For quite a few years now, when I ask my colleague Herve Jacquet about what is going on in his field, he tells me something like: “Maybe someone will soon be able to prove the Fundamental Lemma”. This is a bit of a joke since the terminology “lemma” is supposed to refer to an easy to prove, simple technical result needed on the way to proving a real theorem. In this case the name “Fundamental Lemma” has ended up getting attached to a crucial conjecture that is part of the so-called “Langlands Program”, and this conjecture has resisted all attempts to prove it for more than twenty years.

A few weeks ago Jacquet told me that he had heard that two French mathematicians, Gerard Laumon and Bao Chau Ngo, finally had a proof and today a manuscript has appeared on the arXiv. The techniques it uses are way beyond me (and even Jacquet claims he doesn’t understand them), but are related to those used in the so-called “Geometric Langlands Program” that has interesting relations to conformal field theory.

I won’t embarrass myself by trying to explain in any detail the little that I know about this kind of mathematics, but in extremely vague terms the Langlands Program relates representations of the Galois group (which tell one about the number of solutions of arithmetic problems) to representations of algebraic groups like the general linear group. One example of this kind of thing is the Taniyama-Shimura-Weil conjecture that was proved by Wiles and implies Fermat’s Last Theorem. One way of approaching the Langlands program uses generalizations of the Selberg trace formula, and the lack of a proof of the fundamental lemma has evidently been the main obstruction to getting all that one would like out of the trace formula methods. Maybe someday I’ll understand some of this enough to try and write something more, but that will probably take quite a while. In the meantime, one of the few expository papers I’ve found about the fundamental lemma is here.

Right now Fermat already is just a corollary of Taniyama-Shimura-Weil (as proved by Wiles-Taylor). It is quite possible that at some point Taniyama-Shimura-Weil will be a corollary of a very general theorem coming out of the Langlands program. However, this seems very far off at this point.

Would this mean the Fermat theorem would be reduced to a minor corollary? That’s my uniform expectation about all these famous problems; they are just special cases in temporary lack of an enveloping general theory.

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