Timeline for The fundamental lemma and the conjecture of Birch and Swinnerton-Dyer
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 27, 2010 at 1:38 | vote | accept | Minhyong Kim | ||
| Sep 13, 2010 at 4:57 | comment | added | Emerton | Dear Rob, Thanks for the link to the preprint, and the reference to Theorem 7.1.1. It does indeed seem that Morel and Shin's work gives what Skinner and Urban need. | |
| Sep 13, 2010 at 4:19 | comment | added | Minhyong Kim | Thanks for the quick and useful answer. Your advice is wanted and should have been solicited. | |
| Sep 13, 2010 at 3:46 | comment | added | Rob Harron | Re Conjecture 4.1.1 of Skinner–Urban ICM paper: unless I'm misreading something, Theorem 7.1.1 of the Skinner–Urban preprint (math.columbia.edu/%7Eurban/eurp/MC.pdf) states that that conjecture indeed follows from the work of Morel & Shin. | |
| Sep 13, 2010 at 3:39 | comment | added | Emerton | precisely what Shin has generalized, by using the flexibility that the fundamental lemma provides (namely, the ability to deal with endoscopy) to work on more general (but still compact) unitary Shimura varieties and thus construct Galois representations in great generality than Harris--Taylor, in particular, in sufficient generality to remove the multiplicative reduction hypothesis.) | |
| Sep 13, 2010 at 3:37 | comment | added | Rob Harron | Chandan: indeed this mentioned is in the intro to the BLGHT paper (math.harvard.edu/~rtaylor/cy2fin.pdf). In fact, Sato–Tate is true for elliptic curves over totally real fields and for modular forms of arbitrary weight $\geq2$. | |
| Sep 13, 2010 at 3:37 | comment | added | Emerton | Yes; see recent preprints of Barnet-Lamb--Geraghty--Harris--Taylor (who even prove Sato--Tate for all non-CM eigenforms of weight at least 2) and Barnet-Lamb--Gee--Geraghty (who prove the analogous result even for Hilbert modular forms). But as I imply in my answer, for elliptic curves one can simply take the written proof in the multiplicative reduction case, add Shin's work, and get the proof in the general case. (The point being that the only place where the restriction on reduction comes from is in the appeal to the construction of Galois reps. by Harris--Taylor, and this is ... | |
| Sep 13, 2010 at 3:27 | comment | added | Chandan Singh Dalawat | Ah! So the Sato-Tate conjecture is now proved for all elliptic curves over ${\bf Q}$ (even if they don't have a place of multiplicative reduction) ? | |
| Sep 13, 2010 at 3:20 | history | answered | Emerton | CC BY-SA 2.5 |