Worked solution: Ngô's proof via the Hitchin fibration and geometric support theorem (2008)

Step 1 of 9: State the fundamental lemma as an orbital-integral identity

In plain words

Take a symmetry group $G$ and a smaller companion group $H$ built from a piece of $G$ 's data. For a typical element $\gamma$ of $G$ , one can count, in a precise integral sense, how $G$ moves $\gamma$ around; the fundamental lemma predicts that this count, weighted by a sign-like factor $\kappa$ , exactly matches a similar count done inside the smaller group $H$ .

It is a bit like predicting that shuffling a full deck of cards a certain way always produces the same statistics as a simpler shuffle of a smaller, related deck — a coincidence that, once proved, lets mathematicians replace hard computations in $G$ by easier ones in $H$ .

O_\gamma^\kappa(\mathbf{1}_{K}) \ = \ \Delta(\gamma,\gamma^H)\,SO_{\gamma^H}(\mathbf{1}_{K^H})

Detailed analysis

For a regular semisimple element $\gamma$ in a reductive group $G$ (or its Lie algebra) over a local field, the orbital integral $O_\gamma(\mathbf{1}_K)$ measures the volume, with respect to a compact open subgroup $K$ , of the conjugacy class of $\gamma$ . Because $\gamma$ may split into several ordinary conjugacy classes inside one larger stable conjugacy class, these classes are indexed by a finite abelian group $A$ , and a character $\kappa$ of $A$ singles out a $\kappa$ -weighted orbital integral $O_\gamma^\kappa(\mathbf{1}_K)$ . The fundamental lemma (FL), conjectured by Robert Langlands and Diana Shelstad in 1983 (Hales 2011, §1.3–1.4), asserts that $O_\gamma^\kappa(\mathbf{1}_K)$ equals a stable orbital integral $SO_{\gamma^H}(\mathbf{1}_{K^H})$ on an associated endoscopic group $H$ , up to explicit transfer factors.

This is exactly the local building block needed to compare the geometric side of the Arthur–Selberg trace formula for $G$ with that of $H$ , which is how Langlands's functoriality program stabilizes the trace formula and classifies automorphic representations. Because the identity has to hold at every place and for every reductive group, and orbital integrals are genuinely hard $p$ -adic integrals, the FL resisted a general proof for nearly thirty years, with only rank-one and a handful of small-rank cases checked by hand (Hales 2011, Introduction).

The next step records the first real simplification: Jean-Loup Waldspurger showed the statement about groups follows from an analogous statement about Lie algebras, which is where Ngô Bảo Châu's geometric attack actually takes place.

Terms in this step

orbital integral: For an element $\gamma$ of a group $G$ , the integral $\int f(g^{-1}\gamma g)\,dg$ of a test function $f$ over the conjugates of $\gamma$ ; it measures how $f$ 'sees' the orbit of $\gamma$ under conjugation.
endoscopic group $H$: A reductive group of smaller dimension than $G$ , built from a subset of $G$ 's root data determined by $\gamma$ and $\kappa$ , used to package the 'missing' conjugacy classes inside a stable class.

Knowledge used in this step

Representation theory