# The fundamental lemma and the conjecture of Birch and Swinnerton-Dyer

Here is a rather pathetic question. In a comment on Tim Gower's weblog, I tentatively stated that the fundamental lemma was necessary for the work of Skinner and Urban relating ranks of Selmer groups of elliptic curves to the vanishing of their $p$-adic $L$-functions. Now, I believe it is correct that some endoscopic version of transfer from a unitary group to a general linear group is necessary for the construction of their $\Lambda$-adic representations. However, having a really poor understanding of the actual techniques, I don't know which version is crucial. That is to say, it's entirely likely that some earlier special case is sufficient for Skinner-Urban. Could I trouble some expert to give a brief outline of the situation?

The pathetic part of this is that the journalist I mentioned in the comment will call in about 4 hours, so it would be nice to know before that. Of course I shouldn't have agreed to speak about something I know so little about, but it was hard to refuse under the circumstances. Oh, in case you're worried that I'm going to discuss Skinner-Urban with the fellow, don't. I just want to bone up on the background.

For people who like the idea of linguistic diversity in mathematics, I am including a link to a report written (with Sugwoo Shin) for the Korean Mathematical Society that expands on the comment to the journalist.

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This is completely irrelevant for the actual mathematical question and I don't want to appear impertinent, but surely you can always ask the journalist to call a day later? It's not like he's flying in to speak to you and needs to return the same day. –  Victor Protsak Sep 13 '10 at 4:39

Dear Minhyong,

My understanding, based on recalling talks of Skinner and also briefly looking over the ICM paper that you linked to is that, yes, they do rely on the fundamental lemma, namely, the fundamental lemma for unitary groups as proved by Laumon and Ngo. Unfortunately, I'm not sufficiently educated in their work, or in the field of unitary Shimura varieties in general, to be sure whether their Conjecture 4.1.1 is actually a theorem at this point or not. Certainly related results have been proved by Sug Woo Shin and by Sophie Morel, both relying on Laumon--Ngo. But whether these results address the precise unitary groups considered by Skinner--Urban, I'm not sure. (Just looking briefly at what Skinner and Urban write, it looks as if they are considering quasi-split groups, or at least groups that are not anisotropic, so that their Shimura varieties are non-compact; thus the work of Morel (who considers the non-compact case) may be more directly relevant than that of Shin (who considers the compact case); but both Morel and Shin's work come with certain technical restrictions, so that the precise requirements of Conjecture 4.1.1 (in particular, precise local-global compatibility away from the residue characteristic) may not follow in the full generality considered by Skinner and Urban from the work of either.)

I'm sorry that I can't say more, but I do think that it's safe to say that the fundamental lemma is a crucial (albeit highly technical) ingredient in their program.

At the risk of adding unsolicited and unwanted advice: You could also mention the complete proof of Sato--Tate for elliptic curves over totally real fields (with no requirement that the $j$-invariant be non-integral, unlike in the original work of Clozel--Harris-Shepherd-Barron-Taylor), which follows from the work of CHSBT + that of Shin (the work on compact unitary Shimura varieties mentioned above, which as I already remarked requires Laumon--Ngo as an ingredient).

This is a pretty nice Diophantine statement (technical for a journalist obviously, but presumably one can convey the gist) with the fundamental lemma forming one of the pillars that supports it.

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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) ? –  Chandan Singh Dalawat Sep 13 '10 at 3:27
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 ... –  Emerton Sep 13 '10 at 3:37
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 $\geq2$. –  Rob Harron Sep 13 '10 at 3:37
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.) –  Emerton Sep 13 '10 at 3:39
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. –  Rob Harron Sep 13 '10 at 3:46