will see that he was also focused on modularity, and now is focused on integral models of Shimura varieties. Both authors use a lot of p-adic Hodge theory. | |
| Dec 27, 2014 at 1:55 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Dec 26, 2014 at 8:25 | comment | added | Kevin.lijh | Dear @tracing, first of all, I do not want to be offensive (I am worried about my poor English makes you angry). Of course, I know that I can learn things on the theory through reading experts' works, and maybe that is the only way that I can do at this moment. However, at least for me, it seems quite far from getting intution on the theory for applications. | |
| Dec 24, 2014 at 23:17 | comment | added | Tom Lovering | It plays a key role in proving modularity lifting theorems by making possible the study of local deformation rings for an l-adic representation at p=l. One can look at the section on Fontaine-Laffaille modules in Darmon-Diamond-Taylor for the start of this story I guess. | |
| Dec 22, 2014 at 17:20 | comment | added | tracing | Some of the world experts in this area are Christophe Breuil, Frank Calegari, Pierre Colmez, Toby Gee, Mark Kisin, Peter Scholze, and Richard Taylor. Have you tried looking at their work (say by visiting their web-pages)? | |
| Dec 22, 2014 at 15:59 | comment | added | Simon Pepin Lehalleur | One type of application of p-adic Hodge theory, including some integral Hodge theory, which gives crisp statements is to the study of smooth projective varieties over $\mathbb{Q}$ with everywhere good reduction (or very little ramification). The earliest result of this kind is Fontaine's theorem: there is no abelian variety over $\mathbb{Q}$ with everywhere good reduction, but there are generalizations due to Fontaine and Abrashkin. See e.g. arxiv.org/abs/1003.2905. | |
| Dec 22, 2014 at 15:28 | comment | added | user62675 | @Piotr I guess you're right. I'm not a specialist in arithmetic geometry, and I don't think I can be of much help here. :-) | |
| Dec 22, 2014 at 6:18 | comment | added | Piotr Achinger | The notes by Brinon and Conrad are great, but I personally didn't get much motivation out of them. "An abelian variety has good reduction if and only if the associated Galois rep is crystalline" didn't seem to me like a good enough "application", as the definition of "crystalline" is complicated. I'd love to see here, say, new theorems about varieties over $\mathbb{C}$ proved using $p$-adic comparison theorems. | |
| Dec 22, 2014 at 4:41 | comment | added | user62675 | Have you looked at math.stanford.edu/~conrad/papers/notes.pdf? | |
| Dec 22, 2014 at 2:29 | comment | added | Gjergji Zaimi | I'd imagine this would work better if it was made communiti-wiki, and one application per answer. | |
| Dec 22, 2014 at 2:19 | history | asked | Kevin.lijh | CC BY-SA 3.0 |